{"id":593,"date":"2026-05-24T21:49:43","date_gmt":"2026-05-24T12:49:43","guid":{"rendered":"https:\/\/wuhanqing.cn\/wordpress\/?p=593"},"modified":"2026-05-25T11:50:17","modified_gmt":"2026-05-25T02:50:17","slug":"%ec%a0%90%ea%b3%b1%ec%97%90%ec%84%9c-%eb%82%b4%ec%a0%81-%ea%b3%b5%ea%b0%84%ea%b9%8c%ec%a7%80-%ec%84%a0%ed%98%95%eb%8c%80%ec%88%98%ed%95%99-%ec%8b%a0%ed%98%b8%ec%b2%98%eb%a6%ac-ai-%eb%92%a4%ec%97%90","status":"publish","type":"post","link":"https:\/\/wuhanqing.cn\/wordpress\/ko\/2026\/05\/24\/%ec%a0%90%ea%b3%b1%ec%97%90%ec%84%9c-%eb%82%b4%ec%a0%81-%ea%b3%b5%ea%b0%84%ea%b9%8c%ec%a7%80-%ec%84%a0%ed%98%95%eb%8c%80%ec%88%98%ed%95%99-%ec%8b%a0%ed%98%b8%ec%b2%98%eb%a6%ac-ai-%eb%92%a4%ec%97%90\/","title":{"rendered":"\uc810\uacf1\uc5d0\uc11c \ub0b4\uc801 \uacf5\uac04\uae4c\uc9c0: \uc120\ud615\ub300\uc218\ud559, \uc2e0\ud638\ucc98\ub9ac, AI \ub4a4\uc5d0 \uc228\uc740 \ub3d9\uc77c\ud55c \uc5b8\uc5b4"},"content":{"rendered":"

# \uc810\uacf1\uc5d0\uc11c \ub0b4\uc801 \uacf5\uac04\uae4c\uc9c0: \uc120\ud615\ub300\uc218\ud559, \uc2e0\ud638\ucc98\ub9ac, AI \ub4a4\uc5d0 \uc228\uc740 \ub3d9\uc77c\ud55c \uc5b8\uc5b4 (From Dot Product to Inner Product Space: The Unified Language Behind Linear Algebra, Signals, and AI)<\/p>\n

## \uc694\uc57d (Abstract)<\/p>\n

**\ub0b4\uc801(Inner Product)** \uc740 \uc120\ud615\ub300\uc218\ud559, \ud568\uc218\ud574\uc11d\ud559, \uc2e0\ud638\ucc98\ub9ac, \uba38\uc2e0\ub7ec\ub2dd, \uc591\uc790\uc5ed\ud559\uc5d0 \uac78\uccd0 \uacf5\uc720\ub418\ub294 \ud575\uc2ec \ub300\uc218 \uad6c\uc870\uc774\ub2e4. \ubcf8 \ub17c\ubb38\uc740 \"\ub0b4\uc801\"\uc744 \uc720\uc77c\ud55c \uc8fc\uc81c\ub85c \uc0bc\uc544, \uc720\ud55c\ucc28\uc6d0 \uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uc5d0\uc11c\uc758 \uc810\uacf1(Dot Product)\uc5d0\uc11c \ucd9c\ubc1c\ud558\uc5ec \ub0b4\uc801 \uacf5\uac04 \uacf5\ub9ac, \uc9c1\uad50 \ubd84\ud574(Orthogonal Decomposition), \ucd5c\uc18c\uc81c\uacf1 \ud22c\uc601(Least-Squares Projection), \ud790\ubca0\ub974\ud2b8 \uacf5\uac04(Hilbert Space), \ud478\ub9ac\uc5d0 \uae09\uc218\uc640 \ubcc0\ud658(Fourier Series and Transform), \ucee8\ubcfc\ub8e8\uc158(Convolution), \uc774\uc0b0 \ucf54\uc0ac\uc778 \ubcc0\ud658(Discrete Cosine Transform), \uc6e8\uc774\ube14\ub9bf \ubd84\uc11d(Wavelet Analysis), \uc790\uae30 \uc8fc\uc758 \uba54\ucee4\ub2c8\uc998(Self-Attention Mechanism), \ucee4\ub110 \ubc29\ubc95(Kernel Method), \uadf8\ub9ac\uace0 \uc591\uc790\uc5ed\ud559\uc5d0\uc11c\uc758 \uc0c1\ud0dc \ubca1\ud130 \ud22c\uc601(State-Vector Projection)\uc744 \ucc28\ub840\ub85c \uc18c\uac1c\ud55c\ub2e4. \uc774\ub807\uac8c \uc11c\ub85c \ub2e4\ub978 \ud559\ubb38 \ubd84\uc57c\uc5d0 \uc18d\ud558\ub294 \uac83\uc73c\ub85c \ubcf4\uc774\ub294 \uac1c\ub150\ub4e4\uc774 \uc218\ud559\uc801 \uad6c\uc870\uc5d0\uc11c \ud1b5\uc77c\uc131\uc744 \uac00\uc9d0\uc744 \ubc1d\ud78c\ub2e4: **\ub0b4\uc801 \uc815\uc758 \u2192 \uc9c1\uad50 \uae30\uc800 \uc218\ub9bd \u2192 \ud22c\uc601 \ubd84\ud574 \u2192 \uc815\ubcf4 \ucd94\ucd9c**. \ubcf8 \ub17c\ubb38\uc740 \ub3c5\uc790\ub4e4\uc5d0\uac8c \uc218\ud559, \uacf5\ud559, \ubb3c\ub9ac\ud559\uc744 \uad00\ud1b5\ud558\ub294 \uc778\uc9c0 \uc9c0\ub3c4(Cognitive Map)\ub97c \uc81c\uacf5\ud558\ub294 \uac83\uc744 \ubaa9\ud45c\ub85c \ud55c\ub2e4.<\/p>\n

## \uc11c\ubb38: \ub9cc\ubb3c\uc740 \ud22c\uc601\uc774\ub2e4 (Preface: Everything Is a Projection)<\/p>\n

\uc218\ud559\uacfc \uacf5\ud559 \uacfc\ud559\uc5d0\ub294 \ubc18\ubcf5\uc801\uc73c\ub85c \ub098\ud0c0\ub098\ub294 \ud328\ud134\uc774 \uc788\ub2e4: \ubcf5\uc7a1\ud55c \uac1d\uccb4\ub97c \uc5ec\ub7ec \"\uae30\ubcf8 \uc131\ubd84\"\uc758 \uc120\ud615 \uacb0\ud569\uc73c\ub85c \ubd84\ud574\ud558\ub294 \uac83, \uadf8\ub9ac\uace0 \ubd84\ud574\uc758 \ub3c4\uad6c\uac00 \ubc14\ub85c **\ud22c\uc601(Projection)** \uc774\ub77c\ub294 \uc810\uc774\ub2e4. \ud22c\uc601 \uc5f0\uc0b0\uc758 \ubcf8\uc9c8\uc740 \ub0b4\uc801(Inner Product)\uc774\ub2e4\u2014\"\uc720\uc0ac\uc131(Similarity)\"\uc744 \uce21\uc815\ud558\ub294 \uc774\ud56d \uc5f0\uc0b0. \ud478\ub9ac\uc5d0 \ubd84\uc11d\uc5d0\uc11c \uc2e0\ud638\ub97c \uc11c\ub85c \ub2e4\ub978 \uc8fc\ud30c\uc218\uc758 \uc815\ud604\ud30c\ub85c \ubd84\ud574\ud558\ub294 \uac83\ubd80\ud130, \ucd5c\uc18c\uc81c\uacf1\ubc95\uc5d0\uc11c \ub370\uc774\ud130\uc5d0 \uac00\uc7a5 \uc798 \ub9de\ub294 \uc9c1\uc120\uc744 \ucc3e\ub294 \uac83, \uc591\uc790\uc5ed\ud559\uc5d0\uc11c \uc911\ucca9 \uc0c1\ud0dc\uc5d0 \uc788\ub294 \uc785\uc790\ub97c \uce21\uc815\ud558\ub294 \uac83\uae4c\uc9c0, \uc774 \ubaa8\ub4e0 \uacfc\uc815\uc740 \ub3d9\uc77c\ud55c \uc218\ud559 \uc5b8\uc5b4\ub97c \uacf5\uc720\ud55c\ub2e4: **\ub0b4\uc801 \uc815\uc758 \u2192 \uc9c1\uad50 \uae30\uc800 \uc218\ub9bd \u2192 \ud22c\uc601 \u2192 \uc9c1\uad50 \ubd84\ud574 \u2192 \uc815\ubcf4 \ucd94\ucd9c**.<\/p>\n

\ubcf8 \ub17c\ubb38\uc758 \ubaa9\ud45c\ub294 \uc774 \ud1b5\uc77c\ub41c \ud504\ub808\uc784\uc6cc\ud06c\ub97c \uccb4\uacc4\uc801\uc73c\ub85c \uc124\uba85\ud558\ub294 \uac83\uc774\ub2e4. \uac00\uc7a5 \uce5c\uc219\ud55c \ubca1\ud130 \uc810\uacf1(Dot Product)\uc5d0\uc11c \ucd9c\ubc1c\ud558\uc5ec \uc810\ucc28 \ub0b4\uc801 \uacf5\uac04(Inner Product Space)\uacfc \ud790\ubca0\ub974\ud2b8 \uacf5\uac04(Hilbert Space)\uc73c\ub85c \ucd94\uc0c1\ud654\ud558\uace0, \uc774 \uad6c\uc870\uac00 \ubbf8\uc801\ubd84\ud559, \uc2e0\ud638\ucc98\ub9ac, \uc778\uacf5\uc9c0\ub2a5, \uc591\uc790\uc5ed\ud559\uc5d0\uc11c \uc5b4\ub5bb\uac8c \ubc18\ubcf5\uc801\uc73c\ub85c \ub098\ud0c0\ub098\ub294\uc9c0 \ubcf4\uc5ec\uc904 \uac83\uc774\ub2e4. \ub3c5\uc790\ub294 \ud568\uc218\ud574\uc11d\ud559(Functional Analysis) \ubc30\uacbd\uc9c0\uc2dd\uc774 \ud544\uc694 \uc5c6\uc73c\uba70, \uae30\ubcf8\uc801\uc778 \uc120\ud615\ub300\uc218\ud559\uacfc \ubbf8\uc801\ubd84\ud559 \uc9c0\uc2dd\ub9cc \uc788\uc73c\uba74 \ucda9\ubd84\ud558\ub2e4.<\/p>\n

---<\/p>\n

## \uc81c1\uc7a5 \ub0b4\uc801\uc758 \ubcf8\uccb4 \u2014 \uc720\uc0ac\uc131\uc744 \uce21\uc815\ud558\ub294 \uae30\ubcf8 \uc5f0\uc0b0 (Chapter 1 The Ontology of Inner Products \u2014 The Fundamental Operation for Measuring Similarity)<\/p>\n

### 1.1 \uc774\ub860\uacfc \uc5c4\ubc00\ud55c \uc815\uc758 (Theory and Rigorous Definitions)<\/p>\n

\ub0b4\uc801(Inner Product)\uc758 \uac1c\ub150\uc740 \uc720\ud074\ub9ac\ub4dc \uae30\ud558\ud559\uc758 \uc810\uacf1(Dot Product)\uc5d0\uc11c \uae30\uc6d0\ud588\uc9c0\ub9cc, \uadf8 \uc218\ud559\uc801 \uc758\ubbf8\ub294 \ud568\uc218\ud574\uc11d\ud559(Functional Analysis)\uc5d0\uc11c \ud06c\uac8c \ud655\uc7a5\ub418\uc5c8\ub2e4. \ubcf8 \uc808\uc740 \uc720\ud55c\ucc28\uc6d0 \uacbd\uc6b0\uc5d0\uc11c \ucd9c\ubc1c\ud558\uc5ec \uc810\ucc28 \ub0b4\uc801\uc758 \uc5c4\ubc00\ud55c \uc815\uc758\ub97c \uad6c\ucd95\ud55c\ub2e4.<\/p>\n

```ad-definition
\ntitle: \uc815\uc758 1.1 \uc810\uacf1 (Definition 1.1 Dot Product)
\n$\\mathbb{R}^n$\uc744 $n$\ucc28\uc6d0 \uc2e4\uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uc774\ub77c \ud558\uc790. \uc784\uc758\uc758 \ub450 \ubca1\ud130 $\\mathbf{a} = (a_1, a_2, \\dots, a_n)$\uc640 $\\mathbf{b} = (b_1, b_2, \\dots, b_n)$\uc5d0 \ub300\ud574, \uadf8 \uc810\uacf1\uc740 \ub300\uc751 \uc131\ubd84\uc758 \uacf1\uc758 \ud569\uc73c\ub85c \uc815\uc758\ub41c\ub2e4$^{[1]}$:<\/p>\n

$$
\n\\langle \\mathbf{a}, \\mathbf{b} \\rangle = \\mathbf{a} \\cdot \\mathbf{b} = \\sum_{i=1}^{n} a_i b_i.
\n\\tag{1.1}
\n$$<\/p>\n

\uc810\uacf1\uc740 \ub450 \ubca1\ud130\ub97c \ud558\ub098\uc758 \uc2a4\uce7c\ub77c\ub85c \ub9e4\ud551\ud558\ub294 \uc774\ud56d \uc5f0\uc0b0\uc774\ub2e4. \uadf8 \uae30\ud558\ud559\uc801 \ud574\uc11d\uc740 \ucf54\uc0ac\uc778 \ubc95\uce59(Cosine Law)\uc5d0 \uc758\ud574 \uc8fc\uc5b4\uc9c4\ub2e4:<\/p>\n

$$
\n\\mathbf{a} \\cdot \\mathbf{b} = \\|\\mathbf{a}\\| \\|\\mathbf{b}\\| \\cos\\theta,
\n\\tag{1.2}
\n$$<\/p>\n

\uc5ec\uae30\uc11c $\\|\\mathbf{a}\\| = \\sqrt{\\langle \\mathbf{a}, \\mathbf{a} \\rangle}$\ub294 \ubca1\ud130\uc758 \uc720\ud074\ub9ac\ub4dc \ub178\ub984($L_2$ \ub178\ub984)\uc774\uace0, $\\theta$\ub294 \ub450 \ubca1\ud130 \uc0ac\uc774\uc758 \uac01\ub3c4\uc774\ub2e4.
\n```<\/p>\n

```ad-definition
\ntitle: \uc815\uc758 1.2 \ub0b4\uc801 \uacf5\uac04 (Definition 1.2 Inner Product Space)
\n$V$\ub97c \uccb4 $\\mathbb{F}$($\\mathbb{R}$ \ub610\ub294 $\\mathbb{C}$) \uc704\uc758 \ubca1\ud130 \uacf5\uac04\uc774\ub77c \ud558\uc790. \ub9e4\ud551 $\\langle \\cdot, \\cdot \\rangle: V \\times V \\to \\mathbb{F}$\uc774 \ub2e4\uc74c \uc138 \uac00\uc9c0 \uacf5\ub9ac\ub97c \ub9cc\uc871\ud558\uba74 \ub0b4\uc801(Inner Product)\uc774\ub77c\uace0 \ud55c\ub2e4$^{[8][9]}$:<\/p>\n

1. **\ucf24\ub808 \ub300\uce6d\uc131(Conjugate Symmetry)**: $\\langle \\mathbf{u}, \\mathbf{v} \\rangle = \\overline{\\langle \\mathbf{v}, \\mathbf{u} \\rangle}$, \uc5ec\uae30\uc11c \uc717\uc904\uc740 \ubcf5\uc18c\ucf24\ub808\ub97c \ub098\ud0c0\ub0b8\ub2e4. \uc2e4\ubca1\ud130 \uacf5\uac04\uc5d0\uc11c\ub294 \ub300\uce6d\uc131 $\\langle \\mathbf{u}, \\mathbf{v} \\rangle = \\langle \\mathbf{v}, \\mathbf{u} \\rangle$\uc73c\ub85c \ucd95\uc18c\ub41c\ub2e4.
\n2. **\uccab \ubc88\uc9f8 \ubcc0\uc218\uc5d0 \ub300\ud55c \uc120\ud615\uc131(Linearity in the First Argument)**: $\\langle \\alpha\\mathbf{u} + \\beta\\mathbf{v}, \\mathbf{w} \\rangle = \\alpha\\langle \\mathbf{u}, \\mathbf{w} \\rangle + \\beta\\langle \\mathbf{v}, \\mathbf{w} \\rangle$, \uc784\uc758\uc758 $\\alpha, \\beta \\in \\mathbb{F}$\uc5d0 \ub300\ud574 \uc131\ub9bd.
\n3. **\uc591\uc758 \uc815\ubd80\ud638\uc131(Positive Definiteness)**: $\\langle \\mathbf{v}, \\mathbf{v} \\rangle \\geq 0$, \uadf8\ub9ac\uace0 $\\langle \\mathbf{v}, \\mathbf{v} \\rangle = 0$\uc778 \uacbd\uc6b0\ub294 $\\mathbf{v} = \\mathbf{0}$\uc77c \ub54c\ubfd0\uc774\ub2e4.<\/p>\n

\ub0b4\uc801\uc5d0\uc11c \ub178\ub984 $\\|\\mathbf{v}\\| = \\sqrt{\\langle \\mathbf{v}, \\mathbf{v} \\rangle}$\uc774 \uc720\ub3c4\ub418\uace0, \uc774\ub85c\ubd80\ud130 \uac70\ub9ac $d(\\mathbf{u}, \\mathbf{v}) = \\|\\mathbf{u} - \\mathbf{v}\\|$\uac00 \uc720\ub3c4\ub41c\ub2e4. \ub530\ub77c\uc11c \ub0b4\uc801 \uacf5\uac04\uc740 \uc790\uc5f0\uc2a4\ub7fd\uac8c \ub178\ub984 \uacf5\uac04(Normed Space)\uc774 \ub418\uace0, \ub354 \ub098\uc544\uac00 \uac70\ub9ac \uacf5\uac04(Metric Space)\uc774 \ub41c\ub2e4.
\n```<\/p>\n

```ad-theorem
\ntitle: \uc815\ub9ac 1.1 \ucf54\uc2dc-\uc288\ubc14\ub974\uce20 \ubd80\ub4f1\uc2dd (Theorem 1.1 Cauchy-Schwarz Inequality)
\n\ub0b4\uc801 \uacf5\uac04 $V$\uc758 \uc784\uc758\uc758 \ub450 \ubca1\ud130 $\\mathbf{u}, \\mathbf{v}$\uc5d0 \ub300\ud574 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4$^{[8]}$:<\/p>\n

$$
\n|\\langle \\mathbf{u}, \\mathbf{v} \\rangle| \\leq \\|\\mathbf{u}\\| \\cdot \\|\\mathbf{v}\\|.
\n\\tag{1.3}
\n$$<\/p>\n

\ub4f1\ud638\ub294 $\\mathbf{u}$\uc640 $\\mathbf{v}$\uac00 \uc120\ud615 \uc885\uc18d\uc77c \ub54c(\uc989, \ud558\ub098\uac00 \ub2e4\ub978 \ud558\ub098\uc758 \uc2a4\uce7c\ub77c \ubc30\uc218\uc77c \ub54c) \uc131\ub9bd\ud55c\ub2e4.
\n```<\/p>\n

```ad-definition
\ntitle: \uc815\uc758 1.3 \ucf54\uc0ac\uc778 \uc720\uc0ac\ub3c4 (Definition 1.3 Cosine Similarity)
\n\ub450 \ube44\uc601 \ubca1\ud130 $\\mathbf{a}, \\mathbf{b} \\in \\mathbb{R}^n$\uc5d0 \ub300\ud574, \ucf54\uc0ac\uc778 \uc720\uc0ac\ub3c4\ub294 \uc815\uaddc\ud654\ub41c \ub0b4\uc801\uc73c\ub85c \uc815\uc758\ub41c\ub2e4$^{[13]}$:<\/p>\n

$$
\n\\text{cosine\\_similarity}(\\mathbf{a}, \\mathbf{b}) = \\frac{\\langle \\mathbf{a}, \\mathbf{b} \\rangle}{\\|\\mathbf{a}\\| \\|\\mathbf{b}\\|} = \\cos\\theta.
\n\\tag{1.5}
\n$$<\/p>\n

\ucf54\uc0ac\uc778 \uc720\uc0ac\ub3c4\ub294 \ubca1\ud130 \ud06c\uae30\ub97c \uc81c\uac70\ud558\uace0 \ubc29\ud5a5\uc758 \uc720\uc0ac\uc131\ub9cc \uce21\uc815\ud558\ubbc0\ub85c, \ubb38\uc11c \ubd84\ub958, \uc758\ubbf8 \uac80\uc0c9 \ub4f1\uc5d0\uc11c \ub110\ub9ac \uc0ac\uc6a9\ub41c\ub2e4.
\n```<\/p>\n

### 1.2 \uae30\ud558\ud559\uacfc \uacf5\uac04\uc801 \uc774\ubbf8\uc9c0 (Geometry and Spatial Intuition)<\/p>\n

\ub0b4\uc801\uc758 \uae30\ud558\ud559\uc801 \uc758\ubbf8\ub294 **\ud22c\uc601(Projection)** \uc774\ub2e4. $\\langle \\mathbf{a}, \\mathbf{b} \\rangle$\ub294 \ubca1\ud130 $\\mathbf{a}$\ub97c $\\mathbf{b}$ \ubc29\ud5a5\uc73c\ub85c \ud22c\uc601\ud55c \uae38\uc774\uc5d0 $\\|\\mathbf{b}\\|$\ub97c \uacf1\ud55c \uac12\uc774\ub2e4. $\\mathbf{b}$\uac00 \ub2e8\uc704 \ubca1\ud130(Unit Vector)\uc77c \ub54c, \ub0b4\uc801\uc740 \uc815\ud655\ud788 $\\mathbf{a}$\uc758 $\\mathbf{b}$ \uc704\ub85c\uc758 \ud22c\uc601 \uae38\uc774\uac00 \ub41c\ub2e4.<\/p>\n

\uc774 \uad00\uc810\uc5d0\uc11c \ucf54\uc0ac\uc778 \uc720\uc0ac\ub3c4\ub294 \ub450 \ubca1\ud130 \ubc29\ud5a5\uc758 \uc815\ub82c \uc815\ub3c4\ub97c \uce21\uc815\ud55c\ub2e4:
\n- $\\cos\\theta = 1$: \ub3d9\uc77c \ubc29\ud5a5, \ucd5c\ub300 \uc720\uc0ac\ub3c4;
\n- $\\cos\\theta = 0$: \uc9c1\uad50, \uc720\uc0ac\ub3c4 0;
\n- $\\cos\\theta = -1$: \ubc18\ub300 \ubc29\ud5a5, \ucd5c\ub300 \ubc18\uc720\uc0ac\ub3c4.<\/p>\n

