# 3. \u77e9\u9635 (Matrices)<\/p>\n
## 3.0 \u5f15\u8a00\uff1a\u77e9\u9635\u7684\u8fd0\u7528 (Introduction: Matrices in Action)<\/p>\n
\u5728\u4e0a\u4e00\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u589e\u5e7f\u77e9\u9635 ($\\text{Augmented Matrix}$) \u6765\u8bb0\u5f55\u7ebf\u6027\u65b9\u7a0b\u7ec4 $\\text{(System of Linear Equations)}$ \u7684\u4fe1\u606f\uff0c\u5e76\u5e2e\u52a9\u7b80\u5316\u6d89\u53ca\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u8ba1\u7b97\u3002\u5728\u672c\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u4e86\u89e3\u5230\u77e9\u9635\u672c\u8eab\u7684\u4ee3\u6570\u6027\u8d28\uff0c\u4ece\u51fd\u6570\u5f15\u5165\u77e9\u9635\uff0c\u4ece\u4ee3\u6570\u5f15\u5165\u77e9\u9635\u64cd\u4f5c\uff0c\u4e86\u89e3\u77e9\u9635\u53d8\u6362\u7684\u672c\u8d28\u3002<\/p>\n
\u5728\u9ad8\u7b49\u6570\u5b66\u4e2d\uff0c\u6211\u4eec\u5df2\u7ecf\u5b66\u4e60\u5230\u4e86\uff0c\u51fd\u6570\u662f\u5bf9\u4e8e\u7a7a\u95f4\u4e0a\u70b9\u7684\u6620\u5c04 $\\text{(Mapping)}$ \u64cd\u4f5c\uff0c\u6211\u4eec\u5b66\u4e60\u5230\u4e86\u4e00\u5143\u51fd\u6570\uff0c\u4ee5\u53ca\u591a\u5143\u51fd\u6570\u3002\u8003\u8651\u5982\u4e0b\u4e8c\u5143\u51fd\u6570
\n$$f(x_1,x_2) = x_1 + 2x_2$$<\/p>\n
\u4e0d\u96be\u770b\u51fa\uff0c\u8fd9\u662f\u5bf9\u4e8e\u5b9e\u6570\u57df $\\mathbb{R}^2$ \u5230\u53e6\u4e00\u4e2a\u5b9e\u6570\u57df $\\mathbb{R}$ \u7684\u6620\u5c04\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u70b9 $X = (-2,1)$ \uff0c\u7ecf\u8fc7\u51fd\u6570 $f$ \u7684\u53d8\u6362\u540e\uff0c\u5f97\u5230\u503c $Y = 0$\u3002<\/p>\n
\u8003\u8651\u4ee5\u4e0b\u65b9\u7a0b
\n$$\\begin{cases}
\ny_1 = x_1 + 2x_2 \\\\
\ny_2 = \\phantom{x_1 + {}} 3x_2
\n\\end{cases}$$<\/p>\n
\u6211\u4eec\u53ef\u4ee5\u5c06\u8fd9\u7ec4\u65b9\u7a0b\uff0c\u89c6\u4e3a\u63cf\u8ff0\u5411\u91cf $\\vec{x} = \\begin{bmatrix}x_1\\\\x_2\\end{bmatrix}$ \u5230 $\\vec{y} = \\begin{bmatrix}y_1\\\\y_2\\end{bmatrix}$ \u7684\u53d8\u6362\u3002\u53ef\u8bb0\u4e3a<\/p>\n
$$\\begin{bmatrix}y_1\\\\y_2\\end{bmatrix} = \\begin{bmatrix}1&2\\\\0&3\\end{bmatrix}\\begin{bmatrix}x_1\\\\x_2\\end{bmatrix}$$<\/p>\n
\u6545\u800c\uff0c\u51fd\u6570\u662f\u70b9\u5bf9\u70b9\u7684\u6620\u5c04\uff0c\u6216\u7406\u89e3\u4e3a\u6807\u91cf $\\text{(Scalar)}$ \u5bf9\u6807\u91cf\u7684\u6620\u5c04\u3002\u800c\u77e9\u9635\u5219\u53ef\u4ee5\u7406\u89e3\u4e3a\uff0c\u5411\u91cf $\\text{(Vector)}$ \u5230\u5411\u91cf\u7684\u6620\u5c04\u3002<\/p>\n
\u4eff\u7167\u51fd\u6570\uff0c\u73b0\u5728\u8ba9\u6211\u4eec\u6765\u5bf9\u4e00\u4e2a\u5411\u91cf\u8fdb\u884c\u4e00\u6b21\u77e9\u9635\u64cd\u4f5c $\\text{(Matrix Operation)}$ \uff0c\u4f53\u4f1a\u77e9\u9635\u53d8\u6362\u3002<\/p>\n
\u5047\u8bbe\u6709\u4e00\u5411\u91cf $\\vec{x} = \\begin{bmatrix}-2\\\\1\\end{bmatrix}$ \uff0c\u90a3\u4e48\u6839\u636e\u4e0a\u8ff0\u5f0f\u5b50\uff0c\u6709 $\\vec{y} = \\begin{bmatrix}0\\\\3\\end{bmatrix}$ \u3002\u53ef\u4ee5\u8bb0\u4f5c
\n$$\\begin{bmatrix}0\\\\3\\end{bmatrix} = \\begin{bmatrix}1&2\\\\0&3\\end{bmatrix}\\begin{bmatrix}-2\\\\1\\end{bmatrix}$$<\/p>\n
## 3.1 \u77e9\u9635\u64cd\u4f5c (Matrix Operations)<\/p>\n
$$\\dots$$<\/p>\n
## 3.5 \u5b50\u7a7a\u95f4\u3001\u57fa\u5e95\u3001\u7ef4\u5ea6\u548c\u79e9(Subspaces, Basis, Dimension, and Rank)<\/p>\n
\u81f3\u6b64\uff0c\u77e9\u9635\u7684\u57fa\u7840\u4ee3\u6570\u8fd0\u7b97\u5df2\u544a\u4e00\u6bb5\u843d\u3002\u4ece\u672c\u8282\u5f00\u59cb\uff0c\u6211\u4eec\u5c06\u89c6\u89d2\u4ece\u201c\u4ee3\u6570\u8ba1\u7b97\u201d\u5207\u6362\u5230\u201c\u51e0\u4f55\u7a7a\u95f4\u201d\uff0c\u6df1\u5165\u63a2\u8ba8\u77e9\u9635\u4e0e\u5411\u91cf\u7a7a\u95f4\u4e4b\u95f4\u7684\u6df1\u523b\u8054\u7cfb\u3002<\/p>\n
\u8ba9\u6211\u4eec\u4ece\u4e00\u4e2a\u76f4\u89c2\u7684\u51e0\u4f55\u73b0\u8c61\u51fa\u53d1\uff1a\u5728\u4e09\u7ef4\u7a7a\u95f4 $\\mathbb{R}^3$ \u4e2d\uff0c\u60f3\u8c61\u4e00\u4e2a\u901a\u8fc7\u539f\u70b9\u7684\u5e73\u9762\u3002\u76f4\u89c9\u544a\u8bc9\u6211\u4eec\u8fd9\u4e2a\u5e73\u9762\u662f\u201c\u4e8c\u7ef4\u201d\u7684\uff0c\u56e0\u4e3a\u5728\u8fd9\u4e2a\u5e73\u9762\u5185\uff0c\u4efb\u610f\u5411\u91cf\u7684\u52a0\u6cd5\u548c\u6807\u91cf\u4e58\u6cd5\u7ed3\u679c\u90fd\u4f1a\u88ab\u201c\u9501\u201d\u5728\u8fd9\u4e2a\u9762\u5185\uff0c\u5f62\u6210\u4e00\u4e2a\u5c01\u95ed\u7684\u8fd0\u7b97\u4f53\u7cfb\u3002\u8fd9\u5c31\u5f15\u51fa\u4e86\u4e00\u4e2a\u95ee\u9898\uff0c\u8fd9\u4e2a\u5e73\u9762\u4e0a\u7684\u5411\u91cf\uff0c\u7a76\u7adf\u662f\u4e8c\u7ef4\u8fd8\u662f\u4e09\u7ef4\u7269\u4f53\uff1f \u5b83\u4eec\u8eab\u5904 $\\mathbb{R}^3$ \u4e4b\u4e2d\uff0c\u62e5\u6709\u4e09\u4e2a\u5750\u6807\u5206\u91cf\uff1b\u4f46\u5b83\u4eec\u7684\u6d3b\u52a8\u8303\u56f4\u5374\u88ab\u4e25\u683c\u9650\u5236\u5728\u4e00\u4e2a\u4e8c\u7ef4\u5e73\u9762\u5185\u3002<\/p>\n