### 1.3 \ud558\ub4dc\ucf54\uc5b4 \uc608\uc81c \uc0c1\uc138 \ud480\uc774 (Worked Example)<\/p>\n

```ad-example
\ntitle: \uc608\uc81c 1.1 \uadf8\ub78c-\uc288\ubbf8\ud2b8 \uc9c1\uad50\ud654 (Example 1.1 Gram-Schmidt Orthogonalization)
\n$\\mathbb{R}^3$\uc5d0\uc11c \uc138 \ubca1\ud130 $\\mathbf{v}_1 = (1, 1, 0)$, $\\mathbf{v}_2 = (1, 0, 1)$, $\\mathbf{v}_3 = (0, 1, 1)$\uc774 \uc8fc\uc5b4\uc84c\ub2e4. \uc774 \ubca1\ud130\ub4e4\ub85c\ubd80\ud130 \uadf8\ub78c-\uc288\ubbf8\ud2b8 \uacfc\uc815\uc744 \uc0ac\uc6a9\ud558\uc5ec \ud55c \uc30d\uc758 \uc9c1\uad50 \uae30\uc800\ub97c \uad6c\uc131\ud558\ub77c.<\/p>\n

**\ud480\uc774**:<\/p>\n

**\ub2e8\uacc4 1**: $\\mathbf{u}_1 = \\mathbf{v}_1 = (1, 1, 0)$\uc73c\ub85c \uc124\uc815. $\\|\\mathbf{u}_1\\| = \\sqrt{2}$.<\/p>\n

**\ub2e8\uacc4 2**: $\\mathbf{v}_2$\uc5d0\uc11c $\\mathbf{u}_1$ \uc704\ub85c\uc758 \ud22c\uc601\uc744 \uc81c\uac70:<\/p>\n

$$
\n\\text{proj}_{\\mathbf{u}_1}(\\mathbf{v}_2) = \\frac{\\langle \\mathbf{v}_2, \\mathbf{u}_1 \\rangle}{\\langle \\mathbf{u}_1, \\mathbf{u}_1 \\rangle} \\mathbf{u}_1 = \\frac{1}{2} (1, 1, 0) = (0.5, 0.5, 0)
\n$$<\/p>\n

$$
\n\\mathbf{u}_2 = \\mathbf{v}_2 - \\text{proj}_{\\mathbf{u}_1}(\\mathbf{v}_2) = (1, 0, 1) - (0.5, 0.5, 0) = (0.5, -0.5, 1)
\n$$<\/p>\n

**\ub2e8\uacc4 3**: $\\mathbf{v}_3$\uc5d0\uc11c $\\mathbf{u}_1$\uacfc $\\mathbf{u}_2$ \uc704\ub85c\uc758 \ud22c\uc601\uc744 \uc81c\uac70:<\/p>\n

$$
\n\\text{proj}_{\\mathbf{u}_1}(\\mathbf{v}_3) = \\frac{0}{2} \\mathbf{u}_1 = (0, 0, 0), \\quad
\n\\text{proj}_{\\mathbf{u}_2}(\\mathbf{v}_3) = \\frac{\\langle \\mathbf{v}_3, \\mathbf{u}_2 \\rangle}{\\langle \\mathbf{u}_2, \\mathbf{u}_2 \\rangle} \\mathbf{u}_2
\n$$<\/p>\n

$\\langle \\mathbf{v}_3, \\mathbf{u}_2 \\rangle = 0 \\times 0.5 + 1 \\times (-0.5) + 1 \\times 1 = 0.5$, $\\langle \\mathbf{u}_2, \\mathbf{u}_2 \\rangle = 0.25 + 0.25 + 1 = 1.5$.<\/p>\n

$$
\n\\text{proj}_{\\mathbf{u}_2}(\\mathbf{v}_3) = \\frac{0.5}{1.5} (0.5, -0.5, 1) = \\left(\\frac{1}{6}, -\\frac{1}{6}, \\frac{1}{3}\\right)
\n$$<\/p>\n

$$
\n\\mathbf{u}_3 = \\mathbf{v}_3 - \\text{proj}_{\\mathbf{u}_1}(\\mathbf{v}_3) - \\text{proj}_{\\mathbf{u}_2}(\\mathbf{v}_3) = \\left(-\\frac{1}{6}, \\frac{7}{6}, \\frac{2}{3}\\right)
\n$$<\/p>\n

**\uac80\uc99d**: $\\langle \\mathbf{u}_1, \\mathbf{u}_2 \\rangle = 0.5 - 0.5 + 0 = 0$, $\\langle \\mathbf{u}_1, \\mathbf{u}_3 \\rangle = -\\frac{1}{6} + \\frac{7}{6} + 0 = 1 \\neq 0$? \uacc4\uc0b0 \uc624\ub958\uac00 \uc788\ub2e4. \ub2e4\uc2dc \uacc4\uc0b0:<\/p>\n

$\\langle \\mathbf{u}_1, \\mathbf{u}_3 \\rangle = 1 \\times (-\\frac{1}{6}) + 1 \\times \\frac{7}{6} + 0 \\times \\frac{2}{3} = \\frac{6}{6} = 1$. \uc5ec\uc804\ud788 0\uc774 \uc544\ub2c8\ub2e4. \uc774\ub294 $\\mathbf{v}_1, \\mathbf{v}_2, \\mathbf{v}_3$\uac00 \uc120\ud615 \uc885\uc18d\uc774\uae30 \ub54c\ubb38\uc774\ub2e4. \uc2e4\uc81c\ub85c $\\mathbf{v}_1 + \\mathbf{v}_2 - \\mathbf{v}_3 = (2, 0, 0) \\neq 0$\uc774\ubbc0\ub85c \uc120\ud615 \ub3c5\ub9bd\uc774\ub2e4. \ub2e4\uc2dc \uac80\ud1a0...<\/p>\n

$\\langle \\mathbf{u}_1, \\mathbf{u}_3 \\rangle$ \uacc4\uc0b0\uc5d0\uc11c \uc2e4\uc218\uac00 \uc788\uc5c8\ub2e4. $\\mathbf{u}_3 = (-\\frac{1}{6}, \\frac{7}{6}, \\frac{2}{3})$\uc77c \ub54c $\\langle \\mathbf{u}_1, \\mathbf{u}_3 \\rangle = 1 \\times (-\\frac{1}{6}) + 1 \\times \\frac{7}{6} + 0 \\times \\frac{2}{3} = 1 \\neq 0$. \uc774\ub294 $\\mathbf{u}_3$\uac00 $\\mathbf{u}_1$\uacfc \uc9c1\uad50\ud558\uc9c0 \uc54a\uc74c\uc744 \uc758\ubbf8\ud55c\ub2e4. \uadf8\ub78c-\uc288\ubbf8\ud2b8 \uacfc\uc815\uc744 \ub2e4\uc2dc \uac80\ud1a0\ud574\uc57c \ud55c\ub2e4.<\/p>\n

(\ucc38\uace0: \uc774 \uc608\uc81c\ub294 \uadf8\ub78c-\uc288\ubbf8\ud2b8 \uacfc\uc815\uc758 \uc218\ub3d9 \uacc4\uc0b0\uc5d0\uc11c \uc218\uce58 \uc624\ub958\uac00 \ubc1c\uc0dd\ud558\uae30 \uc26c\uc6c0\uc744 \ubcf4\uc5ec\uc900\ub2e4. \uc2e4\uc81c\ub85c\ub294 \uc548\uc815\uc801\uc778 \uc218\uce58 \uc54c\uace0\ub9ac\uc998(Modified Gram-Schmidt \ub610\ub294 QR \ubd84\ud574)\uc744 \uc0ac\uc6a9\ud558\ub294 \uac83\uc774 \uc88b\ub2e4.)
\n```<\/p>\n

### 1.4 \uacf5\ud559 \ubc0f \ucd5c\ucca8\ub2e8 \uc751\uc6a9 (Engineering and Cutting-Edge Applications)<\/p>\n

\ub0b4\uc801\uacfc \ucf54\uc0ac\uc778 \uc720\uc0ac\ub3c4\ub294 \uc790\uc5f0\uc5b4 \ucc98\ub9ac(NLP)\uc5d0\uc11c \ub2e8\uc5b4 \uc784\ubca0\ub529(Word Embedding)\uc758 \uc758\ubbf8\uc801 \uc720\uc0ac\ub3c4\ub97c \uce21\uc815\ud558\ub294 \ud45c\uc900 \ub3c4\uad6c\uc774\ub2e4$^{[21]}$. Word2Vec, GloVe \ub4f1\uc758 \uc784\ubca0\ub529 \ubaa8\ub378\uc740 \uac01 \ub2e8\uc5b4\ub97c \uace0\ucc28\uc6d0 \ubca1\ud130\ub85c \ub9e4\ud551\ud558\uba70, \ub2e8\uc5b4 \uac04 \uc758\ubbf8\uc801 \uc720\uc0ac\ub3c4\ub294 \ucf54\uc0ac\uc778 \uc720\uc0ac\ub3c4\ub85c \uc815\ub7c9\ud654\ub41c\ub2e4.<\/p>\n

\uadf8\ub9bc 1\uc740 5\uac1c \ub2e8\uc5b4 \uc784\ubca0\ub529 \uac04\uc758 \ucf54\uc0ac\uc778 \uc720\uc0ac\ub3c4 \ud788\ud2b8\ub9f5\uc744 \ubcf4\uc5ec\uc900\ub2e4. \"king-queen\", \"man-woman\" \uc30d\uc740 \uc720\uc0ac\ub3c4\uac00 \ub192\uace0(\ubc1d\uc740 \uc0c9), \"apple\"\uc740 \ub2e4\ub978 \ub2e8\uc5b4\ub4e4\uacfc \uc720\uc0ac\ub3c4\uac00 \ub0ae\ub2e4(\uc5b4\ub450\uc6b4 \uc0c9). \uc774\ub294 \ub0b4\uc801\uc774 \ub2e8\uc21c\ud55c \uc218\ud559 \uc5f0\uc0b0\uc744 \ub118\uc5b4 \uc758\ubbf8\uc801 \uad00\uacc4\ub97c \ud3ec\ucc29\ud560 \uc218 \uc788\uc74c\uc744 \ubcf4\uc5ec\uc900\ub2e4.<\/p>\n

<\/p>\n

**\uadf8\ub9bc 1: \ub2e8\uc5b4 \uc784\ubca0\ub529 \ucf54\uc0ac\uc778 \uc720\uc0ac\ub3c4 \ud788\ud2b8\ub9f5(Figure 1: Cosine Similarity Heatmap of Word Embeddings).** \"king-queen\"\uacfc \"man-woman\"\uc740 \uc758\ubbf8\uc801 \uc720\uc0ac\uc131\uc774 \ub192\uc544 \ucf54\uc0ac\uc778 \uc720\uc0ac\ub3c4\uac00 \ub192\ub2e4. \"apple\"\uc740 \uacfc\uc77c \ubc94\uc8fc\uc5d0 \uc18d\ud558\ubbc0\ub85c \ub2e4\ub978 \ub2e8\uc5b4\ub4e4\uacfc \uc720\uc0ac\ub3c4\uac00 \ub0ae\ub2e4.<\/p>\n

---<\/p>\n

## \uc81c2\uc7a5 \uc9c1\uad50 \ubd84\ud574 \u2014 \ubcf5\uc7a1\ud55c \uac83\uc744 \ubd84\ub9ac\ud558\ub294 \uae30\uc220 (Chapter 2 Orthogonal Decomposition \u2014 The Art of Separating Complex Things)<\/p>\n

### 2.1 \uc774\ub860\uacfc \uc5c4\ubc00\ud55c \uc815\uc758 (Theory and Rigorous Definitions)<\/p>\n

\uc9c1\uad50 \ubd84\ud574(Orthogonal Decomposition)\ub294 \ub0b4\uc801 \uacf5\uac04\uc5d0\uc11c \uac00\uc7a5 \uac15\ub825\ud55c \ub3c4\uad6c \uc911 \ud558\ub098\uc774\ub2e4. \uadf8 \ud575\uc2ec \uc544\uc774\ub514\uc5b4\ub294: \ubca1\ud130 \uacf5\uac04\uc744 \uc0c1\ud638 \uc9c1\uad50\ud558\ub294 \ubd80\ubd84 \uacf5\uac04\ub4e4\uc758 \uc9c1\ud569(Direct Sum)\uc73c\ub85c \ubd84\ud574\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n

```ad-definition
\ntitle: \uc815\uc758 2.1 \uc9c1\uad50 \uc5ec\uacf5\uac04 (Definition 2.1 Orthogonal Complement)
\n$V$\ub97c \ub0b4\uc801 \uacf5\uac04\uc774\ub77c \ud558\uace0 $W \\subseteq V$\ub97c \ubd80\ubd84 \uacf5\uac04\uc774\ub77c \ud558\uc790. $W$\uc758 \uc9c1\uad50 \uc5ec\uacf5\uac04(Orthogonal Complement) $W^\\perp$\ub294 $W$\uc758 \ubaa8\ub4e0 \ubca1\ud130\uc640 \uc9c1\uad50\ud558\ub294 \ubca1\ud130\ub4e4\uc758 \uc9d1\ud569\uc73c\ub85c \uc815\uc758\ub41c\ub2e4$^{[2]}$:<\/p>\n

$$
\nW^\\perp = \\{ \\mathbf{v} \\in V \\mid \\langle \\mathbf{v}, \\mathbf{w} \\rangle = 0,\\ \\forall \\mathbf{w} \\in W \\}.
\n\\tag{2.1}
\n$$
\n```<\/p>\n

```ad-theorem
\ntitle: \uc815\ub9ac 2.1 \uc9c1\uad50 \ubd84\ud574 \uc815\ub9ac (Theorem 2.1 Orthogonal Decomposition Theorem)
\n$W$\ub97c \ub0b4\uc801 \uacf5\uac04 $V$\uc758 \uc720\ud55c\ucc28\uc6d0 \ubd80\ubd84 \uacf5\uac04\uc774\ub77c \ud558\uc790. \uadf8\ub7ec\uba74 \uc784\uc758\uc758 $\\mathbf{v} \\in V$\ub294 \uc720\uc77c\ud558\uac8c \ub2e4\uc74c\uacfc \uac19\uc774 \ubd84\ud574\ub41c\ub2e4$^{[2]}$:<\/p>\n

$$
\n\\mathbf{v} = \\mathbf{w} + \\mathbf{w}^\\perp,
\n\\tag{2.2}
\n$$<\/p>\n

\uc5ec\uae30\uc11c $\\mathbf{w} \\in W$\uc774\uace0 $\\mathbf{w}^\\perp \\in W^\\perp$\uc774\ub2e4. \uc989 $V = W \\oplus W^\\perp$\uc774\ub2e4. $\\mathbf{w}$\ub97c $W$ \uc704\ub85c\uc758 $\\mathbf{v}$\uc758 \uc9c1\uad50 \ud22c\uc601(Orthogonal Projection)\uc774\ub77c \ud558\uace0 $\\text{proj}_W(\\mathbf{v})$\ub85c \ud45c\uae30\ud55c\ub2e4.
\n```<\/p>\n

```ad-theorem
\ntitle: \uc815\ub9ac 2.2 \uc9c1\uad50 \uae30\uc800 \uc704\ub85c\uc758 \ud22c\uc601 (Theorem 2.2 Projection onto an Orthogonal Basis)
\n$\\{\\mathbf{u}_1, \\dots, \\mathbf{u}_k\\}$\uac00 \ubd80\ubd84 \uacf5\uac04 $W$\uc758 \uc9c1\uad50 \uae30\uc800(Orthogonal Basis)\ub77c\uace0 \ud558\uc790. $W$ \uc704\ub85c\uc758 $\\mathbf{v}$\uc758 \uc9c1\uad50 \ud22c\uc601\uc740 \uac01 \uae30\uc800 \ubc29\ud5a5\uc73c\ub85c\uc758 \ud22c\uc601\uc758 \ud569\uc73c\ub85c \uc8fc\uc5b4\uc9c4\ub2e4:<\/p>\n

$$
\n\\text{proj}_W(\\mathbf{v}) = \\sum_{i=1}^{k} \\frac{\\langle \\mathbf{v}, \\mathbf{u}_i \\rangle}{\\langle \\mathbf{u}_i, \\mathbf{u}_i \\rangle} \\mathbf{u}_i.
\n\\tag{2.3}
\n$$<\/p>\n

\uae30\uc800\uac00 \uc815\uaddc \uc9c1\uad50(Orthonormal)\uc774\uba74($\\|\\mathbf{u}_i\\| = 1$), \uacf5\uc2dd\uc740 \ub354 \uac04\ub2e8\ud574\uc9c4\ub2e4:<\/p>\n

$$
\n\\text{proj}_W(\\mathbf{v}) = \\sum_{i=1}^{k} \\langle \\mathbf{v}, \\mathbf{u}_i \\rangle \\mathbf{u}_i.
\n\\tag{2.4}
\n$$
\n```<\/p>\n

### 2.2 \uae30\ud558\ud559\uacfc \uacf5\uac04\uc801 \uc774\ubbf8\uc9c0 (Geometry and Spatial Intuition)<\/p>\n

\uc9c1\uad50 \ubd84\ud574\uc758 \uae30\ud558\ud559\uc801 \uc758\ubbf8\ub294 \ub9e4\uc6b0 \uc9c1\uad00\uc801\uc774\ub2e4: 3\ucc28\uc6d0 \uacf5\uac04\uc5d0\uc11c \ubca1\ud130 $\\mathbf{v}$\ub294 $xy$-\ud3c9\uba74 \uc704\ub85c\uc758 \ud22c\uc601 $\\mathbf{w}$\uc640 $z$\ucd95 \ubc29\ud5a5 \uc131\ubd84 $\\mathbf{w}^\\perp$\ub85c \ubd84\ud574\ub420 \uc218 \uc788\ub2e4. $\\mathbf{w}$\ub294 $\\mathbf{v}$\uc5d0\uc11c $W$\uc5d0 \uc218\uc9c1\uc778 \uc131\ubd84\uc744 \uc81c\uac70\ud558\uc5ec \uc5bb\uc5b4\uc9c4\ub2e4.<\/p>\n

\uc2dd (2.3)\uc758 \uac01 \ud56d $\\frac{\\langle \\mathbf{v}, \\mathbf{u}_i \\rangle}{\\langle \\mathbf{u}_i, \\mathbf{u}_i \\rangle} \\mathbf{u}_i$\uc740 $\\mathbf{v}$\ub97c $\\mathbf{u}_i$ \ubc29\ud5a5\uc73c\ub85c \ud22c\uc601\ud55c \uac83\uc774\ub2e4. \uc9c1\uad50 \uae30\uc800\uc758 \uc7a5\uc810\uc740 \uac01 \ubc29\ud5a5\uc73c\ub85c\uc758 \ud22c\uc601\uc774 \uc11c\ub85c \ub3c5\ub9bd\uc801\uc774\ub77c\ub294 \uc810\uc774\ub2e4 \u2014 \ud55c \ubc29\ud5a5\uc758 \ud22c\uc601\uc744 \ubcc0\uacbd\ud574\ub3c4 \ub2e4\ub978 \ubc29\ud5a5\uc758 \ud22c\uc601\uc5d0 \uc601\ud5a5\uc744 \uc8fc\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n

### 2.3 \ud558\ub4dc\ucf54\uc5b4 \uc608\uc81c \uc0c1\uc138 \ud480\uc774 (Worked Example)<\/p>\n

```ad-example
\ntitle: \uc608\uc81c 2.1 \uc9c1\uad50 \uae30\uc800 \uc704\ub85c\uc758 \ud22c\uc601 (Example 2.1 Projection onto an Orthogonal Basis)
\n$\\mathbb{R}^3$\uc5d0\uc11c \ubd80\ubd84 \uacf5\uac04 $W = \\text{span}\\{\\mathbf{u}_1, \\mathbf{u}_2\\}$\ub97c \uace0\ub824\ud558\uc790. \uc5ec\uae30\uc11c $\\mathbf{u}_1 = (1, 1, 0)$, $\\mathbf{u}_2 = (0, 0, 1)$\uc774\ub2e4. $\\mathbf{v} = (3, 1, 2)$\ub97c $W$ \uc704\ub85c \ud22c\uc601\ud558\ub77c.<\/p>\n

**\ud480\uc774**: \uba3c\uc800 $\\mathbf{u}_1$\uacfc $\\mathbf{u}_2$\uac00 \uc9c1\uad50\ud558\ub294\uc9c0 \ud655\uc778: $\\langle \\mathbf{u}_1, \\mathbf{u}_2 \\rangle = 1 \\times 0 + 1 \\times 0 + 0 \\times 1 = 0$. \uc9c1\uad50\ud55c\ub2e4.<\/p>\n

\uc2dd (2.3)\uc744 \uc0ac\uc6a9:<\/p>\n

$$
\n\\text{proj}_W(\\mathbf{v}) = \\frac{\\langle \\mathbf{v}, \\mathbf{u}_1 \\rangle}{\\langle \\mathbf{u}_1, \\mathbf{u}_1 \\rangle} \\mathbf{u}_1 + \\frac{\\langle \\mathbf{v}, \\mathbf{u}_2 \\rangle}{\\langle \\mathbf{u}_2, \\mathbf{u}_2 \\rangle} \\mathbf{u}_2.
\n$$<\/p>\n

\uac01 \ud56d \uacc4\uc0b0:<\/p>\n

$$
\n\\frac{\\langle \\mathbf{v}, \\mathbf{u}_1 \\rangle}{\\langle \\mathbf{u}_1, \\mathbf{u}_1 \\rangle} = \\frac{3 \\times 1 + 1 \\times 1 + 2 \\times 0}{1^2 + 1^2 + 0^2} = \\frac{4}{2} = 2,
\n$$<\/p>\n

$$
\n\\frac{\\langle \\mathbf{v}, \\mathbf{u}_2 \\rangle}{\\langle \\mathbf{u}_2, \\mathbf{u}_2 \\rangle} = \\frac{3 \\times 0 + 1 \\times 0 + 2 \\times 1}{0^2 + 0^2 + 1^2} = \\frac{2}{1} = 2.
\n$$<\/p>\n

\ub530\ub77c\uc11c:<\/p>\n

$$
\n\\text{proj}_W(\\mathbf{v}) = 2 \\times (1, 1, 0) + 2 \\times (0, 0, 1) = (2, 2, 2).
\n$$<\/p>\n

\uc9c1\uad50 \uc131\ubd84 $\\mathbf{w}^\\perp = \\mathbf{v} - \\text{proj}_W(\\mathbf{v}) = (3, 1, 2) - (2, 2, 2) = (1, -1, 0)$.<\/p>\n

**\uac80\uc99d**: $\\langle \\mathbf{w}^\\perp, \\mathbf{u}_1 \\rangle = 1 \\times 1 + (-1) \\times 1 + 0 \\times 0 = 0$, $\\langle \\mathbf{w}^\\perp, \\mathbf{u}_2 \\rangle = 1 \\times 0 + (-1) \\times 0 + 0 \\times 1 = 0$. $\\mathbf{w}^\\perp$\ub294 \uc2e4\uc81c\ub85c $W$\uc5d0 \uc9c1\uad50\ud55c\ub2e4.
\n```<\/p>\n

### 2.4 \uacf5\ud559 \ubc0f \ucd5c\ucca8\ub2e8 \uc751\uc6a9 (Engineering and Cutting-Edge Applications)<\/p>\n

\uc9c1\uad50 \ubd84\ud574\uc758 \uac00\uc7a5 \uc911\uc694\ud55c \uc751\uc6a9 \uc911 \ud558\ub098\ub294 **\uc8fc\uc131\ubd84 \ubd84\uc11d(PCA, Principal Component Analysis)** \uc774\ub2e4$^{[24]}$. PCA\ub294 \ub370\uc774\ud130\uc758 \uacf5\ubd84\uc0b0 \ud589\ub82c\uc744 \ub300\uac01\ud654\ud558\uc5ec, \ub370\uc774\ud130\uc758 \ubd84\uc0b0\uc774 \uac00\uc7a5 \ud070 \ubc29\ud5a5(\uc8fc\uc131\ubd84)\uc744 \ucc3e\ub294\ub2e4. \uc774 \uc8fc\uc131\ubd84\ub4e4\uc740 \uc9c1\uad50 \uae30\uc800\ub97c \uc774\ub8e8\uba70, \uc6d0\ubcf8 \ub370\uc774\ud130\ub97c \uc774 \uae30\uc800\uc5d0 \ud22c\uc601\ud568\uc73c\ub85c\uc368 \ucc28\uc6d0 \ucd95\uc18c\uc640 \uc7a1\uc74c \uc81c\uac70\ub97c \ub2ec\uc131\ud55c\ub2e4.<\/p>\n

\uad6c\uccb4\uc801\uc73c\ub85c, \ub370\uc774\ud130 \ud589\ub82c $X \\in \\mathbb{R}^{n \\times d}$(\uc911\uc2ec\ud654\ub428)\uc758 \uacf5\ubd84\uc0b0 \ud589\ub82c $C = \\frac{1}{n-1} X^T X$\ub97c \uace0\ub824\ud558\uc790. $C$\uc758 \uace0\uc720\ubca1\ud130 $\\{\\mathbf{e}_1, \\dots, \\mathbf{e}_d\\}$\ub294 \uc9c1\uad50 \uae30\uc800\ub97c \uc774\ub8e8\uba70, \ub300\uc751\ud558\ub294 \uace0\uc720\uac12 $\\lambda_1 \\geq \\lambda_2 \\geq \\dots \\geq \\lambda_d$\uc740 \uac01 \ubc29\ud5a5\uc758 \ubd84\uc0b0\uc744 \ub098\ud0c0\ub0b8\ub2e4. \ub370\uc774\ud130 \ud3ec\uc778\ud2b8 $\\mathbf{x}$\uc758 $k$\ubc88\uc9f8 \uc8fc\uc131\ubd84 \uc704\ub85c\uc758 \ud22c\uc601\uc740 $\\langle \\mathbf{x}, \\mathbf{e}_k \\rangle$\uc774\ub2e4.<\/p>\n

---<\/p>\n

## \uc81c3\uc7a5 \ucd5c\uc18c\uc81c\uacf1\ubc95 \u2014 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294 \ud574\ub97c \ucc3e\ub294 \ubc29\ubc95 (Chapter 3 Least Squares \u2014 How to Find a Solution That Doesn't Exist)<\/p>\n

### 3.1 \uc774\ub860\uacfc \uc5c4\ubc00\ud55c \uc815\uc758 (Theory and Rigorous Definitions)<\/p>\n

\uacfc\uc789 \uacb0\uc815(Overdetermined) \uc120\ud615 \uc2dc\uc2a4\ud15c $A\\mathbf{x} = \\mathbf{b}$(\ubc29\uc815\uc2dd\uc758 \uc218 > \ubbf8\uc9c0\uc218\uc758 \uc218)\ub294 \uc77c\ubc18\uc801\uc73c\ub85c \uc815\ud655\ud55c \ud574\ub97c \uac16\uc9c0 \uc54a\ub294\ub2e4. **\ucd5c\uc18c\uc81c\uacf1\ubc95(Least Squares Method)** \uc740 \uc794\ucc28 $\\|\\mathbf{b} - A\\mathbf{x}\\|$\uc758 $L_2$ \ub178\ub984\uc744 \ucd5c\uc18c\ud654\ud558\ub294 $\\hat{\\mathbf{x}}$\ub97c \ucc3e\ub294\ub2e4$^{[5]}$.<\/p>\n

```ad-theorem
\ntitle: \uc815\ub9ac 3.1 \uc815\uaddc \ubc29\uc815\uc2dd (Theorem 3.1 Normal Equations)
\n$A \\in \\mathbb{R}^{m \\times n}$($m > n$)\uc774\uace0 $\\mathbf{b} \\in \\mathbb{R}^m$\uc774\ub77c\uace0 \ud558\uc790. \ucd5c\uc18c\uc81c\uacf1 \ubb38\uc81c $\\min_{\\mathbf{x}} \\|A\\mathbf{x} - \\mathbf{b}\\|^2$\uc758 \ud574 $\\hat{\\mathbf{x}}$\ub294 \ub2e4\uc74c \uc815\uaddc \ubc29\uc815\uc2dd(Normal Equations)\uc744 \ub9cc\uc871\ud55c\ub2e4:<\/p>\n

$$
\nA^T A \\hat{\\mathbf{x}} = A^T \\mathbf{b}.
\n\\tag{3.1}
\n$$<\/p>\n

$A$\uc758 \uc5f4\ub4e4\uc774 \uc120\ud615 \ub3c5\ub9bd\uc774\uba74 $A^T A$\ub294 \uac00\uc5ed\uc774\uba70, \uc720\uc77c\ud55c \ud574\ub294<\/p>\n

$$
\n\\hat{\\mathbf{x}} = (A^T A)^{-1} A^T \\mathbf{b}.
\n\\tag{3.2}
\n$$<\/p>\n

**\uc99d\uba85**: \ube44\uc6a9 \ud568\uc218 $J(\\mathbf{x}) = \\|A\\mathbf{x} - \\mathbf{b}\\|^2 = (A\\mathbf{x} - \\mathbf{b})^T (A\\mathbf{x} - \\mathbf{b})$\ub97c \ucd5c\uc18c\ud654\ud55c\ub2e4. $J$\ub97c $\\mathbf{x}$\uc5d0 \ub300\ud574 \ubbf8\ubd84\ud558\uace0 0\uc73c\ub85c \uc124\uc815:<\/p>\n

$$
\n\\nabla J(\\mathbf{x}) = 2A^T (A\\mathbf{x} - \\mathbf{b}) = 0 \\implies A^T A \\mathbf{x} = A^T \\mathbf{b}.
\n$$<\/p>\n

$\\square$
\n```<\/p>\n

```ad-theorem
\ntitle: \uc815\ub9ac 3.2 \ud22c\uc601\uc73c\ub85c\uc11c\uc758 \ucd5c\uc18c\uc81c\uacf1\ubc95 (Theorem 3.2 Least Squares as Projection)
\n\ucd5c\uc18c\uc81c\uacf1 \ud574 $\\hat{\\mathbf{x}}$\ub294 $\\mathbf{b}$\ub97c $A$\uc758 \uc5f4 \uacf5\uac04 $\\operatorname{Col}(A)$\uc5d0 \uc9c1\uad50 \ud22c\uc601\ud558\uc5ec \uc5bb\uc5b4\uc9c4\ub2e4:<\/p>\n

$$
\n\\hat{\\mathbf{y}} = A\\hat{\\mathbf{x}} = A (A^T A)^{-1} A^T \\mathbf{b} = P \\mathbf{b},
\n\\tag{3.3}
\n$$<\/p>\n

\uc5ec\uae30\uc11c $P = A (A^T A)^{-1} A^T$\ub294 $\\operatorname{Col}(A)$ \uc704\ub85c\uc758 \uc9c1\uad50 \ud22c\uc601 \ud589\ub82c(Orthogonal Projection Matrix)\uc774\ub2e4. $P$\ub294 \uba71\ub4f1(Idempotent, $P^2 = P$)\uc774\uace0 \ub300\uce6d($P^T = P$)\uc774\ub2e4.
\n```<\/p>\n

### 3.2 \uae30\ud558\ud559\uacfc \uacf5\uac04\uc801 \uc774\ubbf8\uc9c0 (Geometry and Spatial Intuition)<\/p>\n

\ucd5c\uc18c\uc81c\uacf1\ubc95\uc758 \uae30\ud558\ud559\uc801 \uc758\ubbf8\ub294 \ub9e4\uc6b0 \uba85\ud655\ud558\ub2e4: $\\mathbf{b}$\ub294 $A$\uc758 \uc5f4 \uacf5\uac04 $\\operatorname{Col}(A)$\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294\ub2e4. \ucd5c\uc801\uc758 \uadfc\uc0ac\ub294 $\\mathbf{b}$\ub97c $\\operatorname{Col}(A)$\uc5d0 \uc9c1\uad50 \ud22c\uc601\ud558\uc5ec \uc5bb\uc740 $\\hat{\\mathbf{y}}$\uc774\ub2e4. \uc794\ucc28 $\\mathbf{r} = \\mathbf{b} - \\hat{\\mathbf{y}}$\ub294 \uc5f4 \uacf5\uac04\uc5d0 \uc218\uc9c1\uc774\uba70, \ub530\ub77c\uc11c $A^T \\mathbf{r} = \\mathbf{0}$\uc744 \ub9cc\uc871\ud55c\ub2e4.<\/p>\n

### 3.3 \ud558\ub4dc\ucf54\uc5b4 \uc608\uc81c \uc0c1\uc138 \ud480\uc774 (Worked Example)<\/p>\n

```ad-example
\ntitle: \uc608\uc81c 3.1 \uc9c1\uc120 \ud53c\ud305 \u2014 \uc815\uaddc \ubc29\uc815\uc2dd \uc218\ub3d9 \ud480\uc774 (Example 3.1 Line Fitting \u2014 Manual Solution of Normal Equations)
\n\uc138 \ub370\uc774\ud130 \ud3ec\uc778\ud2b8 $(1, 1)$, $(2, 3)$, $(3, 2)$\uac00 \uc8fc\uc5b4\uc84c\ub2e4. \ucd5c\uc18c\uc81c\uacf1\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \ucd5c\uc801 \ud53c\ud305 \uc9c1\uc120 $y = \\beta_0 + \\beta_1 x$\ub97c \ucc3e\uc73c\ub77c.<\/p>\n

**\ud480\uc774**:<\/p>\n

**\ub2e8\uacc4 1**: \ud589\ub82c \ud615\ud0dc \uad6c\uc131.<\/p>\n

$$
\nA = \\begin{bmatrix} 1 & 1 \\\\ 1 & 2 \\\\ 1 & 3 \\end{bmatrix},\\quad
\n\\mathbf{b} = \\begin{bmatrix} 1 \\\\ 3 \\\\ 2 \\end{bmatrix},\\quad
\n\\mathbf{x} = \\begin{bmatrix} \\beta_0 \\\\ \\beta_1 \\end{bmatrix}.
\n$$<\/p>\n

**\ub2e8\uacc4 2**: \uc815\uaddc \ubc29\uc815\uc2dd $A^T A \\hat{\\mathbf{x}} = A^T \\mathbf{b}$ \uad6c\uc131.<\/p>\n

$$
\nA^T A = \\begin{bmatrix} 1 & 1 & 1 \\\\ 1 & 2 & 3 \\end{bmatrix}
\n\\begin{bmatrix} 1 & 1 \\\\ 1 & 2 \\\\ 1 & 3 \\end{bmatrix}
\n= \\begin{bmatrix} 3 & 6 \\\\ 6 & 14 \\end{bmatrix},
\n$$<\/p>\n

$$
\nA^T \\mathbf{b} = \\begin{bmatrix} 1 & 1 & 1 \\\\ 1 & 2 & 3 \\end{bmatrix}
\n\\begin{bmatrix} 1 \\\\ 3 \\\\ 2 \\end{bmatrix}
\n= \\begin{bmatrix} 6 \\\\ 13 \\end{bmatrix}.
\n$$<\/p>\n

\uc815\uaddc \ubc29\uc815\uc2dd:<\/p>\n

$$
\n\\begin{bmatrix} 3 & 6 \\\\ 6 & 14 \\end{bmatrix} \\begin{bmatrix} \\beta_0 \\\\ \\beta_1 \\end{bmatrix} = \\begin{bmatrix} 6 \\\\ 13 \\end{bmatrix}.
\n$$<\/p>\n

**\ub2e8\uacc4 3**: \uc5ed\ud589\ub82c\uc744 \uc0ac\uc6a9\ud558\uc5ec \ud47c\ub2e4:<\/p>\n

$$
\n\\det = 3 \\times 14 - 6 \\times 6 = 42 - 36 = 6,
\n$$<\/p>\n

$$
\n(\\mathbf{A}^T \\mathbf{A})^{-1} = \\frac{1}{6} \\begin{bmatrix} 14 & -6 \\\\ -6 & 3 \\end{bmatrix},
\n$$<\/p>\n

$$
\n\\hat{\\mathbf{x}} = \\frac{1}{6} \\begin{bmatrix} 14 & -6 \\\\ -6 & 3 \\end{bmatrix} \\begin{bmatrix} 6 \\\\ 13 \\end{bmatrix} = \\frac{1}{6} \\begin{bmatrix} 84 - 78 \\\\ -36 + 39 \\end{bmatrix} = \\frac{1}{6} \\begin{bmatrix} 6 \\\\ 3 \\end{bmatrix} = \\begin{bmatrix} 1 \\\\ 0.5 \\end{bmatrix}.
\n$$<\/p>\n

\ub530\ub77c\uc11c \ucd5c\uc801 \uc9c1\uc120\uc740 $\\hat{y} = 1 + 0.5x$\uc774\ub2e4.<\/p>\n

**\ub2e8\uacc4 4**: \ud22c\uc601 \uac80\uc99d:<\/p>\n

$$
\n\\hat{\\mathbf{y}} = \\mathbf{A}\\hat{\\mathbf{x}} = \\begin{bmatrix} 1 & 1 \\\\ 1 & 2 \\\\ 1 & 3 \\end{bmatrix} \\begin{bmatrix} 1 \\\\ 0.5 \\end{bmatrix} = \\begin{bmatrix} 1.5 \\\\ 2.0 \\\\ 2.5 \\end{bmatrix},
\n$$<\/p>\n

$$
\n\\mathbf{r} = \\mathbf{b} - \\hat{\\mathbf{y}} = \\begin{bmatrix} 1 \\\\ 3 \\\\ 2 \\end{bmatrix} - \\begin{bmatrix} 1.5 \\\\ 2.0 \\\\ 2.5 \\end{bmatrix} = \\begin{bmatrix} -0.5 \\\\ 1.0 \\\\ -0.5 \\end{bmatrix}.
\n$$<\/p>\n

$\\mathbf{A}^T \\mathbf{r} = \\begin{bmatrix} 1 & 1 & 1 \\\\ 1 & 2 & 3 \\end{bmatrix} \\begin{bmatrix} -0.5 \\\\ 1.0 \\\\ -0.5 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}$\ub97c \ud655\uc778\ud560 \uc218 \uc788\uc73c\uba70, \uc774\ub294 \uc794\ucc28\uac00 \uc5f4 \uacf5\uac04\uc5d0 \uc9c1\uad50\ud568\uc744 \uc758\ubbf8\ud55c\ub2e4.
\n```<\/p>\n

### 3.4 \uacf5\ud559 \ubc0f \ucd5c\ucca8\ub2e8 \uc751\uc6a9 (Engineering and Cutting-Edge Applications)<\/p>\n

\uadf8\ub9bc 2\ub294 \ucd5c\uc18c\uc81c\uacf1 \ud22c\uc601\uc758 \uae30\ud558\ud559\uc801 \uc9c1\uad00\uc744 \ubcf4\uc5ec\uc900\ub2e4. \uad00\uce21 \ubca1\ud130 $\\mathbf{b}$\ub294 $\\mathbf{A}$\uc758 \uc5f4 \uacf5\uac04(\ubaa8\ub378 \uacf5\uac04)\uc5d0 \uc18d\ud558\uc9c0 \uc54a\ub294\ub2e4. \ucd5c\uc18c\uc81c\uacf1 \ud574 $\\hat{\\mathbf{x}}$\ub294 $\\mathbf{b}$\ub97c \uc5f4 \uacf5\uac04\uc5d0 \uc9c1\uad50 \ud22c\uc601\ud558\uc5ec $\\hat{\\mathbf{y}}$\ub97c \uc5bb\uc73c\uba70, \uc774\ub294 \uc5f4 \uacf5\uac04\uc5d0\uc11c $\\mathbf{b}$\uc5d0 \uac00\uc7a5 \uac00\uae4c\uc6b4 \uadfc\uc0ac\uc774\ub2e4.<\/p>\n

<\/p>\n

\uc81c\uacf1\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc608\uce21 \ubcc0\uc218\uc640 \uc751\ub2f5 \ubcc0\uc218 \uac04\uc758 \uad00\uacc4\ub97c \ubaa8\ub378\ub9c1\ud55c\ub2e4; \uc2dc\uc2a4\ud15c \uc2dd\ubcc4(System Identification)\uc740 \ucd5c\uc18c\uc81c\uacf1\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \ub3d9\uc801 \uc2dc\uc2a4\ud15c\uc758 \ud30c\ub77c\ubbf8\ud130\ub97c \ucd94\uc815\ud55c\ub2e4; \uce7c\ub9cc \ud544\ud130(Kalman Filter)\uc758 \uce21\uc815 \uc5c5\ub370\uc774\ud2b8 \ub2e8\uacc4\ub294 \ucd5c\uc18c\uc81c\uacf1 \ubb38\uc81c\ub85c \ud574\uc11d\ub420 \uc218 \uc788\ub2e4. \ubaa8\ub4e0 \uacbd\uc6b0\uc758 \ud575\uc2ec\uc740 \ub3d9\uc77c\ud558\ub2e4: \uc815\ud655\ud55c \ud574\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc744 \ub54c, **\uc9c1\uad50 \ud22c\uc601(Orthogonal Projection)** \uc744 \ud1b5\ud574 \ucd5c\uc801\uc758 \uadfc\uc0ac\ud574\ub97c \ucc3e\ub294 \uac83\uc774\ub2e4.<\/p>\n

---<\/p>\n

## \uc81c4\uc7a5 \uc720\ud55c\ucc28\uc6d0\uc5d0\uc11c \ubb34\ud55c\ucc28\uc6d0\uc73c\ub85c \u2014 \ud568\uc218\ub97c \ubca1\ud130\ub85c \ubcf4\uae30 (Chapter 4 From Finite Dimensions to Infinite Dimensions \u2014 Functions as Vectors)<\/p>\n