\u4e3a\u4e86\u5728\u4ee3\u6570\u4e0a\u7cbe\u786e\u523b\u753b\u8fd9\u79cd\u201c\u8eab\u5904\u9ad8\u7ef4\u7a7a\u95f4\uff0c\u5374\u81ea\u6210\u4e00\u4e2a\u4f4e\u7ef4\u5c01\u95ed\u4f53\u7cfb\u201d\u7684\u73b0\u8c61\uff0c\u6211\u4eec\u6b63\u5f0f\u5f15\u5165\u672c\u7ae0\u7684\u6838\u5fc3\u6982\u5ff5\u2014\u2014\u5b50\u7a7a\u95f4 $\\text{(Subspace)}$\u3002<\/p>\n
```ad-definition
\ntitle:\u5b9a\u4e49\uff1a\u5b50\u7a7a\u95f4 (Subspace)
\n\u8bbe $S$ \u662f $\\mathbb{R}^n$ \u4e2d\u7684\u4e00\u4e2a\u975e\u7a7a\u96c6\u5408\u3002\u5982\u679c $S$ \u6ee1\u8db3\u4ee5\u4e0b\u4e09\u4e2a\u6761\u4ef6\uff0c\u5219\u79f0 $S$ \u4e3a $\\mathbb{R}^n$ \u7684\u4e00\u4e2a**\u5b50\u7a7a\u95f4**\uff1a
\n1. **\u5305\u542b\u96f6\u5411\u91cf $\\text{(Zero vector)}$**\uff1a\u96f6\u5411\u91cf $\\vec{0}$ \u5c5e\u4e8e $S$\u3002
\n2. **\u52a0\u6cd5\u5c01\u95ed\u6027 $\\text{(Closed under addition)}$**\uff1a\u5982\u679c $\\vec{u}$ \u548c $\\vec{v}$ \u90fd\u5728 $S$ \u4e2d\uff0c\u90a3\u4e48 $\\vec{u} + \\vec{v}$ \u4e5f\u5fc5\u5b9a\u5728 $S$ \u4e2d\u3002
\n3. **\u6807\u91cf\u4e58\u6cd5\u5c01\u95ed\u6027 $\\text{(Closed under scalar multiplication)}$**\uff1a\u5982\u679c $\\vec{u}$ \u5728 $S$ \u4e2d\uff0c\u4e14 $c$ \u662f\u4efb\u610f\u5b9e\u6570\u6807\u91cf\uff0c\u90a3\u4e48 $c\\vec{u}$ \u4e5f\u5fc5\u5b9a\u5728 $S$ \u4e2d\u3002
\n```<\/p>\n
\u6211\u4eec\u53ef\u4ee5\u628a\u6027\u8d282\u30013\u7ed3\u5408\u8d77\u6765\uff0c\u7b49\u4ef7\u4e8e\u8981\u6c42 $S$ **\u5bf9\u4e8e\u7ebf\u6027\u7ec4\u5408\u5c01\u95ed**\uff1a<\/p>\n
\u5047\u8bbe $\\vec{u_1}, \\vec{u_2}, \\dots, \\vec{u_k}$ \u5728 $S$ \u4e2d\uff0c\u5e76\u4e14 $c_1, c_2, \\dots, c_k$ \u4e3a\u6807\u91cf\uff0c\u90a3\u4e48 $c_1\\vec{u_1}+c_2\\vec{u_2}+\\dots+c_k\\vec{u_k}$ \u4e5f\u5728 $S$
\n\u4e2d\u3002<\/p>\n
```ad-theorem
\ntitle:\u5b9a\u7406: \u5b50\u7a7a\u95f4
\n\u8bbe $\\vec{v}_1, \\vec{v}_2, \\dots, \\vec{v}_k$ \u662f $\\mathbb{R}^n$ \u4e2d\u7684\u5411\u91cf\uff0c\u5219 $\\text{span}(\\vec{v}_1, \\vec{v}_2, \\dots, \\vec{v}_k)$ \u662f $\\mathbb{R}^n$ \u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4\u3002
\n```<\/p>\n
**\u8bc1\u660e\uff1a**<\/p>\n
\u4ee4 $S = \\text{span}(\\vec{v}_1, \\vec{v}_2, \\dots, \\vec{v}_k)$\u3002\u9a8c\u8bc1\u6027\u8d28 (1)\uff0c\u6211\u4eec\u53ea\u9700\u89c2\u5bdf\u5230\u96f6\u5411\u91cf $\\vec{0}$ \u5728 $S$ \u4e2d\uff0c\u56e0\u4e3a $\\vec{0} = 0\\vec{v}_1 + 0\\vec{v}_2 + \\dots + 0\\vec{v}_k$\u3002<\/p>\n
\u8bbe<\/p>\n
$$\\vec{u} = c_1\\vec{v}_1 + c_2\\vec{v}_2 + \\dots + c_k\\vec{v}_k \\quad \\text{\u4e14} \\quad \\vec{v} = d_1\\vec{v}_1 + d_2\\vec{v}_2 + \\dots + d_k\\vec{v}_k$$<\/p>\n
\u4e3a $S$ \u4e2d\u7684\u4e24\u4e2a\u5411\u91cf\u3002\u90a3\u4e48<\/p>\n
$$\\begin{aligned} \\vec{u} + \\vec{v} &= (c_1\\vec{v}_1 + c_2\\vec{v}_2 + \\dots + c_k\\vec{v}_k) + (d_1\\vec{v}_1 + d_2\\vec{v}_2 + \\dots + d_k\\vec{v}_k) \\\\ &= (c_1 + d_1)\\vec{v}_1 + (c_2 + d_2)\\vec{v}_2 + \\dots + (c_k + d_k)\\vec{v}_k \\end{aligned}$$<\/p>\n
\u56e0\u6b64\uff0c$\\vec{u} + \\vec{v}$ \u4e5f\u662f $\\vec{v}_1, \\vec{v}_2, \\dots, \\vec{v}_k$ \u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u6240\u4ee5\u4e5f\u5728 $S$ \u4e2d\u3002\u8fd9\u5c31\u9a8c\u8bc1\u4e86\u6027\u8d28 (2)\u3002<\/p>\n
\u4e3a\u4e86\u9a8c\u8bc1\u6027\u8d28 (3)\uff0c\u8bbe $c$ \u4e3a\u4e00\u4e2a\u6807\u91cf\u3002\u90a3\u4e48<\/p>\n
$$\\begin{aligned} c\\vec{u} &= c(c_1\\vec{v}_1 + c_2\\vec{v}_2 + \\dots + c_k\\vec{v}_k) \\\\ &= (cc_1)\\vec{v}_1 + (cc_2)\\vec{v}_2 + \\dots + (cc_k)\\vec{v}_k \\end{aligned}$$<\/p>\n
\u8fd9\u8868\u660e $c\\vec{u}$ \u4e5f\u662f $\\vec{v}_1, \\vec{v}_2, \\dots, \\vec{v}_k$ \u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u56e0\u6b64\u4e5f\u5728 $S$ \u4e2d\u3002\u6211\u4eec\u5df2\u7ecf\u8bc1\u660e\u4e86 $S$ \u6ee1\u8db3\u6027\u8d28 (1) \u5230 (3)\uff0c\u56e0\u6b64\u5b83\u662f $\\mathbb{R}^n$ \u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4\u3002<\/p>\n
\u6211\u4eec\u5c06 $\\text{span}(\\vec{v}_1, \\vec{v}_2, \\dots, \\vec{v}_k)$ \u79f0\u4e3a**\u7531 $\\vec{v}_1, \\vec{v}_2, \\dots, \\vec{v}_k$ \u751f\u6210\u7684\u5b50\u7a7a\u95f4**\u3002<\/p>\n
### 3.5.1 \u4e0e\u77e9\u9635\u76f8\u5173\u7684\u5b50\u7a7a\u95f4 (Subspaces Associated with Matrices)<\/p>\n
\u6211\u4eec\u5df2\u7ecf\u638c\u63e1\u4e86\u5b50\u7a7a\u95f4 $\\text{(Subspace)}$ \u7684\u6982\u5ff5\uff0c\u73b0\u5728\u8ba9\u6211\u4eec\u4ece\u77e9\u9635\u7684\u89c6\u89d2\uff0c\u4e86\u89e3\u5173\u4e8e\u77e9\u9635\u7684\u5b50\u7a7a\u95f4\u3002\u5176\u4e2d\u6700\u91cd\u8981\u7684\u4e09\u4e2a\u77e9\u9635\u7684\u5b50\u7a7a\u95f4\uff0c\u5305\u62ec\u96f6\u7a7a\u95f4 $\\text{(Zero space)}$ \u3001\u884c\u7a7a\u95f4 $\\text{(Row space)}$ \u548c\u5217\u7a7a\u95f4 $\\text{(Column space)}$ \u3002<\/p>\n