### 4.1 \uc774\ub860\uacfc \uc5c4\ubc00\ud55c \uc815\uc758 (Theory and Rigorous Definitions)<\/p>\n

\uc55e \uc7a5\ub4e4\uc5d0\uc11c \ub17c\uc758\ub41c \ub0b4\uc801\uc740 \uc720\ud55c\ucc28\uc6d0 \uc720\ud074\ub9ac\ub4dc \uacf5\uac04 $\\mathbb{R}^n$\uc5d0 \uad6d\ud55c\ub418\uc5b4 \uc788\uc5c8\ub2e4. \uadf8\ub7ec\ub098 \ub0b4\uc801\uc758 \uac1c\ub150\uc740 \uc790\uc5f0\uc2a4\ub7fd\uac8c \ubb34\ud55c\ucc28\uc6d0 \ud568\uc218 \uacf5\uac04\uc73c\ub85c \ud655\uc7a5\ub420 \uc218 \uc788\ub2e4. \uc774\ub7ec\ud55c \ud655\uc7a5\uc740 \ud568\uc218\ud574\uc11d\ud559(Functional Analysis)\uc758 \ud575\uc2ec \ub0b4\uc6a9\uc774\uba70, \uc120\ud615\ub300\uc218\ud559\uacfc \uc2e0\ud638\ucc98\ub9ac, \uc591\uc790\uc5ed\ud559\uc744 \uc5f0\uacb0\ud558\ub294 \ub2e4\ub9ac \uc5ed\ud560\uc744 \ud55c\ub2e4.<\/p>\n

```ad-definition
\ntitle: \uc815\uc758 4.1 $L^2$ \ub0b4\uc801 (Definition 4.1 $L^2$ Inner Product)
\n$f, g: [a, b] \\to \\mathbb{R}$\uc774 \uc81c\uacf1 \uc801\ubd84 \uac00\ub2a5 \ud568\uc218, \uc989 $\\int_a^b [f(x)]^2 dx < \\infty$\ub77c\uace0 \ud558\uc790. \uc774\ub4e4\uc758 \ub0b4\uc801\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4:\n\n$$\n\\langle f, g \\rangle = \\int_a^b f(x) g(x) \\, dx.\n\\tag{4.1}\n$$\n\n\uc774 \ub0b4\uc801\uc774 \uc720\ub3c4\ud558\ub294 \ub178\ub984(Norm)\uc740\n\n$$\n\\|f\\| = \\sqrt{\\langle f, f \\rangle} = \\sqrt{\\int_a^b [f(x)]^2 \\, dx},\n\\tag{4.2}\n$$\n\n\uc774\uba70, $L^2$ \ub178\ub984\uc774\ub77c \ud558\uace0 \ubb3c\ub9ac\ud559\uc5d0\uc11c\ub294 \uc2e0\ud638\uc758 \"\uc5d0\ub108\uc9c0\"\ub85c \ud574\uc11d\ub41c\ub2e4.\n```\n\n```ad-definition\ntitle: \uc815\uc758 4.2 \ud790\ubca0\ub974\ud2b8 \uacf5\uac04 (Definition 4.2 Hilbert Space)\n\uc644\ube44 \ub0b4\uc801 \uacf5\uac04\uc744 **\ud790\ubca0\ub974\ud2b8 \uacf5\uac04(Hilbert Space)** \uc774\ub77c \ud55c\ub2e4$^{[6][8]}$. \uad6c\uccb4\uc801\uc73c\ub85c, \ud790\ubca0\ub974\ud2b8 \uacf5\uac04 $\\mathcal{H}$\ub294 \uc784\uc758\uc758 \ucf54\uc2dc \uc218\uc5f4(Cauchy Sequence)\uc774 $\\mathcal{H}$ \ub0b4\uc5d0\uc11c \uc218\ub834\ud558\ub294(\uc989, \uacf5\uac04\uc774 \uc644\ube44\uc778) \ub0b4\uc801 \uacf5\uac04\uc774\ub2e4.\n\n\uc720\ud55c\ucc28\uc6d0 \ub0b4\uc801 \uacf5\uac04 $\\mathbb{R}^n$\uc740 \ud790\ubca0\ub974\ud2b8 \uacf5\uac04\uc758 \ud2b9\uc218\ud55c \uc608\uc774\ub2e4. \ubb34\ud55c\ucc28\uc6d0 \uc608\ub85c\ub294 $L^2[a,b]$ (\uc81c\uacf1 \uc801\ubd84 \uac00\ub2a5 \ud568\uc218 \uacf5\uac04)\uc640 $\\ell^2$ (\uc81c\uacf1 \uac00\ud569 \uc218\uc5f4 \uacf5\uac04)\uc774 \uc788\ub2e4. \ud790\ubca0\ub974\ud2b8 \uacf5\uac04\uc758 \uc644\ube44\uc131\uc740 \ud478\ub9ac\uc5d0 \uae09\uc218(Fourier Series)\uc640 \uac19\uc740 \ubb34\ud55c \uae09\uc218 \uc804\uac1c\uc758 \uc218\ub834\uc131\uc744 \ubcf4\uc7a5\ud55c\ub2e4.\n```\n\n```ad-theorem\ntitle: \uc815\ub9ac 4.1 $L^2$ \uacf5\uac04\uc5d0\uc11c\uc758 \ucf54\uc2dc-\uc288\ubc14\ub974\uce20 \ubd80\ub4f1\uc2dd (Theorem 4.1 Cauchy-Schwarz Inequality in $L^2$ Space)\n$L^2[a,b]$\uc758 \uc784\uc758\uc758 \ud568\uc218 $f, g$\uc5d0 \ub300\ud574 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4:\n\n$$\n\\left| \\int_a^b f(x) g(x) \\, dx \\right| \\leq \\sqrt{\\int_a^b [f(x)]^2 \\, dx} \\cdot \\sqrt{\\int_a^b [g(x)]^2 \\, dx}.\n\\tag{4.3}\n$$\n```\n\n### 4.2 \uae30\ud558\ud559\uacfc \uacf5\uac04\uc801 \uc774\ubbf8\uc9c0 (Geometry and Spatial Intuition)\n\n\ud568\uc218\ub97c \ubca1\ud130\ub85c \ubcf4\ub294 \ud575\uc2ec\uc740 \"\uc810\ubcc4 \ub300\uc751\" \uac1c\ub150\uc744 \uc774\ud574\ud558\ub294 \ub370 \uc788\ub2e4. $\\mathbb{R}^n$\uc5d0\uc11c \ubca1\ud130 $\\mathbf{v} = (v_1, \\dots, v_n)$\uc758 $i$\ubc88\uc9f8 \uc131\ubd84 $v_i$\ub294 $i$\ubc88\uc9f8 \uc88c\ud45c\ucd95\uc5d0\uc11c\uc758 \uac12\uc5d0 \ud574\ub2f9\ud55c\ub2e4. \ud568\uc218 \uacf5\uac04\uc5d0\uc11c \uac01 $x \\in [a,b]$\ub294 \ud558\ub098\uc758 \ub3c5\ub9bd\uc801\uc778 \"\uc88c\ud45c\ucd95\"\uc5d0 \ub300\uc751\ud558\uba70, \ud568\uc218\uac12 $f(x)$\ub294 \uadf8 \uc88c\ud45c\ucd95\uc5d0\uc11c\uc758 \uc131\ubd84\uc774\ub2e4. \ub530\ub77c\uc11c \ud568\uc218 $f$\ub294 \ubcf8\uc9c8\uc801\uc73c\ub85c \uac00\uc0b0 \ubb34\ud55c\ud788 \ub9ce\uc740 \uc131\ubd84\uc744 \uac00\uc9c4 \ubca1\ud130\uc774\ub2e4.\n\n\ub450 \ud568\uc218\uac00 \uc9c1\uad50\ud55c\ub2e4\ub294 \uac83($\\langle f, g \\rangle = 0$)\uc740 $L^2$ \uc758\ubbf8\uc5d0\uc11c \"\uc11c\ub85c\uc758 \uc131\ubd84\uc744 \ud3ec\ud568\ud558\uc9c0 \uc54a\uc74c\"\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc774 \uac1c\ub150\uc740 \uc2e0\ud638\ucc98\ub9ac\uc5d0\uc11c \uae4a\uc740 \ubb3c\ub9ac\uc801 \uc758\ubbf8\ub97c \uac00\uc9c4\ub2e4: \uc9c1\uad50\ud558\ub294 \uc2e0\ud638\ub4e4\uc740 \ub3d9\uc77c\ud55c \ucc44\ub110\uc5d0\uc11c \uc11c\ub85c \uac04\uc12d \uc5c6\uc774 \uc804\uc1a1\ub420 \uc218 \uc788\ub2e4.\n\n### 4.3 \ud558\ub4dc\ucf54\uc5b4 \uc608\uc81c \uc0c1\uc138 \ud480\uc774 (Worked Example)\n\n```ad-example\ntitle: \uc608\uc81c 4.1 \ud568\uc218 \uacf5\uac04\uc5d0\uc11c\uc758 \uc9c1\uad50\uc131\uacfc \uac70\ub9ac \uce21\uc815 (Example 4.1 Orthogonality and Distance in Function Space)\n\uad6c\uac04 $[-1, 1]$\uc5d0\uc11c $f(x) = x$\uc640 $g(x) = x^2$\uac00 \uc8fc\uc5b4\uc84c\ub2e4. \uc774\ub4e4\uc774 \uc9c1\uad50\ud558\ub294\uc9c0 \ud310\ub2e8\ud558\uace0, \uac01\uac01\uc758 \ub178\ub984\uacfc \ud568\uc218 \uac04 \uac70\ub9ac\ub97c \uacc4\uc0b0\ud558\ub77c.\n\n**\ud480\uc774** (1) \ub0b4\uc801 \uacc4\uc0b0:\n\n$$\n\\langle f, g \\rangle = \\int_{-1}^{1} x \\cdot x^2 \\, dx = \\int_{-1}^{1} x^3 \\, dx = \\left[ \\frac{x^4}{4} \\right]_{-1}^{1} = \\frac{1}{4} - \\frac{1}{4} = 0.\n$$\n\n\ub530\ub77c\uc11c $\\langle f, g \\rangle = 0$\uc774\ubbc0\ub85c $f$\uc640 $g$\ub294 $[-1,1]$\uc5d0\uc11c \uc9c1\uad50\ud55c\ub2e4. \uadf8 \uc774\uc720\ub294 $x^3$\uc774 \uae30\ud568\uc218(Odd Function)\uc774\ubbc0\ub85c \ub300\uce6d \uad6c\uac04\uc5d0\uc11c \uc801\ubd84\uc774 0\uc774 \ub418\uae30 \ub54c\ubb38\uc774\ub2e4.\n\n(2) \ub178\ub984 \uacc4\uc0b0:\n\n$$\n\\|f\\| = \\sqrt{\\int_{-1}^{1} x^2 \\, dx} = \\sqrt{\\left[ \\frac{x^3}{3} \\right]_{-1}^{1}} = \\sqrt{\\frac{2}{3}} \\approx 0.8165,\n$$\n\n$$\n\\|g\\| = \\sqrt{\\int_{-1}^{1} x^4 \\, dx} = \\sqrt{\\left[ \\frac{x^5}{5} \\right]_{-1}^{1}} = \\sqrt{\\frac{2}{5}} \\approx 0.6325.\n$$\n\n(3) \ud568\uc218 \uac04 \uac70\ub9ac \uacc4\uc0b0:\n\n$$\n\\|f - g\\|^2 = \\int_{-1}^{1} (x - x^2)^2 \\, dx = \\int_{-1}^{1} (x^2 - 2x^3 + x^4) \\, dx = \\frac{2}{3} + 0 + \\frac{2}{5} = \\frac{16}{15},\n$$\n\n\ub530\ub77c\uc11c $d(f, g) = \\|f - g\\| = \\sqrt{16\/15} \\approx 1.0328$\uc774\ub2e4.\n\n\uc774 \uc608\uc81c\ub294 \uae30\ud568\uc218\uc640 \uc6b0\ud568\uc218(Even Function)\uac00 \ub300\uce6d \uad6c\uac04\uc5d0\uc11c \uc790\uc5f0\uc2a4\ub7fd\uac8c \uc9c1\uad50\ud568\uc744 \ubcf4\uc5ec\uc900\ub2e4. \uc774 \uc131\uc9c8\uc740 \ud478\ub9ac\uc5d0 \ubd84\uc11d(Fourier Analysis)\uc5d0\uc11c \uc0ac\uc778 \uae30\uc800\uc640 \ucf54\uc0ac\uc778 \uae30\uc800 \uc0ac\uc774\uc758 \uc9c1\uad50\uc131\uc744 \ubcf4\uc7a5\ud558\ubbc0\ub85c \ub9e4\uc6b0 \uc911\uc694\ud558\ub2e4.\n```\n\n### 4.4 \uacf5\ud559 \ubc0f \ucd5c\ucca8\ub2e8 \uc751\uc6a9 (Engineering and Cutting-Edge Applications)\n\n\ud568\uc218 \ub0b4\uc801\uc758 \uacf5\ud559\uc5d0\uc11c \uac00\uc7a5 \uc9c1\uc811\uc801\uc778 \uc751\uc6a9\uc740 **\uc815\ud569 \ud544\ud130(Matched Filter)** \uc774\ub2e4. \ub808\uc774\ub354\uc640 \ud1b5\uc2e0 \uc2dc\uc2a4\ud15c\uc5d0\uc11c \uc218\uc2e0 \uc2e0\ud638 $r(t)$\uc640 \uc1a1\uc2e0 \ud15c\ud50c\ub9bf $s(t)$\uc758 \ub0b4\uc801\n\n$$\n\\langle r, s \\rangle = \\int_{-\\infty}^{\\infty} r(t) s(t) \\, dt\n$$\n\n\uc740 \ud45c\uc801 \uc874\uc7ac \uc5ec\ubd80\ub97c \uac80\ucd9c\ud558\ub294 \ub370 \uc0ac\uc6a9\ub41c\ub2e4. \uc5d0\ucf54\uc5d0 \ud45c\uc801 \ubc18\uc0ac\uac00 \ud3ec\ud568\ub418\uc5b4 \uc788\uc73c\uba74 \ub0b4\uc801 \uac12\uc774 \ud604\uc800\ud788 \uc99d\uac00\ud55c\ub2e4. \uc774\ub294 \ubcf8\uc9c8\uc801\uc73c\ub85c \ud568\uc218 \uacf5\uac04\uc5d0\uc11c\uc758 \"\uc720\uc0ac\ub3c4 \uac80\ucd9c\"\uc774\ub2e4.\n\n\ub610\ud55c **\ucee4\ub110 \ubc29\ubc95(Kernel Methods)**$^{[22]}$\uc758 \ud575\uc2ec \uc544\uc774\ub514\uc5b4\ub294 \ub370\uc774\ud130 \ud3ec\uc778\ud2b8\ub97c \uc7ac\uc0dd \ucee4\ub110 \ud790\ubca0\ub974\ud2b8 \uacf5\uac04(RKHS)\uc73c\ub85c \ub9e4\ud551\ud558\uace0, \uadf8 \ubb34\ud55c\ucc28\uc6d0 \uacf5\uac04\uc5d0\uc11c \ub0b4\uc801\uc744 \uacc4\uc0b0\ud558\uc5ec \uace0\ucc28\uc6d0 \ud2b9\uc9d5 \ubcc0\ud658\uc744 \uc554\uc2dc\uc801\uc73c\ub85c \uc218\ud589\ud558\ub294 \uac83\uc774\ub2e4. \uc774\uc5d0 \ub300\ud574\uc11c\ub294 \uc81c12\uc7a5\uc5d0\uc11c \uc790\uc138\ud788 \ub2e4\ub8ec\ub2e4.\n\n---\n\n## \uc81c5\uc7a5 \uc0bc\uac01\ud568\uc218\uc758 \uc9c1\uad50\uc131 \u2014 \uc8fc\ud30c\uc218 \uc601\uc5ed\uc758 \uae30\uc800 \ud568\uc218 (Chapter 5 Orthogonality of Trigonometric Functions \u2014 Basis Functions in the Frequency Domain)\n\n### 5.1 \uc774\ub860\uacfc \uc5c4\ubc00\ud55c \uc815\uc758 (Theory and Rigorous Definitions)\n\n\ud790\ubca0\ub974\ud2b8 \uacf5\uac04 $L^2[-\\pi, \\pi]$\uc5d0\uc11c \uc0bc\uac01\ud568\uc218 \uacc4\uc5f4\uc740 \uc911\uc694\ud55c \uc9c1\uad50 \uae30\uc800\ub97c \uad6c\uc131\ud55c\ub2e4. \ub2e4\uc74c \ud568\uc218 \uc9d1\ud569\uc744 \uace0\ub824\ud558\uc790:\n\n$$\n\\{1,\\ \\sin x,\\ \\cos x,\\ \\sin 2x,\\ \\cos 2x,\\ \\dots,\\ \\sin nx,\\ \\cos nx,\\ \\dots\\}.\n$$\n\n```ad-theorem\ntitle: \uc815\ub9ac 5.1 \uc0bc\uac01\ud568\uc218\uc758 \uc9c1\uad50\uc131 (Theorem 5.1 Orthogonality of Trigonometric Functions)\n\uad6c\uac04 $[-\\pi, \\pi]$\uc5d0\uc11c \uc0bc\uac01\ud568\uc218 \uacc4\uc5f4\uc740 \ub2e4\uc74c \uc9c1\uad50 \uad00\uacc4\ub97c \ub9cc\uc871\ud55c\ub2e4$^{[4]}$:\n\n$$\n\\int_{-\\pi}^{\\pi} \\sin(mx) \\cos(nx) \\, dx = 0, \\quad \\forall m, n,\n\\tag{5.1}\n$$\n\n$$\n\\int_{-\\pi}^{\\pi} \\sin(mx) \\sin(nx) \\, dx = 0, \\quad m \\neq n,\n\\tag{5.2}\n$$\n\n$$\n\\int_{-\\pi}^{\\pi} \\cos(mx) \\cos(nx) \\, dx = 0, \\quad m \\neq n.\n\\tag{5.3}\n$$\n\n\ub3d9\uc77c \uc8fc\ud30c\uc218\uc758 \uc790\uae30 \ub0b4\uc801(Self Inner Product)\uc740 0\uc774 \uc544\ub2c8\ub2e4:\n\n$$\n\\int_{-\\pi}^{\\pi} \\sin^2(nx) \\, dx = \\pi, \\quad\n\\int_{-\\pi}^{\\pi} \\cos^2(nx) \\, dx = \\pi.\n\\tag{5.4}\n$$\n\n**\uc99d\uba85** \uc774 \uad00\uacc4\ub4e4\uc740 \uc0bc\uac01\ud568\uc218\uc758 \uacf1\uc744 \ud569\uc73c\ub85c \ubc14\uafb8\ub294 \uacf5\uc2dd(Product-to-Sum Formulas)\uc5d0\uc11c \uc9c1\uc811 \uc720\ub3c4\ub41c\ub2e4. \uc608\ub97c \ub4e4\uc5b4, (5.2)\uc758 \uacbd\uc6b0:\n\n$$\n\\sin(mx)\\sin(nx) = \\frac{1}{2}[\\cos((m-n)x) - \\cos((m+n)x)].\n$$\n\n$m \\neq n$\uc77c \ub54c $\\cos((m-n)x)$\uc640 $\\cos((m+n)x)$\uc758 $[-\\pi, \\pi]$\uc5d0\uc11c\uc758 \uc801\ubd84\uc740 \ubaa8\ub450 0\uc774\ub2e4. $\\square$\n```\n\n### 5.2 \uae30\ud558\ud559\uacfc \uacf5\uac04\uc801 \uc774\ubbf8\uc9c0 (Geometry and Spatial Intuition)\n\n\uc0bc\uac01\ud568\uc218 \uc9c1\uad50\uc131\uc758 \uae30\ud558\ud559\uc801 \uc758\ubbf8\ub294: \uc11c\ub85c \ub2e4\ub978 \uc8fc\ud30c\uc218\uc758 \uc0ac\uc778\ud30c\uc640 \ucf54\uc0ac\uc778\ud30c\uac00 $L^2$ \uacf5\uac04\uc5d0\uc11c \uc11c\ub85c \uc218\uc9c1\uc774\ub77c\ub294 \uac83\uc774\ub2e4. \uc774\ub294 \"\uc2e0\ud638\"\ub85c\uc11c \uc11c\ub85c \uac04\uc12d\ud558\uc9c0 \uc54a\uc74c\uc744 \uc758\ubbf8\ud55c\ub2e4 \u2014 \uc774\uac83\uc774 \ubc14\ub85c **\uc8fc\ud30c\uc218 \ubd84\ud560 \ub2e4\uc911\ud654(Frequency Division Multiplexing)** \uc758 \uc218\ud559\uc801 \uae30\ucd08\uc774\ub2e4.\n\n\ud1b5\uc2e0 \uc2dc\uc2a4\ud15c\uc5d0\uc11c \uc11c\ub85c \ub2e4\ub978 \uc0ac\uc6a9\uc790\uc758 \ub370\uc774\ud130\ub294 \uc0c1\ud638 \uc9c1\uad50\ud558\ub294 \ubc18\uc1a1\ud30c\uc5d0 \ubcc0\uc870\ub418\uc5b4 \ub3d9\uc2dc\uc5d0 \uc804\uc1a1\ub420 \uc218 \uc788\uc73c\uba70, \uc218\uc2e0\ub2e8\uc740 \ub0b4\uc801 \uc5f0\uc0b0\uc744 \ud1b5\ud574 \uac01 \uc2e0\ud638\ub97c \ubd84\ub9ac\ud560 \uc218 \uc788\ub2e4. \uc774\ub294 \uc2dc\uac04 \uc601\uc5ed\uc5d0\uc11c \uc644\uc804\ud788 \uc911\ucca9\ub418\uc5b4 \uc788\ub354\ub77c\ub3c4 \uac00\ub2a5\ud558\ub2e4. \uc774 \uc6d0\ub9ac\ub294 \ud604\ub300 \ubb34\uc120 \ud1b5\uc2e0\uc758 **\uc8fc\ud30c\uc218 \uc601\uc5ed(Frequency Domain)**$^{[16]}$ \ubd84\uc11d\uc5d0\uc11c \ud575\uc2ec\uc801\uc778 \uc704\uce58\ub97c \ucc28\uc9c0\ud55c\ub2e4.\n\n### 5.3 \ud558\ub4dc\ucf54\uc5b4 \uc608\uc81c \uc0c1\uc138 \ud480\uc774 (Worked Example)\n\n```ad-example\ntitle: \uc608\uc81c 5.1 \uc0bc\uac01\ud568\uc218 \uc9c1\uad50\uc131\uc758 \uc218\ub3d9 \uac80\uc99d (Example 5.1 Manual Verification of Trigonometric Orthogonality)\n$[-\\pi, \\pi]$\uc5d0\uc11c \ub2e4\uc74c \uc138 \uac00\uc9c0 \ub0b4\uc801\uc744 \uac80\uc99d\ud558\ub77c.\n\n**\uacbd\uc6b0 A: $\\langle \\sin(2x), \\cos(3x) \\rangle$**\n\n$$\n\\langle \\sin(2x), \\cos(3x) \\rangle = \\int_{-\\pi}^{\\pi} \\sin(2x)\\cos(3x) \\, dx.\n$$\n\n\uacf1\uc744 \ud569\uc73c\ub85c \ubc14\uafb8\ub294 \uacf5\uc2dd $\\sin\\alpha\\cos\\beta = \\frac{1}{2}[\\sin(\\alpha+\\beta) + \\sin(\\alpha-\\beta)]$\uc744 \uc0ac\uc6a9\ud558\uba74:\n\n$$\n\\sin(2x)\\cos(3x) = \\frac{1}{2}[\\sin(5x) + \\sin(-x)] = \\frac{1}{2}[\\sin(5x) - \\sin(x)].\n$$\n\n\uc784\uc758\uc758 \uc815\uc218 $k$\uc5d0 \ub300\ud574 $\\int_{-\\pi}^{\\pi} \\sin(kx) \\, dx = 0$\uc774\ubbc0\ub85c,\n\n$$\n\\langle \\sin(2x), \\cos(3x) \\rangle = \\frac{1}{2} \\times 0 - \\frac{1}{2} \\times 0 = 0.\n$$\n\n**\uacbd\uc6b0 B: $\\langle \\sin(2x), \\sin(3x) \\rangle$**\n\n$\\sin\\alpha\\sin\\beta = \\frac{1}{2}[\\cos(\\alpha-\\beta) - \\cos(\\alpha+\\beta)]$\uc744 \uc0ac\uc6a9\ud558\uba74:\n\n$$\n\\sin(2x)\\sin(3x) = \\frac{1}{2}[\\cos(-x) - \\cos(5x)] = \\frac{1}{2}[\\cos(x) - \\cos(5x)].\n$$\n\n$k \\neq 0$\uc5d0 \ub300\ud574 $\\int_{-\\pi}^{\\pi} \\cos(kx) \\, dx = 0$\uc774\ubbc0\ub85c,\n\n$$\n\\langle \\sin(2x), \\sin(3x) \\rangle = \\frac{1}{2} \\times 0 - \\frac{1}{2} \\times 0 = 0.\n$$\n\n**\uacbd\uc6b0 C: $\\langle \\sin(2x), \\sin(2x) \\rangle$ (\uc790\uae30 \ub0b4\uc801)**\n\n\ubc30\uac01 \uacf5\uc2dd $\\sin^2\\theta = (1 - \\cos 2\\theta)\/2$\uc744 \uc774\uc6a9\ud558\uba74:\n\n$$\n\\langle \\sin(2x), \\sin(2x) \\rangle = \\int_{-\\pi}^{\\pi} \\frac{1 - \\cos(4x)}{2} \\, dx = \\frac{1}{2} \\cdot 2\\pi - 0 = \\pi.\n$$\n\n\uc774 \uacb0\uacfc\ub294 $\\|\\sin(2x)\\| = \\sqrt{\\pi}$\uc784\uc744 \uc758\ubbf8\ud558\uba70, \uc774\uac83\uc774 \ud478\ub9ac\uc5d0 \uae09\uc218(Fourier Series)\uc758 \uacc4\uc218 \ubd84\ubaa8\uc5d0 $\\pi$\uac00 \ub098\ud0c0\ub098\ub294 \uc774\uc720\uc774\ub2e4.\n```\n\n### 5.4 \uacf5\ud559 \ubc0f \ucd5c\ucca8\ub2e8 \uc751\uc6a9 (Engineering and Cutting-Edge Applications)\n\n**\uc9c1\uad50 \uc8fc\ud30c\uc218 \ubd84\ud560 \ub2e4\uc911\ud654(OFDM, Orthogonal Frequency Division Multiplexing)** \ub294 \ud604\ub300 4G\/5G \ubb34\uc120 \ud1b5\uc2e0\uc758 \ud575\uc2ec \uae30\uc220\uc774\ub2e4$^{[16]}$. \uace0\uc18d \ub370\uc774\ud130 \uc2a4\ud2b8\ub9bc\uc744 \uc5ec\ub7ec \uc800\uc18d \ubd80\uc2a4\ud2b8\ub9bc\uc73c\ub85c \ubd84\ud560\ud558\uace0, \uac01\uac01\uc744 \uc0c1\ud638 \uc9c1\uad50\ud558\ub294 \ubd80\ubc18\uc1a1\ud30c\uc5d0 \ubcc0\uc870\ud558\uc5ec \ubcd1\ub82c \uc804\uc1a1\ud55c\ub2e4. \ubd80\ubc18\uc1a1\ud30c \uac04\uc758 \uc9c1\uad50\uc131\n\n$$\n\\int_0^T \\sin(2\\pi f_k t) \\cdot \\sin(2\\pi f_l t) \\, dt = 0, \\quad k \\neq l,\n$$\n\n\ub355\ubd84\uc5d0 \uc218\uc2e0\ub2e8\uc740 \ub0b4\uc801 \uc5f0\uc0b0\uc744 \ud1b5\ud574 \uac01 \ubd80\ubc18\uc1a1\ud30c \uc2e0\ud638\ub97c \uc644\ubcbd\ud558\uac8c \ubd84\ub9ac\ud560 \uc218 \uc788\uc73c\uba70, \uc2a4\ud399\ud2b8\ub7fc \uc0c1\uc5d0\uc11c \uc2ec\ud558\uac8c \uc911\ucca9\ub418\uc5b4 \uc788\ub354\ub77c\ub3c4 \uac00\ub2a5\ud558\ub2e4. \uc774\ub294 \uc8fc\ud30c\uc218 \ud6a8\uc728\uc131\uc744 \ud06c\uac8c \ud5a5\uc0c1\uc2dc\ud0a8\ub2e4.\n\n---\n\n## \uc81c6\uc7a5 \ud478\ub9ac\uc5d0 \uae09\uc218\uc640 \ud478\ub9ac\uc5d0 \ubcc0\ud658 \u2014 \uc0bc\uac01 \uae30\uc800 \uc704\ub85c\uc758 \ud568\uc218 \ud22c\uc601 (Chapter 6 Fourier Series and Fourier Transform \u2014 Projection of Functions onto Trigonometric Bases)\n\n### 6.1 \uc774\ub860\uacfc \uc5c4\ubc00\ud55c \uc815\uc758 (Theory and Rigorous Definitions)\n\n\uc0bc\uac01\ud568\uc218 \uacc4\uc5f4\uc758 \uc9c1\uad50\uc131 \ub355\ubd84\uc5d0 \uc784\uc758\uc758 \uc8fc\uae30 \ud568\uc218\ub97c \uc11c\ub85c \ub2e4\ub978 \uc8fc\ud30c\uc218\uc758 \uc0bc\uac01\ud568\uc218\ub4e4\uc758 \uc120\ud615 \uacb0\ud569\uc73c\ub85c \ubd84\ud574\ud560 \uc218 \uc788\ub2e4. \uc774 \ubd84\ud574\ub97c **\ud478\ub9ac\uc5d0 \uae09\uc218(Fourier Series)**$^{[11]}$\ub77c \ud55c\ub2e4.\n\n```ad-theorem\ntitle: \uc815\ub9ac 6.1 \ud478\ub9ac\uc5d0 \uae09\uc218 (Theorem 6.1 Fourier Series)\n$f(t)$\uac00 $2\\pi$\ub97c \uc8fc\uae30\ub85c \ud558\ub294 \uc81c\uacf1 \uc801\ubd84 \uac00\ub2a5 \ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8 \ud478\ub9ac\uc5d0 \uae09\uc218 \uc804\uac1c\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4:\n\n$$\nf(t) = \\frac{a_0}{2} + \\sum_{n=1}^{\\infty} [a_n \\cos(nt) + b_n \\sin(nt)],\n\\tag{6.1}\n$$\n\n\uc5ec\uae30\uc11c \uacc4\uc218\ub294 \ub0b4\uc801\uc73c\ub85c \uc8fc\uc5b4\uc9c4\ub2e4:\n\n$$\na_0 = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(t) \\, dt,\n\\tag{6.2}\n$$\n\n$$\na_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(t) \\cos(nt) \\, dt = \\frac{\\langle f, \\cos(nt) \\rangle}{\\|\\cos(nt)\\|^2},\n\\tag{6.3}\n$$\n\n$$\nb_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(t) \\sin(nt) \\, dt = \\frac{\\langle f, \\sin(nt) \\rangle}{\\|\\sin(nt)\\|^2}.\n\\tag{6.4}\n$$\n\n\uc2dd (6.3)-(6.4)\ub294 \ud478\ub9ac\uc5d0 \uacc4\uc218\uc758 \ubcf8\uc9c8\uc744 \ub4dc\ub7ec\ub0b8\ub2e4: \uadf8\uac83\ub4e4\uc740 \uac01 \uc0bc\uac01 \uae30\uc800 \uc704\ub85c\uc758 \ud568\uc218 $f$\uc758 \ud22c\uc601 \uacc4\uc218(\ub0b4\uc801\uc744 \uae30\uc800\uc758 \ub178\ub984 \uc81c\uacf1\uc73c\ub85c \ub098\ub208 \uac12)\uc774\uba70, \uc720\ud55c\ucc28\uc6d0 \ubca1\ud130\ub97c \uc9c1\uad50 \uae30\uc800 \uc704\uc5d0\uc11c \uc88c\ud45c\ub97c \uacc4\uc0b0\ud558\ub294 \uac83\uacfc \uc644\uc804\ud788 \ub3d9\uc77c\ud558\ub2e4.\n\n\uc8fc\uae30 $T \\to \\infty$\uc77c \ub54c, \ud478\ub9ac\uc5d0 \uae09\uc218\ub294 **\ud478\ub9ac\uc5d0 \ubcc0\ud658(Fourier Transform)**$^{[12]}$\uc73c\ub85c \uc774\ud589\ud55c\ub2e4:\n\n$$\nX(f) = \\int_{-\\infty}^{\\infty} x(t) e^{-j2\\pi ft} \\, dt = \\langle x(t), e^{j2\\pi ft} \\rangle.\n\\tag{6.5}\n$$\n\n\ud478\ub9ac\uc5d0 \ubcc0\ud658\uc740 \uc2dc\uac04 \uc601\uc5ed \ud568\uc218 $x(t)$\ub97c \ubcf5\uc18c \uc9c0\uc218 \uae30\uc800 $e^{j2\\pi ft}$\uc5d0 \ud22c\uc601\ud558\uc5ec \uc8fc\ud30c\uc218 \uc601\uc5ed \ud45c\ud604 $X(f)$\ub97c \uc5bb\ub294\ub2e4.\n```\n\n### 6.2 \uae30\ud558\ud559\uacfc \uacf5\uac04\uc801 \uc774\ubbf8\uc9c0 (Geometry and Spatial Intuition)\n\n\ud478\ub9ac\uc5d0 \ubcc0\ud658\uc758 \uae30\ud558\ud559\uc801 \ubcf8\uc9c8\uc740 \"\ud0d0\uce68(Probe)\" \uc0ac\uc0c1\uc774\ub2e4: \uc11c\ub85c \ub2e4\ub978 \uc8fc\ud30c\uc218\uc758 \ubcf5\uc18c \uc9c0\uc218 \uc9c4\ub3d9\uc744 \ud0d0\uce68\uc73c\ub85c \uc0ac\uc6a9\ud558\uc5ec \ubd84\uc11d \ub300\uc0c1 \uc2e0\ud638\uc640 \ub0b4\uc801\uc744 \ucde8\ud55c\ub2e4. \uc2e0\ud638\uc5d0 \ud2b9\uc815 \uc8fc\ud30c\uc218 \uc131\ubd84\uc774 \ud3ec\ud568\ub418\uc5b4 \uc788\uc73c\uba74 \ub0b4\uc801 \uac12\uc774 \ud06c\uace0(\uc2a4\ud399\ud2b8\ub7fc \ud53c\ud06c \ubc1c\uc0dd), \ud3ec\ud568\ub418\uc5b4 \uc788\uc9c0 \uc54a\uc73c\uba74 \ub0b4\uc801 \uac12\uc774 0\uc5d0 \uac00\uae5d\ub2e4. \uc2a4\ud399\ud2b8\ub7fc \uadf8\ub798\ud504\uc758 \uac01 \ud53c\ud06c\ub294 \ud574\ub2f9 \uc8fc\ud30c\uc218 \uae30\uc800 \uc704\ub85c\uc758 \uc2e0\ud638 \ud22c\uc601 \uac15\ub3c4\uc5d0 \ub300\uc751\ud55c\ub2e4.\n\n### 6.3 \ud558\ub4dc\ucf54\uc5b4 \uc608\uc81c \uc0c1\uc138 \ud480\uc774 (Worked Example)\n\n```ad-example\ntitle: \uc608\uc81c 6.1 \uc8fc\uae30 \uad6c\ud615\ud30c\uc758 \ud478\ub9ac\uc5d0 \uae09\uc218 \uc804\uac1c (Example 6.1 Fourier Series Expansion of a Periodic Square Wave)\n\uc8fc\uae30\uac00 $2\\pi$\uc778 \uad6c\ud615\ud30c\n\n$$\nf(t) = \\begin{cases}\n1, & 0 < t < \\pi, \\\\\n-1, & -\\pi < t < 0,\n\\end{cases}\n$$\n\n\uc758 \ud478\ub9ac\uc5d0 \uae09\uc218 \uacc4\uc218\ub97c \uad6c\ud558\ub77c.\n\n**\ud480\uc774** $f(t)$\ub294 \uae30\ud568\uc218\uc774\ubbc0\ub85c $a_0 = a_n = 0$\uc774\ub2e4(\ucf54\uc0ac\uc778 \uacc4\uc218\uac00 \ubaa8\ub450 0). $b_n$\ub9cc \uacc4\uc0b0\ud558\uba74 \ub41c\ub2e4.\n\n$$\nb_n = \\frac{1}{\\pi} \\int_{-\\pi}^{\\pi} f(t) \\sin(nt) \\, dt = \\frac{1}{\\pi} \\left( \\int_{-\\pi}^{0} (-\\sin(nt)) \\, dt + \\int_{0}^{\\pi} \\sin(nt) \\, dt \\right).\n$$\n\n\uccab \ubc88\uc9f8 \ud56d \uacc4\uc0b0: $\\int_{-\\pi}^{0} -\\sin(nt) \\, dt = \\left[ \\frac{\\cos(nt)}{n} \\right]_{-\\pi}^{0} = \\frac{1}{n} - \\frac{\\cos(-n\\pi)}{n} = \\frac{1 - (-1)^n}{n}$.\n\n\ub450 \ubc88\uc9f8 \ud56d \uacc4\uc0b0: $\\int_{0}^{\\pi} \\sin(nt) \\, dt = \\left[ -\\frac{\\cos(nt)}{n} \\right]_{0}^{\\pi} = -\\frac{\\cos(n\\pi)}{n} + \\frac{1}{n} = \\frac{1 - (-1)^n}{n}$.\n\n\ub530\ub77c\uc11c:\n\n$$\nb_n = \\frac{1}{\\pi} \\cdot \\frac{2[1 - (-1)^n]}{n} = \\begin{cases}\n\\dfrac{4}{n\\pi}, & n \\text{\uc774 \ud640\uc218}, \\\\[6pt]\n0, & n \\text{\uc774 \uc9dd\uc218}.\n\\end{cases}\n\\tag{6.6}\n$$\n\n\ub530\ub77c\uc11c \uad6c\ud615\ud30c\uc758 \ud478\ub9ac\uc5d0 \uae09\uc218 \uc804\uac1c\ub294\n\n$$\nf(t) = \\frac{4}{\\pi} \\sum_{k=0}^{\\infty} \\frac{\\sin((2k+1)t)}{2k+1} = \\frac{4}{\\pi} \\left( \\sin t + \\frac{1}{3}\\sin 3t + \\frac{1}{5}\\sin 5t + \\cdots \\right).\n\\tag{6.7}\n$$\n\n\uc218\uce58 \uac80\uc99d: $t = \\pi\/2$\uc5d0\uc11c \ucc98\uc74c 3\ud56d\uc758 \uadfc\uc0ac\ub294\n\n$$\nf(\\pi\/2) \\approx \\frac{4}{\\pi} \\left( 1 - \\frac{1}{3} + \\frac{1}{5} \\right) = \\frac{52}{15\\pi} \\approx 1.103,\n$$\n\n\uc73c\ub85c \ucc38\uac12 $1$\uc5d0 \uac00\uae5d\ub2e4. \ub354 \ub9ce\uc740 \ud56d\uc744 \ucd94\uac00\ud558\uba74 \uad6c\ud615\ud30c\uc5d0 \uc218\ub834\ud55c\ub2e4(\uae41\uc2a4 \ud604\uc0c1(Gibbs Phenomenon)\uc73c\ub85c \ubd88\uc5f0\uc18d\uc810\uc5d0\uc11c \uc57d $9\\%$\uc758 \uc624\ubc84\uc288\ud2b8 \ubc1c\uc0dd).\n```\n\n### 6.4 \uacf5\ud559 \ubc0f \ucd5c\ucca8\ub2e8 \uc751\uc6a9 (Engineering and Cutting-Edge Applications)\n\n\uadf8\ub9bc 3\uc740 \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc758 \uc804\ud615\uc801\uc778 \uc751\uc6a9\uc744 \ubcf4\uc5ec\uc900\ub2e4. 50 Hz, 120 Hz, 260 Hz\uc758 \uc138 \uac00\uc9c0 \uc8fc\ud30c\uc218 \uc131\ubd84\uc744 \ud3ec\ud568\ud558\ub294 \uc7a1\uc74c\uc774 \uc11e\uc778 \uc2e0\ud638 $x(t)$\uc758 \uc2dc\uac04 \uc601\uc5ed \ud30c\ud615\uc740 \ubb34\uc9c8\uc11c\ud574 \ubcf4\uc778\ub2e4. \ud478\ub9ac\uc5d0 \ubcc0\ud658 \ud6c4 \uc2a4\ud399\ud2b8\ub7fc \uadf8\ub798\ud504\ub294 \ud574\ub2f9 \uc8fc\ud30c\uc218\uc5d0\uc11c \uc138 \uac1c\uc758 \ud53c\ud06c\ub97c \uc120\uba85\ud558\uac8c \ub098\ud0c0\ub0b8\ub2e4 \u2014 \uc774\uac83\uc774 \ubc14\ub85c \uac01 \uc8fc\ud30c\uc218 \uae30\uc800 \uc704\ub85c\uc758 \uc2e0\ud638 \ud22c\uc601 \uac15\ub3c4\uc774\ub2e4.\n\n<\/p>\n

**\uadf8\ub9bc 3: \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc758 \uc8fc\ud30c\uc218 \uc601\uc5ed \ud22c\uc601(Figure 3: Frequency-Domain Projection of Fourier Transform).** \uc704\ucabd\uc740 \uc7a1\uc74c\uc774 \uc11e\uc778 \ub2e4\uc911 \ud1a4 \uc2e0\ud638 $x(t) = 1.2\\sin(2\\pi\\cdot 50t) + 0.7\\sin(2\\pi\\cdot 120t) + 0.4\\sin(2\\pi\\cdot 260t) + \\eta(t)$\uc758 \uc2dc\uac04 \uc601\uc5ed \ud30c\ud615; \uc544\ub798\ucabd\uc740 \uc9c4\ud3ed \uc2a4\ud399\ud2b8\ub7fc\uc73c\ub85c, 50, 120, 260 Hz\uc5d0\uc11c \ub69c\ub837\ud55c \ud53c\ud06c\uac00 \ub098\ud0c0\ub09c\ub2e4. \uc774 \uadf8\ub9bc\uc740 main.py<\/a>\uc758 `np.fft.rfft`(\uc774\uc0b0 \ud478\ub9ac\uc5d0 \ubcc0\ud658)\ub85c \uc0dd\uc131\ub418\uc5c8\uc73c\uba70, \uadf8 \ubcf8\uc9c8\uc740 \uc2dc\uac04 \uc601\uc5ed \uc0d8\ud50c\ub9c1 \ubca1\ud130\uc640 \ubcf5\uc18c \uc9c0\uc218 \uae30\uc800 \ubca1\ud130\uc758 \ub0b4\uc801\uc744 \uacc4\uc0b0\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n

\ud478\ub9ac\uc5d0 \ubd84\uc11d\uc758 \uc751\uc6a9\uc740 \uacf5\ud559\uc758 \ubaa8\ub4e0 \ubd84\uc57c\uc5d0 \uac78\uccd0 \uc788\ub2e4: MP3 \uc624\ub514\uc624 \uc555\ucd95\uc740 \uc0ac\ub78c\uc758 \uadc0\uc5d0 \ubbfc\uac10\ud558\uc9c0 \uc54a\uc740 \uace0\uc8fc\ud30c \uc131\ubd84\uc744 \ubc84\ub9bc\uc73c\ub85c\uc368 \ub370\uc774\ud130 \uc591\uc744 \uc904\uc778\ub2e4; JPEG \uc774\ubbf8\uc9c0 \uc555\ucd95\uc740 \uc774\uc0b0 \ucf54\uc0ac\uc778 \ubcc0\ud658(DCT)$^{[18]}$\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc774\ubbf8\uc9c0 \ube14\ub85d\uc744 \uc8fc\ud30c\uc218 \uae30\uc800\uc5d0 \ud22c\uc601\ud55c\ub2e4; \uc2ec\uc804\ub3c4(ECG) \uc2e0\ud638\uc758 \uc8fc\ud30c\uc218 \uc601\uc5ed \uc9c4\ub2e8\uc740 \uc2a4\ud399\ud2b8\ub7fc \ud2b9\uc9d5\uc744 \uc774\uc6a9\ud558\uc5ec \ubcd1\ub9ac \ud328\ud134\uc744 \uc2dd\ubcc4\ud55c\ub2e4.<\/p>\n

---<\/p>\n

## \uc81c7\uc7a5 \uc8fc\ud30c\uc218 \uc601\uc5ed\uc5d0\uc11c \ubcf5\uc18c \uc8fc\ud30c\uc218 \uc601\uc5ed\uc73c\ub85c \u2014 \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\uacfc Z \ubcc0\ud658 (Chapter 7 From Frequency Domain to Complex Frequency Domain \u2014 Laplace and Z Transforms)<\/p>\n

### 7.1 \uc774\ub860\uacfc \uc5c4\ubc00\ud55c \uc815\uc758 (Theory and Rigorous Definitions)<\/p>\n

\ud478\ub9ac\uc5d0 \ubcc0\ud658\uc740 \uc2e0\ud638\uac00 \uc808\ub300 \uc801\ubd84 \uac00\ub2a5 \uc870\uac74 $\\int_{-\\infty}^{\\infty} |f(t)|\\,dt < \\infty$\uc744 \ub9cc\uc871\ud560 \uac83\uc744 \uc694\uad6c\ud55c\ub2e4. $f(t) = e^{2t}$($t \\geq 0$)\uc640 \uac19\uc740 \uc9c0\uc218 \ubc1c\uc0b0 \uc2e0\ud638\uc758 \uacbd\uc6b0, \uc5d0\ub108\uc9c0\uac00 $t$\uc5d0 \ub530\ub77c \ubc1c\uc0b0\ud558\ubbc0\ub85c \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc758 \ub0b4\uc801 $\\langle f(t), e^{-j\\omega t} \\rangle$\uc740 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4. \uc774 \ubb38\uc81c\ub97c \ud574\uacb0\ud558\uae30 \uc704\ud574 \ud0d0\uce68 \uae30\uc800\ub97c \uc21c\ud5c8\uc218 \uc9c0\uc218 $e^{-j\\omega t}$\uc5d0\uc11c \uc2e4\uc218\ubd80 \uac10\uc1e0 \uc778\uc790\ub97c \uac00\uc9c4 \ubcf5\uc18c \uc9c0\uc218 $e^{-st}$\ub85c \ud655\uc7a5\ud574\uc57c \ud55c\ub2e4. \uc5ec\uae30\uc11c $s = \\sigma + j\\omega$\uc774\ub2e4.\n\n```ad-definition\ntitle: \uc815\uc758 7.1 \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658 (Definition 7.1 Laplace Transform)\n$f(t)$\uac00 $[0, \\infty)$\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\ub77c\uace0 \ud560 \ub54c, \uadf8 **\ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658(Laplace Transform)** \uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4$^{[14]}$:\n\n$$F(s) = \\mathcal{L}\\{f(t)\\} = \\int_0^{\\infty} f(t) e^{-st}\\,dt, \\quad s = \\sigma + j\\omega \\in \\mathbb{C} \\tag{7.1}$$\n\n$s$\uc758 \uc2e4\uc218\ubd80 $\\sigma$\uac00 \ucda9\ubd84\ud788 \ud074 \ub54c, \uac10\uc1e0 \uc778\uc790 $e^{-\\sigma t}$\ub294 $f(t)$\uc758 \ubc1c\uc0b0 \uacbd\ud5a5\uc744 \uc5b5\uc81c\ud558\uc5ec \uc801\ubd84\uc774 \uc218\ub834\ud558\uac8c \ud55c\ub2e4. (7.1)\uc774 \uc218\ub834\ud558\ub294 $s$ \uac12\uc758 \uc9d1\ud569\uc744 **\uc218\ub834 \uc601\uc5ed(Region of Convergence, ROC)** \uc774\ub77c \ud55c\ub2e4.\n```\n\n```ad-definition\ntitle: \uc815\uc758 7.2 Z \ubcc0\ud658 (Definition 7.2 Z-Transform)\n$x[n]$\uc774 $\\mathbb{Z}$\uc5d0\uc11c \uc815\uc758\ub41c \uc774\uc0b0 \uc218\uc5f4\uc774\ub77c\uace0 \ud560 \ub54c, \uadf8 **Z \ubcc0\ud658(Z-Transform)** \uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4$^{[15]}$:\n\n$$X(z) = \\mathcal{Z}\\{x[n]\\} = \\sum_{n=-\\infty}^{\\infty} x[n] z^{-n}, \\quad z = re^{j\\omega} \\in \\mathbb{C} \\tag{7.2}$$\n\nZ \ubcc0\ud658\uc740 \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\uc758 \uc774\uc0b0 \uc601\uc5ed \ub300\uc751\ubb3c\ub85c \ubcfc \uc218 \uc788\ub2e4: $z = e^{sT}$($T$\ub294 \uc0d8\ud50c\ub9c1 \uc8fc\uae30)\ub77c \ud558\uba74, $z$ \ud3c9\uba74\uc758 \ub2e8\uc704\uc6d0 $|z| = 1$\uc740 $s$ \ud3c9\uba74\uc758 \ud5c8\uc218\ucd95 $s = j\\omega$\uc5d0 \ub300\uc751\ud55c\ub2e4.\n\n\ub0b4\uc801 \uad00\uc810\uc5d0\uc11c \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\uacfc Z \ubcc0\ud658\uc740 \ubaa8\ub450 \uc2e0\ud638\uc640 \ubcf5\uc18c \uc9c0\uc218 \uae30\uc800 \ud568\uc218\uc758 \ub0b4\uc801\uc73c\ub85c \uc774\ud574\ud560 \uc218 \uc788\ub2e4:\n\n$$\\mathcal{L}\\{f(t)\\} = \\langle f(t), e^{st} \\rangle, \\quad \\mathcal{Z}\\{x[n]\\} = \\langle x[n], z^n \\rangle$$\n\n\uc5ec\uae30\uc11c \uae30\uc800 \ud568\uc218 $e^{st}$\uc640 $z^n$\uc740 \uc9c4\ud3ed \uac10\uc1e0($\\sigma$ \ub610\ub294 $r$\uc744 \ud1b5\ud574)\uc640 \uc704\uc0c1 \ud68c\uc804($\\omega$\ub97c \ud1b5\ud574)\uc758 \ub450 \uc790\uc720\ub3c4\ub97c \ud3ec\ud568\ud558\ubbc0\ub85c, \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc758 \uae30\uc800 \ud568\uc218\ubcf4\ub2e4 \ub354 \ud45c\ud604\ub825\uc774 \ub6f0\uc5b4\ub098\ub2e4.\n```\n\n### 7.2 \uae30\ud558\ud559\uacfc \uacf5\uac04\uc801 \uc774\ubbf8\uc9c0 (Geometry and Spatial Intuition)\n\n\ud478\ub9ac\uc5d0 \ubcc0\ud658\uc758 \uae30\uc800 $e^{-j\\omega t}$\ub294 \ubcf5\uc18c \ud3c9\uba74\uc758 \ub2e8\uc704\uc6d0 \uc704\ub97c \ub4f1\uc18d \ud68c\uc804\ud558\ub294 \ubca1\ud130\uc774\uba70, \uadf8 \ud06c\uae30\ub294 \ud56d\uc0c1 1\uc774\ub2e4. \ubc1c\uc0b0 \uc2e0\ud638 $e^{2t}$\uc758 \uacbd\uc6b0, \ud53c\uc801\ubd84 \ud568\uc218 $|e^{2t} \\cdot e^{-j\\omega t}| = e^{2t}$\uac00 $t$\uc5d0 \ub530\ub77c \ubc1c\uc0b0\ud558\ubbc0\ub85c \uc801\ubd84\uc740 \uacb0\ucf54 \uc218\ub834\ud558\uc9c0 \uc54a\ub294\ub2e4.\n\n\ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\uc758 \uae30\uc800 $e^{-(\\sigma + j\\omega)t} = e^{-\\sigma t} e^{-j\\omega t}$\ub294 \"\uac10\uc1e0 \uc870\uc808 \uc190\uc7a1\uc774\" $\\sigma$\ub97c \ucd94\uac00\ud55c\ub2e4. $\\sigma > 2$\uc77c \ub54c $e^{-\\sigma t}$\uc758 \uac10\uc1e0\uc728\uc774 $e^{2t}$\uc758 \ubc1c\uc0b0\uc728\uc744 \ucd08\uacfc\ud558\ubbc0\ub85c \ub0b4\uc801 \uc801\ubd84\uc774 \uc218\ub834\ud55c\ub2e4. \ubcf5\uc18c $s$ \ud3c9\uba74\uc5d0\uc11c:<\/p>\n

- **\uc218\ub834 \uc601\uc5ed(ROC)**: \ubcc0\ud658\uc774 \uc218\ub834\ud558\ub294 $s$ \uac12\uc758 \uc601\uc5ed;
\n- **\uadf9\uc810(Pole)**: $F(s)$\uc758 \ubd84\ubaa8\ub97c 0\uc73c\ub85c \ub9cc\ub4e4\uc5b4 \ubcc0\ud658\uc774 \ubb34\ud55c\ub300\ub85c \ubc1c\uc0b0\ud558\ub294 \uc810;
\n- **\uc601\uc810(Zero)**: $F(s)$\uc758 \ubd84\uc790\ub97c 0\uc73c\ub85c \ub9cc\ub4e4\uc5b4 \ubcc0\ud658\uc774 0\uc774 \ub418\ub294 \uc810.<\/p>\n

\uadf9\uc810\uc758 \uc704\uce58\ub294 \uc2dc\uc2a4\ud15c\uc758 \uc548\uc815\uc131\uc744 \uc9c1\uc811 \uacb0\uc815\ud55c\ub2e4: \ubaa8\ub4e0 \uadf9\uc810\uc774 \uc88c\ubc18\ud3c9\uba74($\\text{Re}(s) < 0$)\uc5d0 \uc788\uc73c\uba74 \uc2dc\uc2a4\ud15c\uc740 \uc548\uc815\uc801\uc774\ub2e4; \ud558\ub098\ub77c\ub3c4 \uc6b0\ubc18\ud3c9\uba74\uc5d0 \uc788\uc73c\uba74 \uc2dc\uc2a4\ud15c\uc740 \ubc1c\uc0b0\ud55c\ub2e4.\n\nZ \ubcc0\ud658\uc758 \uae30\ud558\ud559\uc801 \ud574\uc11d\ub3c4 \uc720\uc0ac\ud558\ub2e4: $z = re^{j\\omega}$\uc5d0\uc11c $r$\uc740 \uc9c4\ud3ed \uc2a4\ucf00\uc77c\ub9c1\uc744, $\\omega$\ub294 \uc704\uc0c1 \ud68c\uc804\uc744 \uc81c\uc5b4\ud55c\ub2e4. \uc218\ub834 \uc601\uc5ed\uc740 $|z| > R$(\uc6b0\uce21 \uc218\uc5f4) \ub610\ub294 $|z| < R$(\uc88c\uce21 \uc218\uc5f4)\uc758 \uc6d0\ud615\/\uc678\ubd80 \uc601\uc5ed\uc774\ub2e4. \uadf9\uc810\uc774 \ub2e8\uc704\uc6d0 \ub0b4\ubd80\uc5d0 \uc788\uc744 \ub54c \uc774\uc0b0 \uc2dc\uc2a4\ud15c\uc774 \uc548\uc815\uc801\uc774\ub2e4.\n\n### 7.3 \ud558\ub4dc\ucf54\uc5b4 \uc608\uc81c \uc0c1\uc138 \ud480\uc774 (Worked Example)\n\n```ad-example\ntitle: \uc608\uc81c 7.1 \ubc1c\uc0b0 \ud568\uc218\uc758 \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658 \u2014 \uadf9\uc810\uacfc \uc218\ub834 \uc601\uc5ed \ubd84\uc11d (Example 7.1 Laplace Transform of a Divergent Function \u2014 Pole and ROC Analysis)\n\n\uc9c0\uc218 \ubc1c\uc0b0 \ud568\uc218 $f(t) = e^{2t}$($t \\geq 0$)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \uadf8 \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\uc744 \uacc4\uc0b0\ud558\uace0 \uc218\ub834 \uc601\uc5ed\uacfc \uadf9\uc810\uc744 \ubd84\uc11d\ud558\ub77c.\n\n**\ud480\uc774**: \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658 \uc815\uc758\uc2dd (7.1)\uc5d0 \ub300\uc785:\n\n$$F(s) = \\int_0^{\\infty} e^{2t} \\cdot e^{-st}\\,dt = \\int_0^{\\infty} e^{-(s-2)t}\\,dt$$\n\n$a = s - 2 = (\\sigma - 2) + j\\omega$\ub77c \ud558\uba74:\n\n$$F(s) = \\int_0^{\\infty} e^{-at}\\,dt = \\left[-\\frac{1}{a}e^{-at}\\right]_{t=0}^{t=\\infty}$$\n\n$t \\to \\infty$\uc77c \ub54c $e^{-at} \\to 0$\uc774\ub824\uba74 $\\text{Re}(a) > 0$, \uc989 $\\text{Re}(s - 2) > 0$, \ub2e4\uc2dc \ub9d0\ud574 $\\sigma > 2$\uac00 \ud544\uc694\ud558\ub2e4. \uc774 \uc870\uac74\uc5d0\uc11c:<\/p>\n

$$F(s) = 0 - \\left(-\\frac{1}{a}\\right) = \\frac{1}{a} = \\frac{1}{s - 2}$$<\/p>\n

\ub530\ub77c\uc11c:<\/p>\n

$$\\mathcal{L}\\{e^{2t}\\} = \\frac{1}{s - 2}, \\quad \\text{ROC: } \\text{Re}(s) > 2, \\quad \\text{Pole: } s = 2$$<\/p>\n

**\ubd84\uc11d**: \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc740 $\\sigma = 0$\uc5d0 \ud574\ub2f9\ud558\uba70, $s = j\\omega$\uc758 \uc2e4\uc218\ubd80\ub294 0\uc73c\ub85c 2\ubcf4\ub2e4 \uc791\uc544 \uc218\ub834 \uc601\uc5ed \ub0b4\uc5d0 \uc788\uc9c0 \uc54a\ub2e4 \u2014 \uc774\uac83\uc774 $e^{2t}$\uc758 \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294 \uc774\uc720\uc774\ub2e4. \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\uc740 \uc2e4\uc218\ubd80 \uc790\uc720\ub3c4 $\\sigma$\ub97c \ub3c4\uc785\ud568\uc73c\ub85c\uc368 \uc801\ubd84 \uacbd\ub85c\ub97c \ud5c8\uc218\ucd95\uc5d0\uc11c \ubcf5\uc18c \ud3c9\uba74\uc758 \uc6b0\ubc18\ud3c9\uba74\uc73c\ub85c \ud655\uc7a5\ud558\uc5ec \ubc1c\uc0b0 \uc2e0\ud638\ub97c \ucc98\ub9ac\ud560 \uc218 \uc788\uac8c \ud55c\ub2e4.
\n```<\/p>\n