```ad-definition
\ntitle:\u5b9a\u4e49: \u884c\u7a7a\u95f4\u4e0e\u5217\u7a7a\u95f4
\n\u8bbe $A$ \u662f\u4e00\u4e2a $m \\times n$ \u77e9\u9635\u3002
\n1. **\u884c\u7a7a\u95f4 $\\text{(Row Space)}$**\uff1a\u7531\u77e9\u9635 $A$ \u7684\u6240\u6709**\u884c\u5411\u91cf**\u5f20\u6210\u7684 $\\mathbb{R}^n$ \u7684\u5b50\u7a7a\u95f4\uff0c\u8bb0\u4f5c $\\text{row}(A)$\u3002
\n2. **\u5217\u7a7a\u95f4 $\\text{(Column Space)}$**\uff1a\u7531\u77e9\u9635 $A$ \u7684\u6240\u6709**\u5217\u5411\u91cf**\u5f20\u6210\u7684 $\\mathbb{R}^m$ \u7684\u5b50\u7a7a\u95f4\uff0c\u8bb0\u4f5c $\\text{col}(A)$\u3002
\n```<\/p>\n
```ad-example
\ntitle:\u4f8b\u9898\uff1a\u5224\u5b9a\u5411\u91cf\u662f\u5426\u5c5e\u4e8e\u884c\/\u5217\u7a7a\u95f4<\/p>\n
\u5df2\u77e5\u77e9\u9635 $A = \\begin{bmatrix} 1 & -1 \\\\ 0 & 1 \\\\ 3 & -3 \\end{bmatrix}$\u3002<\/p>\n
**\u95ee\u9898 (a)\uff1a\u5224\u65ad\u5411\u91cf $\\mathbf{b} = \\begin{bmatrix} 1 \\\\ 2 \\\\ 3 \\end{bmatrix}$ \u662f\u5426\u5728 $A$ \u7684\u5217\u7a7a\u95f4 $\\text{col}(A)$ \u4e2d\uff1f**
\n* **\u89e3\u9898\u76f4\u89c9**\uff1a\u8f6c\u5316\u4e3a\u89e3\u65b9\u7a0b $A\\mathbf{x} = \\mathbf{b}$\u3002
\n* **\u8ba1\u7b97\u8fc7\u7a0b**\uff1a\u6784\u9020\u5de6\u53f3\u589e\u5e7f\u77e9\u9635\uff08\u6ce8\u610f\u7ad6\u7ebf\u5206\u5272\uff09\u5e76\u8fdb\u884c\u884c\u5316\u7b80\uff1a
\n $$ [A \\mid \\mathbf{b}] = \\left[ \\begin{array}{cc|c} 1 & -1 & 1 \\\\ 0 & 1 & 2 \\\\ 3 & -3 & 3 \\end{array} \\right] \\xrightarrow{\\text{\u884c\u5316\u7b80}} \\left[ \\begin{array}{cc|c} 1 & 0 & 3 \\\\ 0 & 1 & 2 \\\\ 0 & 0 & 0 \\end{array} \\right] $$
\n* **\u7ed3\u8bba**\uff1a\u6700\u540e\u4e00\u884c\u5168\u4e3a 0\uff0c\u65b9\u7a0b\u4e00\u81f4\uff08\u6709\u89e3\uff09\u3002\u56e0\u6b64\uff0c$\\mathbf{b}$ **\u5728**\u5217\u7a7a\u95f4 $\\text{col}(A)$ \u4e2d\u3002<\/p>\n
**\u95ee\u9898 (b)\uff1a\u5224\u65ad\u5411\u91cf $\\mathbf{w} = \\begin{bmatrix} 4 & 5 \\end{bmatrix}$ \u662f\u5426\u5728 $A$ \u7684\u884c\u7a7a\u95f4 $\\text{row}(A)$ \u4e2d\uff1f**
\n* **\u89e3\u9898\u76f4\u89c9**\uff1a\u5982\u679c $\\mathbf{w}$ \u662f $A$ \u7684\u884c\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u90a3\u4e48\u628a\u5b83\u62fc\u5728 $A$ \u5e95\u90e8\uff0c\u7528\u4e0a\u65b9\u884c\u53d8\u6362\u6d88\u5143\uff0c\u5fc5\u7136\u80fd\u628a\u5b83\u6d88\u6210\u5168 0\u3002