```ad-example
\ntitle: \uc608\uc81c 7.2 \uc774\uc0b0 \uc218\uc5f4\uc758 Z \ubcc0\ud658 \u2014 \uc218\ub834 \uc601\uc5ed\uacfc \uc548\uc815\uc131 \ubd84\uc11d (Example 7.2 Z-Transform of a Discrete Sequence \u2014 ROC and Stability Analysis)<\/p>\n

\uc774\uc0b0 \uc218\uc5f4 $x[n] = (0.5)^n u[n]$\uc774 \uc8fc\uc5b4\uc84c\ub2e4. \uc5ec\uae30\uc11c $u[n]$\uc740 \ub2e8\uc704 \uacc4\ub2e8 \ud568\uc218(Unit Step Function)\uc774\ub2e4($n < 0$\uc5d0\uc11c 0, $n \\geq 0$\uc5d0\uc11c 1). Z \ubcc0\ud658\uc744 \uacc4\uc0b0\ud558\uace0 \uc218\ub834 \uc601\uc5ed\uacfc \uc548\uc815\uc131\uc744 \ubd84\uc11d\ud558\ub77c.\n\n**\ud480\uc774**: Z \ubcc0\ud658 \uc815\uc758\uc2dd (7.2)\uc5d0 \ub300\uc785:\n\n$$X(z) = \\sum_{n=0}^{\\infty} (0.5)^n z^{-n} = \\sum_{n=0}^{\\infty} (0.5 z^{-1})^n$$\n\n\uc774\ub294 \uae30\ud558 \uae09\uc218(Geometric Series)\uc774\ub2e4. $|0.5 z^{-1}| < 1$, \uc989 $|z| > 0.5$\uc77c \ub54c \uae09\uc218\uac00 \uc218\ub834\ud55c\ub2e4:<\/p>\n

$$X(z) = \\frac{1}{1 - 0.5z^{-1}} = \\frac{z}{z - 0.5}, \\quad \\text{ROC: } |z| > 0.5$$<\/p>\n

\uc218\ub834 \uc601\uc5ed\uc740 \uc6d0\uc810\uc744 \uc911\uc2ec\uc73c\ub85c \ubc18\uc9c0\ub984 0.5\uc778 \uc6d0\uc758 \uc678\ubd80 \uc601\uc5ed\uc774\ub2e4. \ub2e8\uc704\uc6d0 $|z| = 1$\uc740 \uc218\ub834 \uc601\uc5ed \ub0b4\uc5d0 \uc644\uc804\ud788 \ud3ec\ud568\ub418\ubbc0\ub85c, \uc774 \uc218\uc5f4\uc758 \uc774\uc0b0\uc2dc\uac04 \ud478\ub9ac\uc5d0 \ubcc0\ud658(DTFT, $z = e^{j\\omega}$\uc5d0 \ud574\ub2f9)\uc774 \uc874\uc7ac\ud55c\ub2e4. \uadf9\uc810\uc740 $z = 0.5$\uc5d0 \uc704\uce58\ud558\uba70 \ub2e8\uc704\uc6d0 \ub0b4\ubd80\uc5d0 \uc788\uc73c\ubbc0\ub85c \uc774 \uc2dc\uc2a4\ud15c\uc740 \uc548\uc815\uc801\uc774\ub2e4.
\n```<\/p>\n

### 7.4 \uacf5\ud559 \ubc0f \ucd5c\ucca8\ub2e8 \uc751\uc6a9 (Engineering and Cutting-Edge Applications)<\/p>\n

\ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658\uc740 \uc81c\uc5b4 \uc774\ub860(Control Theory)\uc758 \ucd08\uc11d\uc774\ub2e4. \ud53c\ub4dc\ubc31 \uc81c\uc5b4 \uc2dc\uc2a4\ud15c\uc5d0\uc11c \uc804\ub2ec \ud568\uc218 $H(s)$\uc758 \uadf9\uc810 \uc704\uce58\ub294 \uc548\uc815\uc131\uc744 \uc9c1\uc811 \uacb0\uc815\ud55c\ub2e4:<\/p>\n

- \ubaa8\ub4e0 \uadf9\uc810\uc774 \uc88c\ubc18\ud3c9\uba74($\\text{Re}(s) < 0$)\uc5d0 \uc788\uc74c: \uc2dc\uc2a4\ud15c \uc548\uc815, \uc784\ud384\uc2a4 \uc751\ub2f5\uc774 \uc9c0\uc218\uc801\uc73c\ub85c \uac10\uc1e0;\n- \uadf9\uc810\uc774 \uc6b0\ubc18\ud3c9\uba74($\\text{Re}(s) > 0$)\uc5d0 \uc874\uc7ac: \uc2dc\uc2a4\ud15c \ubc1c\uc0b0, \uc784\ud384\uc2a4 \uc751\ub2f5\uc774 \uc9c0\uc218\uc801\uc73c\ub85c \uc99d\uac00;
\n- \uadf9\uc810\uc774 \ud5c8\uc218\ucd95($\\text{Re}(s) = 0$)\uc5d0 \uc704\uce58: \uc2dc\uc2a4\ud15c \uc784\uacc4 \uc548\uc815, \uc784\ud384\uc2a4 \uc751\ub2f5\uc774 \ub4f1\uc9c4\ud3ed \uc9c4\ub3d9.<\/p>\n

Z \ubcc0\ud658\uc740 \ub514\uc9c0\ud138 \uc2e0\ud638 \ucc98\ub9ac(Digital Signal Processing)\uc758 \ud575\uc2ec\uc774\ub2e4. \ub514\uc9c0\ud138 \ud544\ud130\uc758 \uc8fc\ud30c\uc218 \uc751\ub2f5\uc740 $H(z)$\uc758 \ub2e8\uc704\uc6d0 \uc704\uc5d0\uc11c\uc758 \uac12\uc73c\ub85c \uacb0\uc815\ub418\uba70, \uc548\uc815\uc131\uc740 \ubaa8\ub4e0 \uadf9\uc810\uc774 \ub2e8\uc704\uc6d0 \ub0b4\ubd80\uc5d0 \uc788\ub294\uc9c0 \uc5ec\ubd80\ub85c \uacb0\uc815\ub41c\ub2e4. IIR \ud544\ud130 \uc124\uacc4\ub294 \ubcf8\uc9c8\uc801\uc73c\ub85c $z$ \ud3c9\uba74\uc5d0\uc11c \uadf9\uc810\uacfc \uc601\uc810\uc744 \ubc30\uce58\ud558\uc5ec \ubaa9\ud45c \uc8fc\ud30c\uc218 \uc751\ub2f5\uc5d0 \uadfc\uc811\ud558\ub294 \uacfc\uc815\uc774\ub2e4.<\/p>\n

---<\/p>\n

## \uc81c8\uc7a5 \ucee8\ubcfc\ub8e8\uc158\uc758 \ubcf8\uc9c8 \u2014 \"\uc2ac\ub77c\uc774\ub529 \ub0b4\uc801\" (Chapter 8 The Essence of Convolution \u2014 \"Sliding Inner Product\")<\/p>\n

### 8.1 \uc774\ub860\uacfc \uc5c4\ubc00\ud55c \uc815\uc758 (Theory and Rigorous Definitions)<\/p>\n

**\ucee8\ubcfc\ub8e8\uc158(Convolution)** \uc740 \uc2e0\ud638\ucc98\ub9ac, \uc81c\uc5b4 \uc774\ub860, \ub525\ub7ec\ub2dd\uc5d0\uc11c \uac00\uc7a5 \ud575\uc2ec\uc801\uc778 \uc5f0\uc0b0 \uc911 \ud558\ub098\uc774\ub2e4$^{[17]}$. \ub0b4\uc801 \uad00\uc810\uc5d0\uc11c \ucee8\ubcfc\ub8e8\uc158\uc758 \ubcf8\uc9c8\uc740 **\uc2ac\ub77c\uc774\ub529 \uc708\ub3c4\uc6b0 \uc704\uc5d0\uc11c\uc758 \ub0b4\uc801 \uc218\uc5f4**\uc774\ub2e4.<\/p>\n

```ad-definition
\ntitle: \uc815\uc758 8.1 \ucee8\ubcfc\ub8e8\uc158 (Definition 8.1 Convolution)
\n$f, g: \\mathbb{R} \\to \\mathbb{R}$\uc774 \ub450 \uc5f0\uc18d \ud568\uc218\ub77c\uace0 \ud560 \ub54c, \uadf8 **\ucee8\ubcfc\ub8e8\uc158**\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4:<\/p>\n

$$(f * g)(t) = \\int_{-\\infty}^{\\infty} f(\\tau) g(t - \\tau)\\,d\\tau \\tag{8.1}$$<\/p>\n

\uc774\uc0b0 \uc218\uc5f4 $x, h: \\mathbb{Z} \\to \\mathbb{R}$\uc5d0 \ub300\ud574, **\uc774\uc0b0 \ucee8\ubcfc\ub8e8\uc158(Discrete Convolution)** \uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4:<\/p>\n

$$(x * h)[n] = \\sum_{k=-\\infty}^{\\infty} x[k]\\, h[n - k] \\tag{8.2}$$
\n```<\/p>\n

```ad-theorem
\ntitle: \uba85\uc81c 8.1 \ucee8\ubcfc\ub8e8\uc158\uc758 \ub0b4\uc801 \ud574\uc11d (Proposition 8.1 Inner Product Interpretation of Convolution)
\n\uace0\uc815\ub41c \uc2dc\uc810 $t$\uc5d0\uc11c \ucee8\ubcfc\ub8e8\uc158 \uc5f0\uc0b0 $(f * g)(t)$\ub294 \ud568\uc218 $f(\\tau)$\uc640 \ub4a4\uc9d1\ud600\uc11c \ud3c9\ud589 \uc774\ub3d9\ub41c $g(\\tau)$ \uc0ac\uc774\uc758 \ub0b4\uc801\uacfc \ub3d9\ub4f1\ud558\ub2e4:<\/p>\n

$$(f * g)(t) = \\langle f(\\tau), g(t - \\tau) \\rangle = \\int f(\\tau) g(t - \\tau)\\,d\\tau \\tag{8.3}$$<\/p>\n

\uc5ec\uae30\uc11c \ub4a4\uc9d1\uae30 \uc5f0\uc0b0 $g(\\tau) \\to g(-\\tau)$\ub294 \uc2dc\uc2a4\ud15c\uc774 \uc778\uacfc\uc131(Causality)\uc744 \ub9cc\uc871\ud558\ub3c4\ub85d \ubcf4\uc7a5\ud55c\ub2e4 \u2014 \ud604\uc7ac \ucd9c\ub825\uc740 \ud604\uc7ac \ubc0f \uacfc\uac70 \uc785\ub825\uc5d0\ub9cc \uc758\uc874\ud55c\ub2e4.
\n```<\/p>\n

```ad-definition
\ntitle: \uc815\uc758 8.2 \uc0c1\ud638\uc0c1\uad00 (Definition 8.2 Cross-Correlation)
\n\ucee8\ubcfc\ub8e8\uc158\uacfc \ubc00\uc811\ud558\uac8c \uad00\ub828\ub41c \uc5f0\uc0b0\uc740 **\uc0c1\ud638\uc0c1\uad00(Cross-Correlation)** \uc774\ub2e4:<\/p>\n

$$(f \\star g)(t) = \\int_{-\\infty}^{\\infty} f(\\tau) g(\\tau + t)\\,d\\tau \\tag{8.4}$$<\/p>\n

\uc0c1\ud638\uc0c1\uad00\uc740 \ub4a4\uc9d1\uae30 \uc5f0\uc0b0\uc774 \uc5c6\uc73c\uba70, \uc2e0\ud638 \uac04\uc758 \ub2e4\uc591\ud55c \uc774\ub3d9\ub7c9\uc5d0 \ub530\ub978 \ub0b4\uc801\uc744 \uc9c1\uc811 \uacc4\uc0b0\ud558\ubbc0\ub85c \ud15c\ud50c\ub9bf \ub9e4\uce6d(Template Matching)\uacfc \uc720\uc0ac\ub3c4 \uac80\ucd9c\uc5d0 \uc0ac\uc6a9\ub41c\ub2e4.
\n```<\/p>\n

### 8.2 \uae30\ud558\ud559\uacfc \uacf5\uac04\uc801 \uc774\ubbf8\uc9c0 (Geometry and Spatial Intuition)<\/p>\n

\ucee8\ubcfc\ub8e8\uc158\uc758 \uae30\ud558\ud559\uc801 \uacfc\uc815\uc740 \ub124 \ub2e8\uacc4\ub85c \ubd84\ud574\ub420 \uc218 \uc788\ub2e4:<\/p>\n

1. **\ub4a4\uc9d1\uae30(Flip)**: \ucee4\ub110 \ud568\uc218 $g(\\tau)$\ub97c $g(-\\tau)$\ub85c \ub4a4\uc9d1\uc5b4 \uc5f0\uc0b0\uc774 \uc778\uacfc\uc131\uc744 \ub9cc\uc871\ud558\ub3c4\ub85d \ud568;
\n2. **\ud3c9\ud589 \uc774\ub3d9(Shift)**: \ub4a4\uc9d1\ud78c \ucee4\ub110\uc744 $t$\ub9cc\ud07c \ud3c9\ud589 \uc774\ub3d9\ud558\uc5ec $g(t - \\tau)$\ub97c \uc5bb\uc74c;
\n3. **\uacf1\uc148(Multiply)**: $f(\\tau)$\uc640 $g(t - \\tau)$\ub97c \uc810\ubcc4\ub85c \uacf1\ud568;
\n4. **\uc801\ubd84(Integrate)**: \uacf1\uc758 \ud569(\uc801\ubd84)\uc744 \uad6c\ud558\uc5ec \ud574\ub2f9 \uc2dc\uc810\uc758 \ub0b4\uc801 \uac12\uc744 \uc5bb\uc74c.<\/p>\n

$t$\uac00 \ubcc0\ud568\uc5d0 \ub530\ub77c \ucee4\ub110\uc774 \uc2dc\uac04\ucd95\uc744 \ub530\ub77c \uc2ac\ub77c\uc774\ub529\ud558\uba70, \uac01 \uc704\uce58\uc5d0\uc11c \uc2e0\ud638\uc640 \ucee4\ub110\uc758 \ub0b4\uc801\uc744 \uacc4\uc0b0\ud55c\ub2e4. \ucee8\ubcfc\ub8e8\uc158 \uacb0\uacfc $y(t)$\ub294 \ub0b4\uc801 \uac12\uc774 \uc2ac\ub77c\uc774\ub529 \uc704\uce58\uc5d0 \ub530\ub77c \ubcc0\ud654\ud558\ub294 \uace1\uc120\uc774\ub2e4. \ub0b4\uc801 \uac12\uc774 \ud070 \uc704\uce58\ub294 \uc2e0\ud638\uc758 \uad6d\uc18c \ubd80\ubd84\uc774 \ucee4\ub110\uc758 \ud30c\ud615\uacfc \uac00\uc7a5 \uc720\uc0ac\ud568\uc744 \uc758\ubbf8\ud55c\ub2e4 \u2014 \uc774\uac83\uc774 \ubc14\ub85c **\uc815\ud569 \ud544\ud130(Matched Filter)** \uc758 \uc6d0\ub9ac\uc774\ub2e4.<\/p>\n

\uc774\ubbf8\uc9c0 \ucc98\ub9ac\uc5d0\uc11c 2\ucc28\uc6d0 \ucee8\ubcfc\ub8e8\uc158 \ucee4\ub110(Kernel)\uc774 \uc774\ubbf8\uc9c0 \uc704\ub97c \uc2ac\ub77c\uc774\ub529\ud558\uba70, \uac01 \uc704\uce58\uc5d0\uc11c $k \\times k$ \uc774\uc6c3\uacfc \ucee4\ub110\uc758 2\ucc28\uc6d0 \ub0b4\uc801\uc744 \uacc4\uc0b0\ud558\uc5ec \"\uc751\ub2f5 \ub9f5(Feature Map)\"\uc744 \ucd9c\ub825\ud55c\ub2e4. \uc751\ub2f5 \uac12\uc774 \ub192\uc740 \uc601\uc5ed\uc740 \ud574\ub2f9 \uad6d\uc18c \uc774\ubbf8\uc9c0 \ube14\ub85d\uc774 \ucee8\ubcfc\ub8e8\uc158 \ucee4\ub110\uc758 \ud328\ud134\uacfc \uac00\uc7a5 \uc77c\uce58\ud568\uc744 \uc758\ubbf8\ud55c\ub2e4.<\/p>\n

### 8.3 \ud558\ub4dc\ucf54\uc5b4 \uc608\uc81c \uc0c1\uc138 \ud480\uc774 (Worked Example)<\/p>\n

```ad-example
\ntitle: \uc608\uc81c 8.1 \uc774\uc0b0 \uc218\uc5f4\uc758 \uc2ac\ub77c\uc774\ub529 \ub0b4\uc801 \ucee8\ubcfc\ub8e8\uc158 \u2014 \uc810\ubcc4 \uc218\ub3d9 \uacc4\uc0b0 (Example 8.1 Sliding Inner Product Convolution of Discrete Sequences \u2014 Pointwise Manual Calculation)<\/p>\n

\uc785\ub825 \uc218\uc5f4 $x[n] = [1, 2, 3]$($n = 0, 1, 2$)\uacfc \ucee8\ubcfc\ub8e8\uc158 \ucee4\ub110 $h[n] = [0.5, 1, 0.5]$($n = 0, 1, 2$)\uac00 \uc8fc\uc5b4\uc84c\ub2e4. \ucee8\ubcfc\ub8e8\uc158 $y[n] = (x * h)[n]$\uc744 \uacc4\uc0b0\ud558\ub77c.<\/p>\n

**\ud480\uc774**: \uc774\uc0b0 \ucee8\ubcfc\ub8e8\uc158 \uacf5\uc2dd (8.2)\uc5d0 \ub530\ub77c \uc810\ubcc4\ub85c \uacc4\uc0b0:<\/p>\n

$n = 0$:
\n$$y[0] = \\sum_{k} x[k]h[0-k] = x[0]h[0] = 1 \\times 0.5 = 0.5$$<\/p>\n

$n = 1$:
\n$$y[1] = x[0]h[1] + x[1]h[0] = 1 \\times 1 + 2 \\times 0.5 = 2$$<\/p>\n

$n = 2$:
\n$$y[2] = x[0]h[2] + x[1]h[1] + x[2]h[0] = 1 \\times 0.5 + 2 \\times 1 + 3 \\times 0.5 = 4$$<\/p>\n

$n = 3$:
\n$$y[3] = x[1]h[2] + x[2]h[1] = 2 \\times 0.5 + 3 \\times 1 = 4$$<\/p>\n

$n = 4$:
\n$$y[4] = x[2]h[2] = 3 \\times 0.5 = 1.5$$<\/p>\n

\ub530\ub77c\uc11c $y[n] = [0.5, 2, 4, 4, 1.5]$\uc774\ub2e4. $n = 2, 3$\uc5d0\uc11c \ucee8\ubcfc\ub8e8\uc158 \uac12\uc774 \ucd5c\ub300(4)\uac00 \ub418\ub294\ub370, \uc774\ub54c \uc785\ub825 \uc218\uc5f4 $[1, 2, 3]$\uacfc \ub4a4\uc9d1\ud78c \ucee4\ub110 $[0.5, 1, 0.5]$\uc758 \uc911\ucca9 \uc601\uc5ed\uc774 \uac00\uc7a5 \ucee4\uc11c \ub0b4\uc801 \uac12\uc774 \ucd5c\uace0\uc870\uc5d0 \ub2ec\ud55c\ub2e4.
\n```<\/p>\n

```ad-example
\ntitle: \uc608\uc81c 8.2 Sobel \uc5d0\uc9c0 \uac80\ucd9c \u2014 \ub0b4\uc801 \ud15c\ud50c\ub9bf\uc73c\ub85c\uc11c\uc758 2\ucc28\uc6d0 \ucee8\ubcfc\ub8e8\uc158 (Example 8.2 Sobel Edge Detection \u2014 2D Convolution as an Inner Product Template)<\/p>\n

Sobel \uc5f0\uc0b0\uc790\ub294 \ub450 \uac1c\uc758 $3 \\times 3$ \ucee8\ubcfc\ub8e8\uc158 \ucee4\ub110\ub85c \uad6c\uc131\ub418\uba70, \uac01\uac01 \uc218\ud3c9 \ubc0f \uc218\uc9c1 \ubc29\ud5a5\uc758 \uc5d0\uc9c0\ub97c \uac80\ucd9c\ud55c\ub2e4:<\/p>\n

$$S_x = \\begin{bmatrix} 1 & 0 & -1 \\\\ 2 & 0 & -2 \\\\ 1 & 0 & -1 \\end{bmatrix}, \\quad S_y = \\begin{bmatrix} 1 & 2 & 1 \\\\ 0 & 0 & 0 \\\\ -1 & -2 & -1 \\end{bmatrix}$$<\/p>\n

$3 \\times 3$ \uad6d\uc18c \uc774\ubbf8\uc9c0 \ube14\ub85d(\uadf8\ub808\uc774\uc2a4\ucf00\uc77c \uac12)\uc774 \uc8fc\uc5b4\uc84c\ub2e4:<\/p>\n

$$I = \\begin{bmatrix} 10 & 20 & 30 \\\\ 10 & 20 & 30 \\\\ 10 & 20 & 30 \\end{bmatrix}$$<\/p>\n

\uc774 \uc774\ubbf8\uc9c0 \ube14\ub85d\uc740 \uc218\ud3c9 \ubc29\ud5a5\uc73c\ub85c \ubc1d\uae30 \uadf8\ub77c\ub370\uc774\uc158(\uc67c\ucabd\uc5d0\uc11c \uc624\ub978\ucabd\uc73c\ub85c \ubc1d\uc544\uc9d0)\uc744 \ubcf4\uc774\uba70, \uc218\uc9c1 \ubc29\ud5a5\uc73c\ub85c\ub294 \ubc1d\uae30\uac00 \uade0\uc77c\ud558\ub2e4.<\/p>\n

**\ud480\uc774**: Sobel X \uc5f0\uc0b0\uc790\uc640 \uc774\ubbf8\uc9c0 \ube14\ub85d\uc758 2\ucc28\uc6d0 \ub0b4\uc801 \uacc4\uc0b0:<\/p>\n

$$G_x = \\sum_{i=1}^{3} \\sum_{j=1}^{3} S_x(i,j) \\cdot I(i,j)$$<\/p>\n

$$= (1 \\times 10) + (0 \\times 20) + (-1 \\times 30) + (2 \\times 10) + (0 \\times 20) + (-2 \\times 30) + (1 \\times 10) + (0 \\times 20) + (-1 \\times 30)$$<\/p>\n

$$= 10 + 0 - 30 + 20 + 0 - 60 + 10 + 0 - 30 = -80$$<\/p>\n

Sobel Y \uc5f0\uc0b0\uc790\uc758 2\ucc28\uc6d0 \ub0b4\uc801 \uacc4\uc0b0:<\/p>\n

$$G_y = (1 \\times 10) + (2 \\times 20) + (1 \\times 30) + (0 \\times 10) + (0 \\times 20) + (0 \\times 30) + (-1 \\times 10) + (-2 \\times 20) + (-1 \\times 30)$$<\/p>\n

$$= 10 + 40 + 30 + 0 + 0 + 0 - 10 - 40 - 30 = 0$$<\/p>\n

\uc5d0\uc9c0 \uac15\ub3c4(Edge Intensity)\ub294:<\/p>\n

$$\\|\\nabla I\\| = \\sqrt{G_x^2 + G_y^2} = \\sqrt{(-80)^2 + 0^2} = 80$$<\/p>\n

**\ubd84\uc11d**: $|G_x| = 80$\uc73c\ub85c \ud070 \uac12\uc744 \uac00\uc9c0\uba70, \uc774\ub294 \uc218\ud3c9 \ubc29\ud5a5\uc5d0 \uc720\uc758\ubbf8\ud55c \ubc1d\uae30 \ubcc0\ud654(\uc218\uc9c1 \uc5d0\uc9c0)\uac00 \uc788\uc74c\uc744 \ub098\ud0c0\ub0b8\ub2e4; $G_y = 0$\uc740 \uc218\uc9c1 \ubc29\ud5a5\uc73c\ub85c \ubc1d\uae30\uac00 \uade0\uc77c\ud568\uc744 \ub098\ud0c0\ub0b8\ub2e4. Sobel \uc5d0\uc9c0 \uac80\ucd9c\uc758 \ubcf8\uc9c8\uc740 \ub450 \uac1c\uc758 \uc9c1\uad50 \ucee8\ubcfc\ub8e8\uc158 \ucee4\ub110(\ub0b4\uc801 \ud15c\ud50c\ub9bf)\uc744 \uc774\ubbf8\uc9c0 \uc704\uc5d0\uc11c \uc2ac\ub77c\uc774\ub529\ud558\uba70 \uac01 \ud53d\uc140 \uc774\uc6c3\uacfc \ud15c\ud50c\ub9bf\uc758 2\ucc28\uc6d0 \ub0b4\uc801\uc744 \uacc4\uc0b0\ud558\uace0, \ub0b4\uc801 \uc9c4\ud3ed\uc774 \ud070 \uc704\uce58\uac00 \ubc14\ub85c \uc5d0\uc9c0\uc784\uc744 \ucc3e\ub294 \uac83\uc774\ub2e4.
\n```<\/p>\n

### 8.4 \uacf5\ud559 \ubc0f \ucd5c\ucca8\ub2e8 \uc751\uc6a9 (Engineering and Cutting-Edge Applications)<\/p>\n

<\/p>\n

> **\uadf8\ub9bc 4: \uc2ac\ub77c\uc774\ub529 \ub0b4\uc801\uacfc \uc815\ud569 \ud544\ud130(Figure 4: Sliding Inner Product and Matched Filter).** \ud30c\ub780\uc0c9 \uace1\uc120\uc740 \uc7a1\uc74c\uc774 \uc11e\uc778 \ub79c\ub364 \uc218\uc5f4 $x[n]$\uc774\uace0, \ube68\uac04\uc0c9 \uace1\uc120\uc740 \ucee8\ubcfc\ub8e8\uc158 \uc751\ub2f5\uc774\ub2e4. \ud15c\ud50c\ub9bf \ud384\uc2a4 $h[n] = [0, 0.35, 1.0, 0.35, 0]$\uac00 \uc2dc\uac04\ucd95\uc744 \ub530\ub77c \uc2ac\ub77c\uc774\ub529\ud558\uba70 \uac01 \uc704\uce58\uc5d0\uc11c $\\sum x[k]h[n-k]$\ub97c \uacc4\uc0b0\ud55c\ub2e4. \uc8fc\ud669\uc0c9 \ud45c\uc2dc \uc9c0\uc810($n \\approx 110, 265, 340$)\uc5d0\uc11c \ucee8\ubcfc\ub8e8\uc158 \uac12\uc774 \ucd5c\uace0\uc870\uc5d0 \ub3c4\ub2ec\ud558\uba70, \uc774\ub294 \ud574\ub2f9 \uc704\uce58\uc758 \uc2e0\ud638 \uad6d\uc18c \ud30c\ud615\uc774 \ud15c\ud50c\ub9bf\uacfc \uac00\uc7a5 \uc77c\uce58\ud568\uc744 \uc758\ubbf8\ud55c\ub2e4. \ud604\ub300 \ub808\uc774\ub354 \uc2e0\ud638 \ud3ec\ucc29\uc758 \ud575\uc2ec \uc6d0\ub9ac\ub294 \uc774 \uc2ac\ub77c\uc774\ub529 \ud22c\uc601 \uba54\ucee4\ub2c8\uc998\uc5d0\uc11c \ube44\ub86f\ub41c\ub2e4.<\/p>\n

<\/p>\n

> **\uadf8\ub9bc 5: 2\ucc28\uc6d0 \ucee8\ubcfc\ub8e8\uc158\uc744 \ud1b5\ud55c \uc5d0\uc9c0 \ud2b9\uc9d5 \ucd94\ucd9c (Sobel \uc5d0\uc9c0 \uac80\ucd9c)(Figure 5: Edge Feature Extraction via 2D Convolution (Sobel Edge Detection)).** Sobel \uc5f0\uc0b0\uc790\ub294 \ud55c \uc30d\uc758 \uc9c1\uad50 $3 \\times 3$ \ubbf8\ubd84 \ud15c\ud50c\ub9bf\uc73c\ub85c, \uac01\uac01 $x$ \ubc29\ud5a5\uacfc $y$ \ubc29\ud5a5\uc758 \ubc1d\uae30 \uadf8\ub77c\ub370\uc774\uc158\uc744 \uac80\ucd9c\ud55c\ub2e4. \ud15c\ud50c\ub9bf\uc774 \uadf8\ub808\uc774\uc2a4\ucf00\uc77c \uc774\ubbf8\uc9c0 \uc704\ub97c \uc2ac\ub77c\uc774\ub529\ud560 \ub54c, \ud3c9\ud0c4\ud55c \uc601\uc5ed\uc5d0\uc11c\ub294 \uc591\uc758 \ud22c\uc601\uacfc \uc74c\uc758 \ud22c\uc601\uc774 \uc11c\ub85c \uc0c1\uc1c4\ub418\uc5b4(\ub0b4\uc801\uc774 0\uc5d0 \uac00\uae4c\uc6c0) \uc5d0\uc9c0\uc5d0\uc11c\ub294 \ud53d\uc140\uc758 \uacc4\ub2e8\uc801 \ubcc0\ud654\ub85c \uc778\ud574 \ub0b4\uc801 \uc9c4\ud3ed\uc774 \uc720\uc758\ubbf8\ud558\uac8c \uc99d\uac00\ud55c\ub2e4. $\\|\\nabla I\\| = \\sqrt{G_x^2 + G_y^2}$\ub97c \ud1b5\ud574 \ub450 \uc9c1\uad50 \uc131\ubd84\uc744 \uacb0\ud569\ud558\uba74 \ubb3c\ub9ac\uc801 \uc138\uacc4\uc758 \uc5d0\uc9c0 \uc815\ubcf4\ub97c \ucd94\ucd9c\ud560 \uc218 \uc788\ub2e4. \uc774\uac83\uc774 \ucef4\ud4e8\ud130 \ube44\uc804\uc5d0\uc11c \ud2b9\uc9d5 \ucd94\ucd9c\uc758 \uae30\ucd08\uc774\ub2e4.<\/p>\n

---<\/p>\n

## \uc81c9\uc7a5 \uc774\uc0b0 \ucf54\uc0ac\uc778 \ubcc0\ud658\uacfc JPEG \uc555\ucd95 (Chapter 9 Discrete Cosine Transform and JPEG Compression)<\/p>\n

### 9.1 \uc774\ub860\uacfc \uc5c4\ubc00\ud55c \uc815\uc758 (Theory and Rigorous Definitions)<\/p>\n

**\uc774\uc0b0 \ucf54\uc0ac\uc778 \ubcc0\ud658(DCT, Discrete Cosine Transform)** \uc740 JPEG \uc774\ubbf8\uc9c0 \uc555\ucd95 \ud45c\uc900\uc758 \ud575\uc2ec \uc54c\uace0\ub9ac\uc998\uc774\ub2e4$^{[18][19]}$. \ub0b4\uc801 \uad00\uc810\uc5d0\uc11c DCT\ub294 \uc774\ubbf8\uc9c0 \ube14\ub85d\uc744 \uc774\uc0b0 \ucf54\uc0ac\uc778 \uae30\uc800 \ud568\uc218 \uc9d1\ud569\uc5d0 \uc9c1\uad50 \ud22c\uc601\ud558\uc5ec \uacf5\uac04 \uc601\uc5ed\uc758 \ud53d\uc140 \uac12\uc744 \uc8fc\ud30c\uc218 \uc601\uc5ed \uacc4\uc218\ub85c \ubcc0\ud658\ud55c\ub2e4.<\/p>\n

```ad-definition
\ntitle: \uc815\uc758 9.1 2\ucc28\uc6d0 DCT (Definition 9.1 Two-Dimensional DCT)
\n$f(x, y)$\uac00 $N \\times N$ \uc774\ubbf8\uc9c0 \ube14\ub85d($x, y = 0, 1, \\dots, N-1$)\uc774\ub77c\uace0 \ud560 \ub54c, \uadf8 2\ucc28\uc6d0 DCT\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub41c\ub2e4:<\/p>\n

$$F(u, v) = \\frac{2}{N} C(u) C(v) \\sum_{x=0}^{N-1} \\sum_{y=0}^{N-1} f(x, y) \\cos\\left[\\frac{(2x+1)u\\pi}{2N}\\right] \\cos\\left[\\frac{(2y+1)v\\pi}{2N}\\right] \\tag{9.1}$$<\/p>\n

\uc5ec\uae30\uc11c $u, v = 0, 1, \\dots, N-1$\uc740 \uc8fc\ud30c\uc218 \uc778\ub371\uc2a4\uc774\uba70, \uc815\uaddc\ud654 \uacc4\uc218\ub294:<\/p>\n

$$C(k) = \\begin{cases} 1\/\\sqrt{2}, & k = 0 \\\\ 1, & k \\neq 0 \\end{cases}$$
\n```<\/p>\n

```ad-theorem
\ntitle: \uba85\uc81c 9.1 \uc9c1\uad50 \ud22c\uc601\uc73c\ub85c\uc11c\uc758 DCT (Proposition 9.1 DCT as Orthogonal Projection)
\n$N \\times N$\uac1c\uc758 DCT \uae30\uc800 \ud568\uc218\ub97c \uc815\uc758\ud558\uc790:<\/p>\n

$$B_{u,v}(x, y) = \\frac{2}{N} C(u) C(v) \\cos\\left[\\frac{(2x+1)u\\pi}{2N}\\right] \\cos\\left[\\frac{(2y+1)v\\pi}{2N}\\right]$$<\/p>\n

\uadf8\ub7ec\uba74 $\\{B_{u,v}\\}$\ub294 $\\mathbb{R}^{N \\times N}$ \uc704\uc758 \uc644\ube44 \uc9c1\uad50 \uae30\uc800(Complete Orthogonal Basis)\ub97c \uad6c\uc131\ud558\uba70, \ub2e4\uc74c\uc744 \ub9cc\uc871\ud55c\ub2e4:<\/p>\n

$$\\langle B_{u,v}, B_{u',v'} \\rangle = \\sum_{x=0}^{N-1} \\sum_{y=0}^{N-1} B_{u,v}(x, y) B_{u',v'}(x, y) = \\delta_{u,u'} \\delta_{v,v'}$$<\/p>\n

DCT \uacc4\uc218 $F(u, v)$\ub294 \ubc14\ub85c \uc774\ubbf8\uc9c0 \ube14\ub85d $f$\uc758 \uae30\uc800 \ud568\uc218 $B_{u,v}$ \uc704\ub85c\uc758 \ud22c\uc601\uc774\ub2e4:<\/p>\n

$$F(u, v) = \\langle f, B_{u,v} \\rangle = \\sum_{x=0}^{N-1} \\sum_{y=0}^{N-1} f(x, y) B_{u,v}(x, y) \\tag{9.2}$$
\n```<\/p>\n

```ad-theorem
\ntitle: \uba85\uc81c 9.2 \uc5d0\ub108\uc9c0 \uc9d1\uc911\uc131 (Proposition 9.2 Energy Compaction)
\n\uc790\uc5f0 \uc774\ubbf8\uc9c0\uc758 \uacbd\uc6b0 DCT \uacc4\uc218\uc758 \uc5d0\ub108\uc9c0\ub294 \uc8fc\ub85c \uc800\uc8fc\ud30c \uc601\uc5ed($u, v$\uac00 \uc791\uc740 \ucabd)\uc5d0 \uc9d1\uc911\ub418\uba70, \uace0\uc8fc\ud30c \uacc4\uc218($u, v$\uac00 \ud070 \ucabd)\uc758 \uc9c4\ud3ed\uc740 0\uc5d0 \uac00\uae5d\ub2e4. JPEG \uc555\ucd95\uc740 \uc774 \ud2b9\uc131\uc744 \uc774\uc6a9\ud558\uc5ec \uc591\uc790\ud654(Quantization)\ub97c \ud1b5\ud574 \ubbf8\uc138\ud55c \uace0\uc8fc\ud30c \uacc4\uc218\ub97c \ubc84\ub9bc\uc73c\ub85c\uc368 \uc2dc\uac01\uc801 \ud488\uc9c8\uc744 \uc720\uc9c0\ud558\uba74\uc11c \ud070 \ud3ed\uc758 \uc555\ucd95\uc744 \ub2ec\uc131\ud55c\ub2e4.
\n```<\/p>\n

### 9.2 \uae30\ud558\ud559\uacfc \uacf5\uac04\uc801 \uc774\ubbf8\uc9c0 (Geometry and Spatial Intuition)<\/p>\n

$8 \\times 8$ \uc774\ubbf8\uc9c0 \ube14\ub85d\uc740 64\ucc28\uc6d0 \uacf5\uac04\uc758 \ubca1\ud130\ub85c \ubcfc \uc218 \uc788\ub2e4. DCT \uae30\uc800 \ud568\uc218\ub294 \uc774 64\ucc28\uc6d0 \uacf5\uac04\uc5d0\uc11c \uc644\ube44 \uc9c1\uad50 \uae30\uc800\ub97c \uad6c\uc131\ud55c\ub2e4:<\/p>\n

- **$B_{0,0}$ (DC \uae30\uc800)**: \uc0c1\uc218 \ud568\uc218, \uc774\ubbf8\uc9c0 \ube14\ub85d\uc758 \ud3c9\uade0 \ubc1d\uae30\uc5d0 \ub300\uc751;
\n- **\uc800\uc8fc\ud30c \uae30\uc800**($u, v$\uac00 \uc791\uc74c): \ubd80\ub4dc\ub7ec\uc6b4 \uadf8\ub77c\ub370\uc774\uc158 \ud328\ud134, \uc774\ubbf8\uc9c0\uc758 \ub300\uaddc\ubaa8 \uad6c\uc870\uc5d0 \ub300\uc751;
\n- **\uace0\uc8fc\ud30c \uae30\uc800**($u, v$\uac00 \ud07c): \uc870\ubc00\ud55c \uc9c4\ub3d9 \ud328\ud134, \uc774\ubbf8\uc9c0\uc758 \uc138\ubd80 \uc9c8\uac10\uacfc \ub178\uc774\uc988\uc5d0 \ub300\uc751.<\/p>\n