\n* **\u8ba1\u7b97\u8fc7\u7a0b**\uff1a\u6784\u9020\u4e0a\u4e0b\u589e\u5e7f\u77e9\u9635\uff08\u6ce8\u610f\u5e95\u90e8\u7684\u6a2a\u7ebf\u5206\u5272\uff09\u5e76\u4ece\u4e0a\u5230\u4e0b\u8fdb\u884c\u884c\u6d88\u5143\uff1a
\n $$ \\begin{bmatrix} A \\\\ \\mathbf{w} \\end{bmatrix} = \\left[ \\begin{array}{cc} 1 & -1 \\\\ 0 & 1 \\\\ 3 & -3 \\\\ \\hline 4 & 5 \\end{array} \\right] \\xrightarrow[\\text{\u6d88\u53bb\u4e0b\u65b9\u5143\u7d20}]{R_3 - 3R_1, R_4 - 4R_1} \\left[ \\begin{array}{cc} 1 & -1 \\\\ 0 & 1 \\\\ 0 & 0 \\\\ \\hline 0 & 9 \\end{array} \\right] \\xrightarrow{R_4 - 9R_2} \\left[ \\begin{array}{cc} 1 & -1 \\\\ 0 & 1 \\\\ 0 & 0 \\\\ \\hline 0 & 0 \\end{array} \\right] $$
\n* **\u7ed3\u8bba**\uff1a\u7ecf\u8fc7\u884c\u53d8\u6362\uff0c\u57ab\u5e95\u7684 $\\mathbf{w}$ \u6210\u529f\u88ab\u5316\u6210\u4e86\u5168 0 \u884c $\\begin{bmatrix} 0 & 0 \\end{bmatrix}$\u3002\u8fd9\u8bc1\u660e\u4e86 $\\mathbf{w}$ \u786e\u5b9e\u662f\u4e0a\u65b9\u884c\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u56e0\u6b64 $\\mathbf{w}$ **\u5728**\u884c\u7a7a\u95f4 $\\text{row}(A)$ \u4e2d\u3002
\n```<\/p>\n","protected":false},"excerpt":{"rendered":"
3. \u77e9\u9635 (Matrices)<\/p>\n
3.0 \u5f15\u8a00\uff1a\u77e9\u9635\u7684\u8fd0\u7528 (Introduction: Matrices in Action)<\/p>\n
\u5728\u4e0a\u4e00\u7ae0\u8282\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u589e\u5e7f\u77e9\u9635 ($\\text{Augm","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"emotion":"","emotion_color":"","title_style":"","license":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-219","post","type-post","status-publish","format-standard","hentry","category-article-cn"],"_links":{"self":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/219","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=219"}],"version-history":[{"count":9,"href":"https:\/\/wuhanqing.cn\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/219\/revisions"}],"predecessor-version":[{"id":233,"href":"https:\/\/wuhanqing.cn\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/219\/revisions\/233"}],"wp:attachment":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=219"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=219"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=219"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}