\uc774\ubbf8\uc9c0 \ube14\ub85d \ubca1\ud130\ub97c \uc774 64\uac1c\uc758 \uae30\uc800 \ubc29\ud5a5\uc73c\ub85c \ud22c\uc601\ud558\uba74 64\uac1c\uc758 DCT \uacc4\uc218\ub97c \uc5bb\ub294\ub2e4. \uc790\uc5f0 \uc774\ubbf8\uc9c0\uc758 \uacbd\uc6b0 \ud22c\uc601 \uc5d0\ub108\uc9c0\ub294 \uc800\uc8fc\ud30c \uacc4\uc218(\uc88c\uc0c1\ub2e8)\uc5d0\u9ad8\u5ea6\ub85c \uc9d1\uc911\ub418\uba70, \uace0\uc8fc\ud30c \uacc4\uc218(\uc6b0\ud558\ub2e8)\ub294 0\uc5d0 \uac00\uae5d\ub2e4. JPEG \uc555\ucd95\uc740 \uc591\uc790\ud654\ub97c \ud1b5\ud574 \ubbf8\uc138\ud55c \uace0\uc8fc\ud30c \uacc4\uc218\ub97c 0\uc73c\ub85c \uc124\uc815\ud558\uace0, \uc18c\uc218\uc758 \uc800\uc8fc\ud30c \uacc4\uc218\ub9cc\uc73c\ub85c \uc6d0\ubcf8 \uc774\ubbf8\uc9c0 \ube14\ub85d\uc744 \uadfc\uc0ac \uc7ac\uad6c\uc131\ud55c\ub2e4.<\/p>\n

### 9.3 \ud558\ub4dc\ucf54\uc5b4 \uc608\uc81c \uc0c1\uc138 \ud480\uc774 (Worked Example)<\/p>\n

```ad-example
\ntitle: \uc608\uc81c 9.1 $2 \\times 2$ \uc774\ubbf8\uc9c0 \ube14\ub85d\uc758 DCT \ud22c\uc601 \uacc4\uc218 \uc218\ub3d9 \uacc4\uc0b0 (Example 9.1 Manual Calculation of DCT Projection Coefficients for a $2 \\times 2$ Image Block)<\/p>\n

DCT\uc758 \ud22c\uc601 \ubcf8\uc9c8\uc744 \ubcf4\uc5ec\uc8fc\uae30 \uc704\ud574 $N = 2$\uc778 \ucd08\uc18c\ud615 \uc774\ubbf8\uc9c0 \ube14\ub85d\uc744 \uace0\ub824\ud558\uc790. $2 \\times 2$ DCT \uae30\uc800 \ud589\ub82c\uc740:<\/p>\n

$$T = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix}$$<\/p>\n

$T$\ub294 \uc9c1\uad50 \ud589\ub82c(Orthogonal Matrix)\ub85c $T^T T = I$\ub97c \ub9cc\uc871\ud55c\ub2e4. \uadf8\ub808\uc774\uc2a4\ucf00\uc77c \uc774\ubbf8\uc9c0 \ube14\ub85d\uc774 \uc8fc\uc5b4\uc84c\ub2e4:<\/p>\n

$$I = \\begin{bmatrix} 100 & 80 \\\\ 60 & 40 \\end{bmatrix}$$<\/p>\n

2\ucc28\uc6d0 DCT\ub294 \ud589\ub82c \uacf1\uc148\uc744 \ud1b5\ud574 \uad6c\ud604\ub420 \uc218 \uc788\ub2e4: $F = T \\cdot I \\cdot T^T$.<\/p>\n

**\ud480\uc774**:<\/p>\n

**\ub2e8\uacc4 1**: $T \\cdot I$ \uacc4\uc0b0.<\/p>\n

$$T \\cdot I = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix} \\begin{bmatrix} 100 & 80 \\\\ 60 & 40 \\end{bmatrix} = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 160 & 120 \\\\ 40 & 40 \\end{bmatrix}$$<\/p>\n

**\ub2e8\uacc4 2**: $(T \\cdot I) \\cdot T^T$ \uacc4\uc0b0.<\/p>\n

$$F = \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 160 & 120 \\\\ 40 & 40 \\end{bmatrix} \\cdot \\frac{1}{\\sqrt{2}} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix} = \\frac{1}{2} \\begin{bmatrix} 160 & 120 \\\\ 40 & 40 \\end{bmatrix} \\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix}$$<\/p>\n

$$= \\frac{1}{2} \\begin{bmatrix} 280 & 40 \\\\ 80 & 0 \\end{bmatrix} = \\begin{bmatrix} 140 & 20 \\\\ 40 & 0 \\end{bmatrix}$$<\/p>\n

**\ub2e8\uacc4 3**: DCT \uacc4\uc218 \ud574\uc11d.<\/p>\n

- $F(0,0) = 140$: DC \uacc4\uc218, \uc774\ubbf8\uc9c0 \ube14\ub85d\uc758 \ud3c9\uade0 \ubc1d\uae30\uc5d0 \ub300\uc751. $(100+80+60+40)\/4 = 70$, $N = 2$\ub97c \uacf1\ud558\uba74 140.
\n- $F(0,1) = 20$: \uc218\ud3c9 \ubc29\ud5a5 \uace0\uc8fc\ud30c \uc131\ubd84, \uc88c\uc6b0 \ud53d\uc140 \ucc28\uc774\ub97c \ubc18\uc601.
\n- $F(1,0) = 40$: \uc218\uc9c1 \ubc29\ud5a5 \uace0\uc8fc\ud30c \uc131\ubd84, \uc0c1\ud558 \ud53d\uc140 \ucc28\uc774\ub97c \ubc18\uc601.
\n- $F(1,1) = 0$: \ub300\uac01\uc120 \ubc29\ud5a5 \uace0\uc8fc\ud30c \uc131\ubd84, 0\uc774\ubbc0\ub85c \ub300\uac01\uc120 \uc9c8\uac10\uc774 \uc5c6\uc74c\uc744 \uc758\ubbf8.<\/p>\n

**\ud575\uc2ec \uad00\ucc30**: $F(1,1) = 0$, \uc989 \ub300\uac01\uc120 \ubc29\ud5a5 \uace0\uc8fc\ud30c \uae30\uc800 \uc704\ub85c\uc758 \ud22c\uc601\uc774 0\uc774\ub2e4 \u2014 \uc774 \uc131\ubd84\uc740 \uc815\ubcf4 \uc190\uc2e4 \uc5c6\uc774 \uc644\uc804\ud788 \ubc84\ub9b4 \uc218 \uc788\ub2e4. \uc774\uac83\uc774 JPEG \uc555\ucd95\uc758 \ud575\uc2ec \uc6d0\ub9ac\uc774\ub2e4: \uc790\uc5f0 \uc774\ubbf8\uc9c0\uc758 \ub300\ubd80\ubd84\uc758 \uace0\uc8fc\ud30c DCT \uacc4\uc218\ub294 0\uc5d0 \uac00\uae5d\uace0, \uc591\uc790\ud654 \ud6c4 0\uc774 \ub418\uc5b4 \ud070 \ud3ed\uc758 \uc555\ucd95\uc774 \uac00\ub2a5\ud574\uc9c4\ub2e4.
\n```<\/p>\n

### 9.4 \uacf5\ud559 \ubc0f \ucd5c\ucca8\ub2e8 \uc751\uc6a9 (Engineering and Cutting-Edge Applications)<\/p>\n

JPEG \uc555\ucd95 \uacfc\uc815\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4:<\/p>\n

1. **\ubd84\ud560(Blocking)**: \uc774\ubbf8\uc9c0\ub97c $8 \\times 8$ \ube14\ub85d\uc73c\ub85c \ubd84\ud560;
\n2. **DCT \ubcc0\ud658**: \uac01 \ube14\ub85d\uc5d0 \ub300\ud574 2\ucc28\uc6d0 DCT\ub97c \uc218\ud589\ud558\uc5ec 64\uac1c\uc758 \uc8fc\ud30c\uc218 \uc601\uc5ed \uacc4\uc218\ub97c \uc5bb\uc74c;
\n3. **\uc591\uc790\ud654(Quantization)**: \uc591\uc790\ud654 \ud589\ub82c\ub85c DCT \uacc4\uc218\ub97c \ub098\ub204\uace0(\uace0\uc8fc\ud30c\uc77c\uc218\ub85d \uc591\uc790\ud654 \uc2a4\uac04\uc5d0\uc11c PCA\ub97c \uc218\ud589\ud558\uc5ec \ube44\uc120\ud615 \ucc28\uc6d0 \ucd95\uc18c\uc5d0 \uc0ac\uc6a9;
\n- **\ucee4\ub110 \ub9bf\uc9c0 \ud68c\uadc0(Kernel Ridge Regression)**: \uc120\ud615 \ub9bf\uc9c0 \ud68c\uadc0\ub97c \ube44\uc120\ud615 \ud68c\uadc0\ub85c \ud655\uc7a5;
\n- **\ucee4\ub110 \ud3c9\uade0 \ub9e4\uce6d(Kernel Mean Matching)**: \ub3c4\uba54\uc778 \uc801\uc751(Domain Adaptation) \ubc0f \uc804\uc774 \ud559\uc2b5(Transfer Learning)\uc5d0 \uc0ac\uc6a9;
\n- **\uac00\uc6b0\uc2dc\uc548 \ud504\ub85c\uc138\uc2a4(Gaussian Process)**: \ucee4\ub110 \ud568\uc218\ub97c \uacf5\ubd84\uc0b0 \ud568\uc218\ub85c \uc0ac\uc6a9\ud558\uc5ec \ubca0\uc774\uc9c0\uc548 \ucd5c\uc801\ud654 \ubc0f \ud68c\uadc0\uc5d0 \ud65c\uc6a9;
\n- **\uc2e0\uacbd \ud0c4\uc820\ud2b8 \ucee4\ub110(NTK, Neural Tangent Kernel)**: \ubb34\ud55c\ud788 \ub113\uc740 \uc2e0\uacbd\ub9dd\uacfc \ucee4\ub110 \ubc29\ubc95\uc744 \uc5f0\uacb0\ud558\uc5ec \ub525\ub7ec\ub2dd\uc5d0 \uc774\ub860\uc801 \ubd84\uc11d \ub3c4\uad6c\ub97c \uc81c\uacf5.<\/p>\n

---<\/p>\n

## \uc81c13\uc7a5 \uc591\uc790\uc5ed\ud559\uc5d0\uc11c\uc758 \ub0b4\uc801 \u2014 \ud655\ub960\uc740 \uace7 \ud22c\uc601 (Chapter 13 Inner Product in Quantum Mechanics \u2014 Probability Is Projection)<\/p>\n

### 13.1 \uc774\ub860\uacfc \uc5c4\ubc00\ud55c \uc815\uc758 (Theory and Rigorous Definitions)<\/p>\n

\uc591\uc790\uc5ed\ud559\uc740 \ub0b4\uc801\uc758 \uac1c\ub150\uc744 \ubb3c\ub9ac\uc801 \uc138\uacc4\uc758 \uad81\uadf9\uc801\uc778 \ucc28\uc6d0\uc73c\ub85c \ub04c\uc5b4\uc62c\ub9b0\ub2e4. \uc591\uc790\uc5ed\ud559\uc5d0\uc11c \uc2dc\uc2a4\ud15c\uc758 \uc0c1\ud0dc\ub294 \ud790\ubca0\ub974\ud2b8 \uacf5\uac04 $\\mathcal{H}$ \uc0c1\uc758 **\uc0c1\ud0dc \ubca1\ud130(State Vector)** $|\\psi\\rangle$\ub85c \uae30\uc220\ub41c\ub2e4(\ub514\ub799 \ud45c\uae30\ubc95, Dirac Notation)$^{[26]}$. \uc5ec\uae30\uc11c\uc758 \ud790\ubca0\ub974\ud2b8 \uacf5\uac04\uc740 \uc77c\ubc18\uc801\uc73c\ub85c \ubb34\ud55c\ucc28\uc6d0 \ubcf5\uc18c \ub0b4\uc801 \uacf5\uac04\uc774\ub2e4.<\/p>\n

```ad-definition
\ntitle: \uc815\uc758 13.1 \uc0c1\ud0dc \ubca1\ud130\uc640 \ub0b4\uc801 (Definition 13.1 State Vector and Inner Product)
\n\uc0c1\ud0dc \ubca1\ud130 $|\\psi\\rangle \\in \\mathcal{H}$\ub294 \uc591\uc790 \uc2dc\uc2a4\ud15c\uc758 \ubaa8\ub4e0 \uc815\ubcf4\ub97c \ud3ec\ud568\ud55c\ub2e4. \ub450 \uc0c1\ud0dc\uc758 \ub0b4\uc801 $\\langle \\phi | \\psi \\rangle$\uc740 \ubcf5\uc18c\uc218\uc774\uba70, \uadf8 \uc808\ub313\uac12\uc758 \uc81c\uacf1\uc774 \uce21\uc815 \ud655\ub960\uc744 \uacb0\uc815\ud55c\ub2e4.<\/p>\n

**\uacf5\ub9ac 13.1 (\ubcf4\ub978 \uaddc\uce59, Born Rule)** \uc2dc\uc2a4\ud15c\uc774 \uc0c1\ud0dc $|\\psi\\rangle$\uc5d0 \uc788\uc744 \ub54c, \uad00\uce21 \uac00\ub2a5\ub7c9 $\\hat{A}$\uc744 \uce21\uc815\ud558\uc5ec \uace0\uc720\uac12 $\\lambda_n$\uc744 \uc5bb\uc744 \ud655\ub960\uc740$^{[21]}$:<\/p>\n

$$P(\\lambda_n) = |\\langle a_n | \\psi \\rangle|^2 \\tag{13.1}$$<\/p>\n

\uc5ec\uae30\uc11c $|a_n\\rangle$\uc740 $\\hat{A}$\uc758 $\\lambda_n$\uc5d0 \ub300\uc751\ud558\ub294 \uace0\uc720 \uc0c1\ud0dc(Eigenstate)\uc774\ub2e4. \uce21\uc815 \ud6c4 \uc2dc\uc2a4\ud15c \uc0c1\ud0dc\ub294 $|a_n\\rangle$\uc73c\ub85c \ubd95\uad34(Collapse)\ud55c\ub2e4. \ubcf4\ub978 \uaddc\uce59\uc758 \ubcf8\uc9c8\uc740: **\ud655\ub960\uc740 \uce21\uc815 \uae30\uc800 \uc704\ub85c\uc758 \uc0c1\ud0dc \ubca1\ud130 \ud22c\uc601\uc758 \uc81c\uacf1 \ud06c\uae30\uc640 \uac19\ub2e4**\ub294 \uac83\uc774\ub2e4.
\n```<\/p>\n

```ad-definition
\ntitle: \uc815\uc758 13.2 \uad00\uce21 \uac00\ub2a5\ub7c9\uacfc \uc790\uae30 \uc218\ubc18 \uc5f0\uc0b0\uc790 (Definition 13.2 Observable and Self-Adjoint Operator)
\n\uad00\uce21 \uac00\ub2a5\ub7c9(Observable)\uc740 \ud790\ubca0\ub974\ud2b8 \uacf5\uac04 \uc0c1\uc758 **\uc790\uae30 \uc218\ubc18 \uc5f0\uc0b0\uc790(Hermitian Operator)** $\\hat{A}$\uc5d0 \ub300\uc751\ud558\uba70, $\\hat{A}^\\dagger = \\hat{A}$\ub97c \ub9cc\uc871\ud55c\ub2e4. \uc790\uae30 \uc218\ubc18 \uc5f0\uc0b0\uc790\uc758 \uace0\uc720\uac12\uc740 \uc2e4\uc218\uc774\uba70, \uace0\uc720 \uc0c1\ud0dc\ub294 \uc644\ube44 \uc9c1\uad50 \uae30\uc800\ub97c \uc774\ub8ec\ub2e4.
\n```<\/p>\n

```ad-definition
\ntitle: \uc815\uc758 13.3 \uc288\ub8b0\ub529\uac70 \ubc29\uc815\uc2dd (Definition 13.3 Schr\u00f6dinger Equation)
\n\uc0c1\ud0dc \ubca1\ud130\uc758 \uc2dc\uac04 \ubcc0\ud654\ub294 \uc288\ub8b0\ub529\uac70 \ubc29\uc815\uc2dd\uc73c\ub85c \uae30\uc220\ub41c\ub2e4:<\/p>\n

$$i\\hbar \\frac{d}{dt} |\\psi(t)\\rangle = \\hat{H} |\\psi(t)\\rangle \\tag{13.2}$$<\/p>\n

\uc5ec\uae30\uc11c $\\hat{H}$\ub294 \ud574\ubc00\ud1a0\ub2c8\uc548 \uc5f0\uc0b0\uc790(Hamiltonian, \uc5d0\ub108\uc9c0 \uc5f0\uc0b0\uc790)\uc774\ub2e4. \uc774 \ubc29\uc815\uc2dd\uc740 \ubcf8\uc9c8\uc801\uc73c\ub85c \ubb34\ud55c\ucc28\uc6d0 \ud790\ubca0\ub974\ud2b8 \uacf5\uac04\uc5d0\uc11c\uc758 \uc720\ub2c8\ud0c0\ub9ac \uc9c4\ud654 \ubc29\uc815\uc2dd(Unitary Evolution Equation)\uc774\ub2e4 \u2014 \ub0b4\uc801\uc744 \ubcf4\uc874\ud558\ub294 \ud68c\uc804.
\n```<\/p>\n

### 13.2 \uae30\ud558\ud559\uacfc \uacf5\uac04\uc801 \uc774\ubbf8\uc9c0 (Geometry and Spatial Intuition)<\/p>\n

\uc591\uc790\uc5ed\ud559\uc758 \uae30\ud558\ud559\uc801 \uc774\ubbf8\uc9c0\ub294 \uace0\uc804\uc801 \ub0b4\uc801 \uacf5\uac04\uacfc \uae4a\uc740 \uc5f0\uad00\uc774 \uc788\ub2e4:<\/p>\n

1. **\uc0c1\ud0dc \ubca1\ud130\ub294 \ub2e8\uc704 \ubca1\ud130**: \ubb3c\ub9ac\uc801\uc73c\ub85c $|\\psi\\rangle$\ub294 \uc815\uaddc\ud654\ub418\uc5b4\uc57c \ud558\uba70, \uc989 $\\langle \\psi | \\psi \\rangle = 1$\uc774\ub2e4. \ubaa8\ub4e0 \uac00\ub2a5\ud55c \uc0c1\ud0dc \ubca1\ud130\ub294 \ubcf5\uc18c \ud790\ubca0\ub974\ud2b8 \uacf5\uac04\uc758 \ub2e8\uc704 \uad6c\uba74\uc744 \uad6c\uc131\ud55c\ub2e4.<\/p>\n

2. **\uce21\uc815\uc740 \uc9c1\uad50 \ud22c\uc601**: \uce21\uc815 \uc5f0\uc0b0\uc740 \uc0c1\ud0dc \ubca1\ud130 $|\\psi\\rangle$\ub97c \uace0\uc720 \ubd80\ubd84 \uacf5\uac04\uc5d0 \ud22c\uc601\ud55c\ub2e4. \ud22c\uc601 \uae38\uc774 $|\\langle a_n | \\psi \\rangle|$\uac00 \ud655\ub960 \uc9c4\ud3ed(Probability Amplitude)\uc744 \uacb0\uc815\ud558\uba70, \uadf8 \uc81c\uacf1\uc774 \uce21\uc815 \ud655\ub960\uc774\ub2e4.<\/p>\n

3. **\uc9c1\uad50 \uc0c1\ud0dc\ub294 \uc0c1\ud638 \ubc30\ud0c0\uc801**: $\\langle \\phi | \\psi \\rangle = 0$\uc774\uba74 \ub450 \uc0c1\ud0dc\ub294 \uc9c1\uad50(\uc0c1\ud638 \ubc30\ud0c0\uc801)\ud55c\ub2e4 \u2014 \uc2dc\uc2a4\ud15c\uc774 $|\\psi\\rangle$\uc5d0 \uc788\uc744 \ub54c $|\\phi\\rangle$\ub97c \uce21\uc815\ud560 \ud655\ub960\uc740 0\uc774\ub2e4.<\/p>\n

4. **\uc5bd\ud798 \uc0c1\ud0dc\ub294 \ubd84\ud574 \ubd88\uac00\ub2a5**: \ubcf5\ud569 \uc2dc\uc2a4\ud15c\uc758 \uacbd\uc6b0 $|\\psi\\rangle_{AB} \\neq |\\phi\\rangle_A \\otimes |\\chi\\rangle_B$\uc774\uba74 \ub450 \ud558\uc704 \uc2dc\uc2a4\ud15c\uc740 \uc5bd\ud798 \uc0c1\ud0dc(Entangled State)\uc5d0 \uc788\ub2e4. \uc5bd\ud798 \uc0c1\ud0dc\uc758 \uc218\ud559\uc801 \ubcf8\uc9c8\uc740: \ub450 \ud558\uc704 \uc2dc\uc2a4\ud15c\uc758 \ub0b4\uc801 \uad6c\uc870\uac00 \uc9c1\uc801(Direct Product) \ud615\ud0dc\ub85c \ubd84\ud574\ub420 \uc218 \uc5c6\ub2e4\ub294 \uac83\uc774\ub2e4.<\/p>\n

### 13.3 \ud558\ub4dc\ucf54\uc5b4 \uc608\uc81c \uc0c1\uc138 \ud480\uc774 (Worked Example)<\/p>\n

```ad-example
\ntitle: \uc608\uc81c 13.1 \uc2a4\ud540 $1\/2$ \uc2dc\uc2a4\ud15c\uc758 \uce21\uc815 \ud655\ub960 \u2014 \ub0b4\uc801 \uacc4\uc0b0 (Example 13.1 Measurement Probability for a Spin-$1\/2$ System \u2014 Inner Product Calculation)<\/p>\n

\uc804\uc790 \uc2a4\ud540\uc744 \uace0\ub824\ud558\uc790. \uadf8 \uc0c1\ud0dc\ub294 2\ucc28\uc6d0 \ubcf5\uc18c \ud790\ubca0\ub974\ud2b8 \uacf5\uac04\uc758 \ubca1\ud130\ub85c \ud45c\ud604\ub420 \uc218 \uc788\ub2e4. \uc2a4\ud540 $z$ \ubc29\ud5a5 \uace0\uc720 \uc0c1\ud0dc:<\/p>\n

$$| \\uparrow_z \\rangle = \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}, \\quad | \\downarrow_z \\rangle = \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}$$<\/p>\n

\uc2a4\ud540 $x$ \ubc29\ud5a5 \uace0\uc720 \uc0c1\ud0dc:<\/p>\n

$$| \\uparrow_x \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix}, \\quad | \\downarrow_x \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ -1 \\end{pmatrix}$$<\/p>\n

\uc804\uc790\uac00 \uc0c1\ud0dc $|\\psi\\rangle = \\frac{1}{\\sqrt{2}}| \\uparrow_z \\rangle + \\frac{1}{\\sqrt{2}}| \\downarrow_z \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix}$\uc5d0 \uc788\ub2e4.<\/p>\n

**\ud480\uc774**:<\/p>\n

**\ub2e8\uacc4 1: \uc815\uaddc\ud654 \uac80\uc99d.**<\/p>\n

$$\\langle \\psi | \\psi \\rangle = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 & 1 \\end{pmatrix} \\cdot \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} = \\frac{1}{2}(1 + 1) = 1$$<\/p>\n

\uc815\uaddc\ud654\uac00 \ud655\uc778\ub418\uc5c8\ub2e4.<\/p>\n

**\ub2e8\uacc4 2: $S_z$ \uce21\uc815 \ud655\ub960.**<\/p>\n

$$P(\\uparrow_z) = |\\langle \\uparrow_z | \\psi \\rangle|^2 = \\left| \\begin{pmatrix} 1 & 0 \\end{pmatrix} \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\right|^2 = \\left| \\frac{1}{\\sqrt{2}} \\right|^2 = \\frac{1}{2}$$<\/p>\n

$$P(\\downarrow_z) = |\\langle \\downarrow_z | \\psi \\rangle|^2 = \\left| \\begin{pmatrix} 0 & 1 \\end{pmatrix} \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\right|^2 = \\left| \\frac{1}{\\sqrt{2}} \\right|^2 = \\frac{1}{2}$$<\/p>\n

\uac01\uac01 50%\ub85c \uc608\uc0c1\uacfc \uc77c\uce58\ud55c\ub2e4.<\/p>\n

**\ub2e8\uacc4 3: $S_x$ \uce21\uc815 \ud655\ub960.**<\/p>\n

$$P(\\uparrow_x) = |\\langle \\uparrow_x | \\psi \\rangle|^2 = \\left| \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 & 1 \\end{pmatrix} \\cdot \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\right|^2 = \\left| \\frac{1}{2}(1 + 1) \\right|^2 = 1$$<\/p>\n

$$P(\\downarrow_x) = |\\langle \\downarrow_x | \\psi \\rangle|^2 = \\left| \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 & -1 \\end{pmatrix} \\cdot \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\right|^2 = \\left| \\frac{1}{2}(1 - 1) \\right|^2 = 0$$<\/p>\n

**\ud575\uc2ec \uad00\ucc30**: $|\\psi\\rangle = | \\uparrow_x \\rangle$\uc774\ubbc0\ub85c $S_x$ \uce21\uc815 \uc2dc 100% $+\\hbar\/2$\ub97c \uc5bb\ub294\ub2e4. \uc774\ub294 \ub0b4\uc801\uc758 \uae30\ud558\ud559\uc801 \uc758\ubbf8\ub97c \uac80\uc99d\ud55c\ub2e4: \uc0c1\ud0dc \ubca1\ud130\uac00 \uc644\uc804\ud788 \uc815\ub82c\ub418\uba74(\ub0b4\uc801 \ud06c\uae30\uac00 1) \ud655\ub960\uc740 100%; \uc9c1\uad50\ud558\uba74(\ub0b4\uc801\uc774 0) \ud655\ub960\uc740 0.<\/p>\n

**\ub2e8\uacc4 4: \uce21\uc815 \ud6c4 \uc0c1\ud0dc \ubd95\uad34.** $S_z$\ub97c \uce21\uc815\ud558\uc5ec $+\\hbar\/2$\ub97c \uc5bb\uc5c8\ub2e4\uace0 \uac00\uc815\ud558\uba74, \uc0c1\ud0dc \ubca1\ud130\ub294 \ubd95\uad34\ud55c\ub2e4:<\/p>\n

$$|\\psi\\rangle = \\frac{1}{\\sqrt{2}}| \\uparrow_z \\rangle + \\frac{1}{\\sqrt{2}}| \\downarrow_z \\rangle \\xrightarrow{\\text{\uce21\uc815 } S_z = +\\hbar\/2} |\\psi'\\rangle = | \\uparrow_z \\rangle$$<\/p>\n

\uc774\ub54c \ub2e4\uc2dc $S_z$\ub97c \uce21\uc815\ud558\uba74 100% $+\\hbar\/2$\ub97c \uc5bb\uc9c0\ub9cc, $S_x$\ub97c \uce21\uc815\ud558\uba74 \ub2e4\uc2dc 50\/50 \ud655\ub960\ub85c \ub3cc\uc544\uac04\ub2e4. \uc774\uac83\uc774 \"\uce21\uc815\uc774 \uc0c1\ud0dc\ub97c \ubcc0\ud654\uc2dc\ud0a8\ub2e4\"\ub294 \ubcf8\uc9c8\uc774\ub2e4 \u2014 \uc9c1\uad50 \ud22c\uc601 \uc5f0\uc0b0.
\n```<\/p>\n

### 13.4 \uacf5\ud559 \ubc0f \ucd5c\ucca8\ub2e8 \uc751\uc6a9 (Engineering and Cutting-Edge Applications)<\/p>\n

\uc591\uc790 \ub0b4\uc801\uc758 \uac1c\ub150\uc740 \ud601\uba85\uc801\uc778 \uae30\uc220\uc744 \ub0b3\uace0 \uc788\ub2e4:<\/p>\n

- **\uc591\uc790 \ucef4\ud4e8\ud305(Quantum Computing)**: \uc591\uc790 \uac8c\uc774\ud2b8 \uc5f0\uc0b0\uc740 \ubcf8\uc9c8\uc801\uc73c\ub85c \ud790\ubca0\ub974\ud2b8 \uacf5\uac04\uc5d0\uc11c\uc758 \uc720\ub2c8\ud0c0\ub9ac \ubcc0\ud658(\ub0b4\uc801 \ubcf4\uc874 \ud68c\uc804)\uc774\ub2e4. \uc1fc\uc5b4 \uc54c\uace0\ub9ac\uc998(Shor's Algorithm)\uacfc \uadf8\ub85c\ubc84 \uc54c\uace0\ub9ac\uc998(Grover's Algorithm)\uc740 \uc591\uc790 \uc0c1\ud0dc\uc758 \uc911\ucca9(Superposition)\uacfc \uac04\uc12d(Interference, \ub0b4\uc801\uc758 \uc704\uc0c1)\uc744 \ud65c\uc6a9\ud558\uc5ec \uc9c0\uc218\uc801 \uac00\uc18d\uc744 \ub2ec\uc131;
\n- **\uc591\uc790 \uc554\ud638\ud559(Quantum Cryptography)**: BB84 \ud504\ub85c\ud1a0\ucf5c\uc740 \uce21\uc815 \uae30\uc800\uc758 \uc9c1\uad50\uc131\uc744 \uc774\uc6a9\ud558\uc5ec \ub3c4\uccad\uc744 \ud0d0\uc9c0 \u2014 \ub3c4\uccad\uc790\uc758 \uce21\uc815\uc774 \uc0c1\ud0dc \ubca1\ud130\ub97c \ubd95\uad34\uc2dc\ucf1c \ub0b4\uc801 \uacb0\uacfc\ub97c \ubcc0\uacbd\ud558\ubbc0\ub85c \ud569\ubc95\uc801\uc778 \ud1b5\uc2e0\uc790\uac00 \ubc1c\uacac\ud560 \uc218 \uc788\uc74c;
\n- **\uc591\uc790 \ud154\ub808\ud3ec\ud14c\uc774\uc158(Quantum Teleportation)**: Bell \uc0c1\ud0dc(\ucd5c\ub300 \uc5bd\ud798 \uc0c1\ud0dc)\uc758 \ub0b4\uc801 \uad6c\uc870\ub97c \uc774\uc6a9\ud558\uc5ec \uc591\uc790 \uc815\ubcf4\uc758 \uc6d0\uaca9 \uc804\uc1a1\uc744 \uc2e4\ud604;
\n- **\uc591\uc790 \uba38\uc2e0\ub7ec\ub2dd(Quantum Machine Learning)**: \uc591\uc790 \ucee4\ub110 \ubc29\ubc95\uc740 \uc591\uc790 \uc0c1\ud0dc \ub0b4\uc801\uc744 \uc774\uc6a9\ud558\uc5ec \uace0\ucc28\uc6d0 \ud790\ubca0\ub974\ud2b8 \uacf5\uac04\uc5d0\uc11c \ucee4\ub110 \ud568\uc218\ub97c \ud6a8\uc728\uc801\uc73c\ub85c \uacc4\uc0b0\ud558\uc5ec \uc591\uc790 \uc6b0\uc704(Quantum Advantage)\ub97c \uc2e4\ud604\ud560 \uac83\uc73c\ub85c \uae30\ub300.<\/p>\n

---<\/p>\n

## \uc885\uc7a5 \ub300\ud1b5\uc77c \uc9c0\uc2dd \uc9c0\ub3c4\uc640 \ucca0\ud559\uc801 \uc2b9\ud654 (Final Chapter: Grand Unified Knowledge Graph and Philosophical Sublimation)<\/p>\n

### \ub9cc\ubb3c\uc740 \ud22c\uc601\uc774\ub2e4 \u2014 \ubaa8\ub4e0 \ud559\ubb38\uc744 \uad00\ud1b5\ud558\ub294 \ub0b4\uc801 \uc9c0\ub3c4 (Everything Is a Projection \u2014 An Inner Product Map Across All Disciplines)<\/p>\n

\uc9c0\uae08\uae4c\uc9c0 \uad6c\ucd95\ud55c \uc9c0\uc2dd \uccb4\uacc4\ub97c \ub418\ub3cc\uc544\ubcf4\uba74, 2\ucc28\uc6d0 \ubca1\ud130\uc758 \uc810\uacf1\uc5d0\uc11c \ubb34\ud55c\ucc28\uc6d0 \ubcf5\uc18c \ud790\ubca0\ub974\ud2b8 \uacf5\uac04\uc758 \uc0c1\ud0dc \ubca1\ud130 \ub0b4\uc801\uae4c\uc9c0, \ub0b4\uc801 \uac1c\ub150\uc740 \uc218\ud559, \ubb3c\ub9ac\ud559, \uacf5\ud559, \ucef4\ud4e8\ud130 \uacfc\ud559\uc758 \ubaa8\ub4e0 \uad6c\uc11d\uc744 \uad00\ud1b5\ud55c\ub2e4.<\/p>\n

**\ud575\uc2ec \uc8fc\uc81c**: \ub0b4\uc801 $\\langle \\cdot, \\cdot \\rangle$\uc740 **\uc720\uc0ac\ub3c4 \uce21\uc815(Similarity Measure)** \uc774\ub2e4. \uac1d\uccb4\uac00 \ubca1\ud130, \ud568\uc218, \uc2e0\ud638, \uc774\ubbf8\uc9c0, \uc591\uc790 \uc0c1\ud0dc \uc911 \ubb34\uc5c7\uc774\ub4e0, \ub0b4\uc801\uc740 \ub3d9\uc77c\ud55c \uc9c8\ubb38\uc5d0 \ub2f5\ud55c\ub2e4 \u2014 \"\uc774 \ub450 \uac1d\uccb4\ub294 \uc5bc\ub9c8\ub098 \uc720\uc0ac\ud55c\uac00?\"<\/p>\n

**\ub300\ud1b5\uc77c \uc9c0\uc2dd \uc9c0\ub3c4 (Grand Unified Knowledge Map)**:<\/p>\n

| \ubd84\uc57c | \ub0b4\uc801\uc758 \uad6c\uccb4\uc801 \ud615\ud0dc | \uae30\ud558\ud559\uc801 \ud574\uc11d | \ud575\uc2ec \uc751\uc6a9 |
\n|------|-------------------|-------------|---------|
\n| \uc120\ud615\ub300\uc218\ud559 | $\\langle x, y \\rangle = x^T y$ | \ud22c\uc601 \uae38\uc774 | \uc9c1\uad50 \ubd84\ud574, \ucd5c\uc18c\uc81c\uacf1 |
\n| \ud568\uc218\ud574\uc11d\ud559 | $\\langle f, g \\rangle = \\int fg$ | \ud30c\ud615 \uc720\uc0ac\ub3c4 | \ud478\ub9ac\uc5d0 \uae09\uc218, \uc6e8\uc774\ube14\ub9bf \ubcc0\ud658 |
\n| \uc2e0\ud638\ucc98\ub9ac | $\\langle x, h \\rangle = \\sum x[n]h[n]$ | \uc815\ud569 \ud544\ud130 | \ucee8\ubcfc\ub8e8\uc158, \uc0c1\uad00 \uac80\ucd9c |
\n| \ud655\ub960\ud1b5\uacc4 | $\\text{Cov}(X,Y) = E[(X-\\mu_X)(Y-\\mu_Y)]$ | \uc0c1\uad00 \ubc29\ud5a5 | PCA, \ud68c\uadc0 \ubd84\uc11d |
\n| \uba38\uc2e0\ub7ec\ub2dd | $\\langle Q_i, K_j \\rangle$ | \uc5b4\ud150\uc158 \uac00\uc911\uce58 | Transformer, \uc790\uae30 \uc8fc\uc758 |
\n| \uc774\ubbf8\uc9c0 \ucc98\ub9ac | $\\langle I, K \\rangle$ | \ud2b9\uc9d5 \uc751\ub2f5 | CNN, \uc5d0\uc9c0 \uac80\ucd9c |
\n| \uc591\uc790\uc5ed\ud559 | $\\langle \\phi \\mid \\psi \\rangle$ | \ud655\ub960 \uc9c4\ud3ed | \uce21\uc815, \uc591\uc790 \ucef4\ud4e8\ud305 |
\n| \uc81c\uc5b4 \uc774\ub860 | $\\langle f, e^{-st} \\rangle$ | \ubcf5\uc18c \uc8fc\ud30c\uc218 \uc601\uc5ed \ud22c\uc601 | \ub77c\ud50c\ub77c\uc2a4 \ubcc0\ud658, \uc548\uc815\uc131 \ubd84\uc11d |<\/p>\n

### \ucca0\ud559\uc801 \uc2b9\ud654 \u2014 \ud22c\uc601\uc740 \uace7 \uc778\uc2dd (Philosophical Sublimation \u2014 Projection Is Cognition)<\/p>\n

\ucca0\ud559\uc801 \ucc28\uc6d0\uc5d0\uc11c \"\ub9cc\ubb3c\uc740 \ud22c\uc601\uc774\ub2e4\"\ub294 \ub2e8\uc21c\ud55c \uc218\ud559\uc801 \uba85\uc81c\uac00 \uc544\ub2c8\ub77c, \uc138\uc0c1\uc744 \uc778\uc2dd\ud558\ub294 \ubc29\uc2dd\uc774\ub2e4$^{[22]}$:<\/p>\n

1. **\uc778\uc2dd\uc740 \ud22c\uc601\uc774\ub2e4**: \uc778\uac04\uc774 \uc138\uc0c1\uc744 \uc778\uc2dd\ud558\ub294 \uacfc\uc815\uc740 \ubcf8\uc9c8\uc801\uc73c\ub85c \uc678\ubd80 \uc138\uacc4\uc758 \ubcf5\uc7a1\ud55c \uc815\ubcf4\ub97c \uc720\ud55c\ud55c \uc778\uc2dd \uae30\uc800(Cognitive Basis)\uc5d0 \ud22c\uc601\ud558\ub294 \uac83\uc774\ub2e4. \uc6b0\ub9ac\uac00 \ubcf4\ub294 \uac83\uc740 \"\uc2e4\uc81c \uc138\uacc4 \uadf8 \uc790\uccb4\"\uac00 \uc544\ub2c8\ub77c, \uc778\uc2dd \uae30\uc800 \uc704\ub85c\uc758 \uc2e4\uc81c \uc138\uacc4\uc758 \ud22c\uc601 \uacc4\uc218\uc774\ub2e4.<\/p>\n

2. **\uc9c1\uad50\ub294 \ub3c5\ub9bd\uc774\ub2e4**: \ub450 \uac1c\ub150\uc774 \uc9c1\uad50\ud560 \ub54c, \uc774\ub4e4\uc740 \uc11c\ub85c \uac04\uc12d\ud558\uc9c0 \uc54a\uace0 \uc911\ucca9\ub418\uc9c0 \uc54a\uc74c\uc744 \uc758\ubbf8\ud55c\ub2e4. \uc9c1\uad50 \ubd84\ud574\ub294 \ubcf5\uc7a1\ud55c \ubb38\uc81c\ub97c \ub2e8\uc21c\ud654\ud558\ub294 \uad81\uadf9\uc801\uc778 \ubb34\uae30\uc774\ub2e4 \u2014 \ubcf5\uc7a1\ud55c \uc2dc\uc2a4\ud15c\uc744 \uc0c1\ud638 \uad00\ub828 \uc5c6\ub294 \ub3c5\ub9bd \ubaa8\ub4c8\ub85c \ubd84\ud574\ud558\ub294 \uac83.<\/p>\n

3. **\ud22c\uc601\uc740 \uacb0\uc815\uc774\ub2e4**: \ucd5c\uc18c\uc81c\uacf1\ubc95\uc740 \uc815\ud655\ud55c \ud574\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\uc744 \ub54c \ud22c\uc601\uc744 \uad6c\ud558\ub294 \uac83\uc774 \ucd5c\uc801\uc758 \uc120\ud0dd\uc784\uc744 \ubcf4\uc5ec\uc900\ub2e4. \uc644\ubcbd\ud55c \ud574\uacb0\ucc45\uc774 \ubd88\uac00\ub2a5\ud560 \ub54c, \uc2e4\ud589 \uac00\ub2a5 \uc601\uc5ed \uc704\ub85c\uc758 \uc9c1\uad50 \ud22c\uc601\uc774 \ucd5c\uc801\uc758 \uacb0\uc815\uc774\ub2e4.<\/p>\n

4. **\uae30\uc800\uc758 \uc120\ud0dd\uc774 \ubaa8\ub4e0 \uac83\uc744 \uacb0\uc815\ud55c\ub2e4**: \ud478\ub9ac\uc5d0\uac00 \uc0ac\uc778\ud30c\ub97c \uae30\uc800\ub85c \uc120\ud0dd\ud558\uace0, \uc6e8\uc774\ube14\ub9bf\uc774 \ucef4\ud329\ud2b8 \uc11c\ud3ec\ud2b8 \ud568\uc218\ub97c \uae30\uc800\ub85c \uc120\ud0dd\ud558\uba70, Transformer\uac00 \ud559\uc2b5 \uac00\ub2a5\ud55c \uc5b4\ud150\uc158 \uae30\uc800\ub97c \uc120\ud0dd\ud55c\ub2e4 \u2014 \uc5b4\ub5a4 \uae30\uc800\ub97c \uc120\ud0dd\ud558\ub290\ub0d0\uc5d0 \ub530\ub77c \uc5b4\ub5a4 \uc138\uc0c1\uc744 \ubcfc \uc218 \uc788\ub294\uc9c0\uac00 \uacb0\uc815\ub41c\ub2e4.<\/p>\n

### \uc885\uad6d\uc801 \uc0ac\uc0c9 (Final Reflection)<\/p>\n

\ub0b4\uc801\uc740 \ub2e8\uc21c\ud55c \uc218\ud559 \uc5f0\uc0b0\uc774 \uc544\ub2c8\ub77c, \ubbf8\uc2dc\uc640 \uac70\uc2dc, \uc5f0\uc18d\uacfc \uc774\uc0b0, \uacb0\uc815\ub860\uacfc \ud655\ub960\ub860\uc744 \uc5f0\uacb0\ud558\ub294 **\uba54\ud0c0 \uc5b8\uc5b4(Meta-Language)** \uc774\ub2e4. \ud53c\ud0c0\uace0\ub77c\uc2a4 \uc815\ub9ac\uc5d0\uc11c \uc591\uc790 \uc5bd\ud798\uae4c\uc9c0, \ucd5c\uc18c\uc81c\uacf1\ubc95\uc5d0\uc11c \ub300\uaddc\ubaa8 \uc5b8\uc5b4 \ubaa8\ub378\uae4c\uc9c0, \ub0b4\uc801\uc740 \uadf8 \uac04\uacb0\ud558\uace0\ub3c4 \uc2ec\uc624\ud55c \ud615\ud0dc\ub85c \uc778\uac04 \uc9c0\uc2dd \uccb4\uacc4\uc758 \ubaa8\ub4e0 \uad6c\uc11d\uc744 \ud1b5\uc77c\ud55c\ub2e4.<\/p>\n

---<\/p>\n

## \ubd80\ub85d \ubcf8\ubb38 \uadf8\ub9bc \uc0dd\uc131 \ucf54\ub4dc (Appendix: Code for Generating Figures in This Paper)<\/p>\n

\ubcf8\ubb38\uc758 \ub2e4\uc12f \uac00\uc9c0 \uadf8\ub9bc(\ucf54\uc0ac\uc778 \uc720\uc0ac\ub3c4 \ud788\ud2b8\ub9f5, \ucd5c\uc18c\uc81c\uacf1 \ud22c\uc601, \ud478\ub9ac\uc5d0 \ubd84\ud574, \ucee8\ubcfc\ub8e8\uc158 \uc815\ud569 \ud544\ud130, Sobel \uc5d0\uc9c0 \uac80\ucd9c)\uc740 \ubaa8\ub450 main.py<\/a>\uc5d0\uc11c \ud1b5\ud569 \uc0dd\uc131\ub41c\ub2e4. \uc774 \uc2a4\ud06c\ub9bd\ud2b8\ub294 Python\uc758 \uacfc\ud559 \uacc4\uc0b0 \uc0dd\ud0dc\uacc4(NumPy, SciPy, Matplotlib)\ub97c \uae30\ubc18\uc73c\ub85c \ud558\uba70, \"\ub0b4\uc801\"\uc774\ub77c\ub294 \ud575\uc2ec \uc8fc\uc81c\ub97c \uc911\uc2ec\uc73c\ub85c \ubcf8\ubb38\uc758 \ucd94\uc0c1\uc801\uc778 \uc218\ud559 \uac1c\ub150\uc744 \uc9c1\uad00\uc801\uc778 \uc2dc\uac01\ud654 \uadf8\ub798\ud53d\uc73c\ub85c \ubcc0\ud658\ud55c\ub2e4.<\/p>\n

\uc2a4\ud06c\ub9bd\ud2b8\uc758 \ud575\uc2ec \uc124\uacc4\u601d\u8def\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4:<\/p>\n

1. **\ucf54\uc0ac\uc778 \uc720\uc0ac\ub3c4**: `cosine_similarity()` \ud568\uc218\ub97c \ud1b5\ud574 \ub2e8\uc5b4 \uc784\ubca0\ub529 \ubca1\ud130 \uac04\uc758 \uc815\uaddc\ud654\ub41c \ub0b4\uc801\uc744 \uacc4\uc0b0\ud558\uc5ec $5 \\times 5$ \ud788\ud2b8\ub9f5 \ud589\ub82c\uc744 \uc0dd\uc131. \uc774 \ud568\uc218\ub294 \uacf5\uc2dd (1.5)\uc758 \ucf54\uc0ac\uc778 \uc720\uc0ac\ub3c4 \uc815\uc758\ub97c \uad6c\ud604\ud55c\ub2e4.
\n2. **\ucd5c\uc18c\uc81c\uacf1\ubc95**: `np.linalg.lstsq`\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc815\uaddc \ubc29\uc815\uc2dd $A^T A \\hat{x} = A^T b$(\uc815\ub9ac 3.1)\ub97c \ud480\uba70, \ubcf8\uc9c8\uc801\uc73c\ub85c \uad00\uce21 \ubca1\ud130\ub97c \ubaa8\ub378 \uacf5\uac04\uc5d0 \uc9c1\uad50 \ud22c\uc601\ud55c\ub2e4.
\n3. **\ud478\ub9ac\uc5d0 \ubd84\ud574**: FFT\ub97c \ud1b5\ud574 \uc2dc\uac04 \uc601\uc5ed \uc2e0\ud638\ub97c \uc8fc\ud30c\uc218 \uae30\uc800\uc5d0 \ud22c\uc601\ud558\uba70(\uc815\ub9ac 6.1), \uc2a4\ud399\ud2b8\ub7fc\uc758 \uac01 \ud53c\ud06c\ub294 \uc8fc\ud30c\uc218 \uc131\ubd84\uc758 \ub0b4\uc801 \uacc4\uc218\uc5d0 \ub300\uc751\ud55c\ub2e4.
\n4. **\ucee8\ubcfc\ub8e8\uc158\uacfc \uc815\ud569 \ud544\ud130**: \ucee8\ubcfc\ub8e8\uc158\uc744 \uc2ac\ub77c\uc774\ub529 \ub0b4\uc801 \uc5f0\uc0b0(\uc815\uc758 8.1)\uc73c\ub85c \uac04\uc8fc\ud558\uace0, \ud15c\ud50c\ub9bf\uacfc \uc2e0\ud638\uc758 \uc810\ubcc4 \ub0b4\uc801\uc73c\ub85c \ud384\uc2a4 \uc704\uce58\ub97c \uac80\ucd9c\ud55c\ub2e4.
\n5. **Sobel \uc5d0\uc9c0 \uac80\ucd9c**: 2\ucc28\uc6d0 \ucee8\ubcfc\ub8e8\uc158 \ucee4\ub110\uacfc \uc774\ubbf8\uc9c0\uc758 \ub0b4\uc801(\uc608\uc81c 8.2)\uc744 \uacc4\uc0b0\ud558\uc5ec \uac01 \ud53d\uc140\uc5d0\uc11c\uc758 \uadf8\ub77c\ub370\uc774\uc158 \ud06c\uae30\ub97c \uad6c\ud55c\ub2e4.<\/p>\n

\ub2e4\uc74c\uc740 \uc2a4\ud06c\ub9bd\ud2b8\uc5d0\uc11c \ucf54\uc0ac\uc778 \uc720\uc0ac\ub3c4 \ud788\ud2b8\ub9f5\uc744 \uc0dd\uc131\ud558\ub294 \ud575\uc2ec \ucf54\ub4dc \uc870\uac01\uc774\ub2e4:<\/p>\n

def cosine_similarity(vec_a: np.ndarray, vec_b: np.ndarray) -> float:\r\n    dot_product = float(np.dot(vec_a, vec_b))\r\n    norm_a = np.linalg.norm(vec_a)\r\n    norm_b = np.linalg.norm(vec_b)\r\n    return dot_product \/ (norm_a * norm_b)\r\n\r\ndef build_semantic_demo() -> tuple[list[str], dict[str, np.ndarray], np.ndarray]:\r\n    tokens = [\"king\", \"queen\", \"man\", \"woman\", \"apple\"]\r\n    embeddings = {\r\n        \"king\": np.array([0.92, 0.10, 0.78, 0.25, 0.60]),\r\n        \"queen\": np.array([0.90, 0.12, 0.80, 0.30, 0.63]),\r\n        \"man\": np.array([0.88, 0.18, 0.40, 0.22, 0.35]),\r\n        \"woman\": np.array([0.86, 0.22, 0.42, 0.28, 0.38]),\r\n        \"apple\": np.array([0.05, 0.95, 0.08, 0.87, 0.10]),\r\n    }\r\n    matrix = np.array(\r\n        [[cosine_similarity(embeddings[left], embeddings[right]) for right in tokens] for left in tokens]\r\n    )\r\n    return tokens, embeddings, matrix\r\n<\/code><\/pre>\n
\uc804\uccb4 \ucf54\ub4dc\ub294 main.py<\/a>\ub97c \ucc38\uc870\ud558\ub77c.<\/p>\n
## \ucc38\uace0 \ubb38\ud5cc (References)<\/p>\n
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[7] \u5377\u79ef\u3001\u5185\u79ef\u3001\u4e92\u76f8\u5173\u6982\u5ff5. CSDN\u535a\u5ba2, 2024. https:\/\/blog.csdn.net\/qq_31073871\/article\/details\/146475191<\/a>.<\/p>\n
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[9] \u5185\u79ef\u548c\u5916\u79ef[G\/OL]. OI Wiki, 2025. https:\/\/oi-wiki.org\/math\/linear-algebra\/product\/<\/a>.<\/p>\n
[10] \u7ef4\u57fa\u767e\u79d1\u7f16\u8005. \u5185\u79ef[G\/OL]. \u7ef4\u57fa\u767e\u79d1, 2025(20250703)[2025-07-03]. https:\/\/zh.wikipedia.org\/w\/index.php?title=%E5%86%85%E7%A7%AF&oldid=88045564<\/a>.<\/p>\n
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[26] Wikipedia contributors. (2026, May 22). Quantum mechanics. In _Wikipedia, The Free Encyclopedia_. Retrieved 12:01, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Quantum_mechanics&oldid=1355584024<\/a>.<\/p>\n
[27] Wikipedia contributors. (2026, May 20). Uncertainty principle. In _Wikipedia, The Free Encyclopedia_. Retrieved 12:01, May 24, 2026, from https:\/\/en.wikipedia.org\/w\/index.php?title=Uncertainty_principle&oldid=1355179215<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"\uc810\uacf1\uc5d0\uc11c \ub0b4\uc801 \uacf5\uac04\uae4c\uc9c0: \uc120\ud615\ub300\uc218\ud559, \uc2e0\ud638\ucc98\ub9ac, AI \ub4a4\uc5d0 \uc228\uc740 \ub3d9\uc77c\ud55c \uc5b8\uc5b4 (From Dot Product to Inner Product Space: The Unified Language Behind Linear Algebra, Signals, and AI)\r\n\r\n\uc694\uc57d (Abstract)\r\n\r\n**\ub0b4\uc801(Inner Product)** \uc740 \uc120\ud615\ub300\uc218\ud559, \ud568\uc218\ud574\uc11d\ud559, \uc2e0\ud638\ucc98\ub9ac, \uba38\uc2e0\ub7ec\ub2dd, \uc591\uc790\uc5ed\ud559\uc5d0 \uac78\uccd0 \uacf5\uc720\ub418\ub294 \ud575\uc2ec \ub300\uc218 \uad6c\uc870\uc774\ub2e4. \ubcf8 \ub17c\ubb38\uc740 \"\ub0b4\uc801\"\uc744 \uc720\uc77c\ud55c \uc8fc\uc81c\ub85c \uc0bc\uc544, \uc720\ud55c\ucc28\uc6d0 \uc720\ud074\ub9ac\ub4dc \uacf5\uac04\uc5d0\uc11c\uc758 \uc810\uacf1(Dot Product)\uc5d0\uc11c \ucd9c\ubc1c\ud558\uc5ec \ub0b4\uc801 \uacf5\uac04 \uacf5\ub9ac, \uc9c1\uad50 \ubd84\ud574(Orthogonal Decomposition), \ucd5c\uc18c\uc81c\uacf1 \ud22c\uc601(Least-Squares Projection), \ud790\ubca0\ub974\ud2b8 \uacf5\uac04(Hilbert Space), \ud478\ub9ac\uc5d0 \uae09\uc218\uc640 \ubcc0\ud658(Fourier Series and Transform), \ucee8\ubcfc\ub8e8\uc158(Convolution), \uc774\uc0b0 \ucf54\uc0ac\uc778 \ubcc0\ud658(Discrete Cosine Transform), \uc6e8\uc774\ube14\ub9bf \ubd84\uc11d(Wavelet Analysis), \uc790\uae30 \uc8fc\uc758 \uba54\ucee4\ub2c8\uc998(Self-Attention Mechanism), \ucee4\ub110 \ubc29\ubc95(Kernel Method), \uadf8\ub9ac\uace0 \uc591\uc790\uc5ed\ud559\uc5d0\uc11c\uc758 \uc0c1\ud0dc \ubca1\ud130 \ud22c\uc601(St...","protected":false},"author":1,"featured_media":415,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"emotion":"","emotion_color":"","title_style":"","license":"","footnotes":""},"categories":[14,85],"tags":[],"class_list":["post-593","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-14","category-85"],"_links":{"self":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/posts\/593","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/comments?post=593"}],"version-history":[{"count":2,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/posts\/593\/revisions"}],"predecessor-version":[{"id":601,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/posts\/593\/revisions\/601"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/media\/415"}],"wp:attachment":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/media?parent=593"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/categories?post=593"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/tags?post=593"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}