{"id":260,"date":"2026-04-29T17:23:24","date_gmt":"2026-04-29T08:23:24","guid":{"rendered":"https:\/\/wuhanqing.cn\/wordpress\/?p=260"},"modified":"2026-05-01T10:04:44","modified_gmt":"2026-05-01T01:04:44","slug":"3-5-%e5%ad%90%e7%a9%ba%e9%97%b4%e3%80%81%e5%9f%ba%e5%ba%95%e3%80%81%e7%bb%b4%e5%ba%a6%e5%92%8c%e7%a7%a9-subspaces-basis-dimension-and-rank","status":"publish","type":"post","link":"https:\/\/wuhanqing.cn\/wordpress\/2026\/04\/29\/3-5-%e5%ad%90%e7%a9%ba%e9%97%b4%e3%80%81%e5%9f%ba%e5%ba%95%e3%80%81%e7%bb%b4%e5%ba%a6%e5%92%8c%e7%a7%a9-subspaces-basis-dimension-and-rank\/","title":{"rendered":"3.5 \u5b50\u7a7a\u95f4\u3001\u57fa\u5e95\u3001\u7ef4\u5ea6\u548c\u79e9 (Subspaces, Basis, Dimension, and Rank)"},"content":{"rendered":"<p>## 3.5 \u5b50\u7a7a\u95f4\u3001\u57fa\u5e95\u3001\u7ef4\u5ea6\u548c\u79e9 (Subspaces, Basis, Dimension, and Rank)<\/p>\n<p>\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<p>\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<p>\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<p>```ad-definition<br \/>\ntitle:\u5b9a\u4e49\uff1a\u5b50\u7a7a\u95f4 $\\text{(Subspace)}$<br \/>\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<br \/>\n1. **\u5305\u542b\u96f6\u5411\u91cf $\\text{(Zero vector)}$**\uff1a\u96f6\u5411\u91cf $\\vec{0}$ \u5c5e\u4e8e $S$\u3002<br \/>\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<br \/>\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<br \/>\n```<\/p>\n<p>\u6211\u4eec\u53ef\u4ee5\u628a\u6027\u8d28 $\\text{(2)}$\u3001$\\text{(3)}$ \u7ed3\u5408\u8d77\u6765\uff0c\u7b49\u4ef7\u4e8e\u8981\u6c42 $S$ **\u5bf9\u4e8e\u7ebf\u6027\u7ec4\u5408\u5c01\u95ed** $\\text{(Closed under Linear Combinations)}$\uff1a<\/p>\n<p>\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 $S$<br \/>\n\u4e2d\u3002<\/p>\n<p>```ad-theorem<br \/>\ntitle:\u5b9a\u74063.19 \u5f20\u6210\u7a7a\u95f4\u4e0e\u5b50\u7a7a\u95f4<br \/>\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<br \/>\n```<\/p>\n<p>**\u8bc1\u660e\uff1a**<\/p>\n<p>\u4ee4 $S = \\text{span}(\\vec{v}_1, \\vec{v}_2, \\dots, \\vec{v}_k)$\u3002\u4e3a\u4e86\u8bc1\u660e $S$ \u662f\u5b50\u7a7a\u95f4\uff0c\u6211\u4eec\u9700\u8981\u9a8c\u8bc1\u5176\u6ee1\u8db3\u5b9a\u4e49\u4e2d\u7684\u4e09\u4e2a\u6027\u8d28\uff1a<\/p>\n<p>$\\text{(1)}$ **\u5305\u542b\u96f6\u5411\u91cf**\uff1a\u89c2\u5bdf\u5230\u96f6\u5411\u91cf $\\vec{0}$ \u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<br \/>\n$$\\vec{0} = 0\\vec{v}_1 + 0\\vec{v}_2 + \\dots + 0\\vec{v}_k$$<br \/>\n\u7531\u4e8e $\\vec{0}$ \u662f $\\vec{v}_1, \\dots, \\vec{v}_k$ \u7684\u4e00\u4e2a\u7ebf\u6027\u7ec4\u5408\uff0c\u56e0\u6b64 $\\vec{0} \\in S$\u3002\u8fd9\u9a8c\u8bc1\u4e86\u6027\u8d28 $\\text{(1)}$\u3002<\/p>\n<p>$\\text{(2)}$ **\u52a0\u6cd5\u5c01\u95ed\u6027**\uff1a<\/p>\n<p>\u8bbe $\\vec{u}$ \u548c $\\vec{v}$ \u662f $S$ \u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u5411\u91cf\uff0c\u5219\u5b83\u4eec\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<br \/>\n$$\\vec{u} = c_1\\vec{v}_1 + c_2\\vec{v}_2 + \\dots + c_k\\vec{v}_k$$<br \/>\n$$\\vec{v} = d_1\\vec{v}_1 + d_2\\vec{v}_2 + \\dots + d_k\\vec{v}_k$$<\/p>\n<p>\u5176\u4e2d $c_i, d_i$ \u4e3a\u6807\u91cf\u3002\u90a3\u4e48\u5b83\u4eec\u7684\u548c\u4e3a\uff1a<br \/>\n$$\\begin{aligned} \\vec{u} + \\vec{v} &= (c_1\\vec{v}_1 + \\dots + c_k\\vec{v}_k) + (d_1\\vec{v}_1 + \\dots + d_k\\vec{v}_k) \\\\ &= (c_1 + d_1)\\vec{v}_1 + (c_2 + d_2)\\vec{v}_2 + \\dots + (c_k + d_k)\\vec{v}_k \\end{aligned}$$<\/p>\n<p>\u7531\u4e8e $\\vec{u} + \\vec{v}$ \u4ecd\u7136\u662f $\\vec{v}_1, \\dots, \\vec{v}_k$ \u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u6240\u4ee5 $\\vec{u} + \\vec{v} \\in S$\u3002\u8fd9\u9a8c\u8bc1\u4e86\u6027\u8d28 $\\text{(2)}$\u3002<\/p>\n<p>$\\text{(3)}$ **\u6807\u91cf\u4e58\u6cd5\u5c01\u95ed\u6027**\uff1a<\/p>\n<p>\u8bbe $c$ \u4e3a\u4efb\u610f\u6807\u91cf\uff0c\u5219\uff1a<br \/>\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<p>\u8fd9\u8868\u660e $c\\vec{u}$ \u4e5f\u662f $\\vec{v}_1, \\dots, \\vec{v}_k$ \u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u56e0\u6b64 $c\\vec{u} \\in S$\u3002\u8fd9\u9a8c\u8bc1\u4e86\u6027\u8d28 $\\text{(3)}$\u3002<\/p>\n<p>**\u7ed3\u8bba\uff1a**<\/p>\n<p>\u7531\u4e8e $S$ \u6ee1\u8db3\u5b50\u7a7a\u95f4\u7684\u6240\u6709\u4e09\u4e2a\u5b9a\u4e49\u6027\u8d28\uff0c\u6211\u4eec\u8bc1\u660e\u4e86 $\\text{span}(\\vec{v}_1, \\vec{v}_2, \\dots, \\vec{v}_k)$ \u662f $\\mathbb{R}^n$ \u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4\u3002\u6211\u4eec\u901a\u5e38\u5c06\u5176\u79f0\u4e3a**\u7531 $\\vec{v}_1, \\vec{v}_2, \\dots, \\vec{v}_k$ \u5f20\u6210\u7684\u5b50\u7a7a\u95f4 $\\text{(The subspace spanned by } \\vec{v}_1, \\dots, \\vec{v}_k)$**\u3002<\/p>\n<p>```ad-example<br \/>\ntitle: \u4f8b\u98983.38 \u8bc1\u660e\u96c6\u5408\u6784\u6210\u5b50\u7a7a\u95f4<br \/>\n\u8bc1\u660e\u7531\u6240\u6709\u6ee1\u8db3\u6761\u4ef6 $x = 3y$ \u4e14 $z = -2y$ \u7684\u5411\u91cf $\\begin{bmatrix} x \\\\ y \\\\ z \\end{bmatrix}$ \u7ec4\u6210\u7684\u96c6\u5408\u6784\u6210\u4e86 $\\mathbb{R}^3$ \u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4\u3002<\/p>\n<p>**\u89e3\uff1a**<br \/>\n\u5c06\u7ed9\u5b9a\u7684\u4e24\u4e2a\u6761\u4ef6\u4ee3\u5165\u5411\u91cf\u5f62\u5f0f\u4e2d\uff1a<br \/>\n$$\\begin{bmatrix} x \\\\ y \\\\ z \\end{bmatrix} = \\begin{bmatrix} 3y \\\\ y \\\\ -2y \\end{bmatrix} = y \\begin{bmatrix} 3 \\\\ 1 \\\\ -2 \\end{bmatrix}$$<\/p>\n<p>\u7531\u4e8e $y$ \u662f\u4efb\u610f\u5b9e\u6570\uff0c\u8be5\u5411\u91cf\u96c6\u5408\u5b9e\u9645\u4e0a\u5c31\u662f\u7531\u5411\u91cf $\\begin{bmatrix} 3 \\\\ 1 \\\\ -2 \\end{bmatrix}$ \u5f20\u6210\u7684\u7a7a\u95f4\uff0c\u5373\uff1a<br \/>\n$$\\text{span} \\left( \\begin{bmatrix} 3 \\\\ 1 \\\\ -2 \\end{bmatrix} \\right)$$<br \/>\n\u6839\u636e\u5b9a\u7406 3.19\uff08\u5f20\u6210\u7a7a\u95f4\u5fc5\u4e3a\u5b50\u7a7a\u95f4\uff09\uff0c\u8be5\u96c6\u5408\u662f $\\mathbb{R}^3$ \u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4\u3002<\/p>\n<p>\u4ece\u51e0\u4f55\u4e0a\u770b\uff0c\u8fd9\u4e2a\u5411\u91cf\u96c6\u5408\u4ee3\u8868\u4e86 $\\mathbb{R}^3$ \u4e2d\u4e00\u6761\u7ecf\u8fc7\u539f\u70b9\u4e14\u65b9\u5411\u5411\u91cf\u4e3a $\\begin{bmatrix} 3 \\\\ 1 \\\\ -2 \\end{bmatrix}$ \u7684**\u76f4\u7ebf**\u3002<br \/>\n```<\/p>\n<p>```ad-example<br \/>\ntitle: \u4f8b\u98983.40 \u5224\u5b9a\u975e\u5b50\u7a7a\u95f4\uff08\u53cd\u4f8b\u6cd5\uff09<br \/>\n\u5224\u5b9a\u6240\u6709\u5f62\u5f0f\u4e3a $\\begin{bmatrix} x \\\\ y \\end{bmatrix}$ \u4e14\u6ee1\u8db3 $y = x^2$ \u7684\u5411\u91cf\u96c6\u5408 $S$ \u662f\u5426\u4e3a $\\mathbb{R}^2$ \u7684\u5b50\u7a7a\u95f4\u3002<\/p>\n<p>**\u89e3\uff1a**<br \/>\n\u8be5\u96c6\u5408\u4e2d\u7684\u5411\u91cf\u5f62\u5f0f\u4e3a $\\begin{bmatrix} x \\\\ x^2 \\end{bmatrix}$\u3002<\/p>\n<p>1. **\u9a8c\u8bc1\u6027\u8d28 $\\text{(1)}$**\uff1a\u5f53 $x = 0$ \u65f6\uff0c$\\vec{0} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}$ \u5c5e\u4e8e $S$\uff0c\u6027\u8d28 $\\text{(1)}$ \u6210\u7acb\u3002<br \/>\n2. **\u9a8c\u8bc1\u6027\u8d28 $\\text{(2)}$\uff08\u52a0\u6cd5\u5c01\u95ed\u6027\uff09**\uff1a<br \/>\n   \u8bbe $\\vec{u} = \\begin{bmatrix} x_1 \\\\ x_1^2 \\end{bmatrix}$ \u548c $\\vec{v} = \\begin{bmatrix} x_2 \\\\ x_2^2 \\end{bmatrix}$ \u662f $S$ \u4e2d\u7684\u4e24\u4e2a\u5411\u91cf\u3002\u5b83\u4eec\u7684\u548c\u4e3a\uff1a<br \/>\n   $$\\vec{u} + \\vec{v} = \\begin{bmatrix} x_1 + x_2 \\\\ x_1^2 + x_2^2 \\end{bmatrix}$$<br \/>\n   \u901a\u5e38\u60c5\u51b5\u4e0b\uff0c$x_1^2 + x_2^2 \\neq (x_1 + x_2)^2$\u3002\u4e3a\u4e86\u5177\u4f53\u8bf4\u660e\uff0c\u6211\u4eec\u627e\u4e00\u4e2a**\u53cd\u4f8b $\\text{(Counterexample)}$**:<br \/>\n   \u4ee4 $\\vec{u} = \\begin{bmatrix} 1 \\\\ 1 \\end{bmatrix}$\uff0c$\\vec{v} = \\begin{bmatrix} 2 \\\\ 4 \\end{bmatrix}$\uff08\u5747\u6ee1\u8db3 $y=x^2$\uff09\u3002<br \/>\n   \u7136\u800c $\\vec{u} + \\vec{v} = \\begin{bmatrix} 3 \\\\ 5 \\end{bmatrix}$\uff0c\u7531\u4e8e $5 \\neq 3^2$\uff0c\u6240\u4ee5 $\\vec{u} + \\vec{v} \\notin S$\u3002<\/p>\n<p>**\u7ed3\u8bba\uff1a**<br \/>\n\u7531\u4e8e\u52a0\u6cd5\u5c01\u95ed\u6027\u5931\u6548\uff0c$S$ \u4e0d\u662f $\\mathbb{R}^2$ \u7684\u5b50\u7a7a\u95f4\u3002<\/p>\n<p>**\u6ce8\u91ca\uff1a**<br \/>\n\u8981\u8bc1\u660e\u4e00\u4e2a\u96c6\u5408 $S$ \u662f\u5b50\u7a7a\u95f4\uff0c\u5fc5\u987b\u8bc1\u660e\u6027\u8d28 $\\text{(1)}$ \u5230 $\\text{(3)}$ **\u666e\u904d\u6210\u7acb**\uff1b\u800c\u8981\u8bc1\u660e\u5b83**\u4e0d\u662f**\u5b50\u7a7a\u95f4\uff0c\u53ea\u9700\u8981\u627e\u5230\u4e00\u4e2a\u53cd\u4f8b\u8bf4\u660e\u5176\u4e2d\u4efb\u610f\u4e00\u4e2a\u6027\u8d28\u5931\u6548\u5373\u53ef\u3002<br \/>\n```<\/p>\n<p>### 3.5.1 \u4e0e\u77e9\u9635\u5173\u8054\u7684\u5b50\u7a7a\u95f4 $\\text{(Subspaces Associated with Matrices)}$<\/p>\n<p>\u5bf9\u4e8e\u4e00\u4e2a $m \\times n$ \u77e9\u9635 $A$\uff0c\u6211\u4eec\u6700\u5173\u6ce8\u5b83\u6240\u643a\u5e26\u7684\u4e09\u4e2a\u5b50\u7a7a\u95f4\uff1a\u884c\u7a7a\u95f4\u3001\u5217\u7a7a\u95f4\u4ee5\u53ca\u96f6\u7a7a\u95f4\u3002<\/p>\n<p>```ad-definition<br \/>\ntitle: \u5b9a\u4e49\uff1a\u884c\u7a7a\u95f4\u3001\u5217\u7a7a\u95f4\u4e0e\u96f6\u7a7a\u95f4<br \/>\n1. **\u884c\u7a7a\u95f4 ($\\text{Row Space}$)**\uff1a\u7531\u77e9\u9635 $A$ \u7684\u884c\u5411\u91cf\u5f20\u6210\u7684 $\\mathbb{R}^n$ \u5b50\u7a7a\u95f4\uff0c\u8bb0\u4e3a $\\text{row}(A)$\u3002<br \/>\n2. **\u5217\u7a7a\u95f4 ($\\text{Column Space}$)**\uff1a\u7531\u77e9\u9635 $A$ \u7684\u5217\u5411\u91cf\u5f20\u6210\u7684 $\\mathbb{R}^m$ \u5b50\u7a7a\u95f4\uff0c\u8bb0\u4e3a $\\text{col}(A)$\u3002<br \/>\n3. **\u96f6\u7a7a\u95f4 ($\\text{Null Space}$)**\uff1a\u9f50\u6b21\u65b9\u7a0b\u7ec4 $A\\vec{x} = \\vec{0}$ \u7684\u6240\u6709\u89e3 $\\vec{x}$ \u6784\u6210\u7684 $\\mathbb{R}^n$ \u5b50\u7a7a\u95f4\uff0c\u8bb0\u4e3a $\\text{null}(A)$\u3002<br \/>\n```<\/p>\n<p>### \u2b50\u77e9\u9635\u6620\u5c04\u4e0e\u4e09\u5927\u5b50\u7a7a\u95f4\u7684\u4ee3\u6570\u63a8\u5bfc\u4e0e\u51e0\u4f55\u672c\u8d28<\/p>\n<p>\u5bf9\u4e8e $m \\times n$ \u77e9\u9635 $A$\uff0c\u7ebf\u6027\u53d8\u6362 $T(\\vec{x}) = A\\vec{x}$ \u662f\u4e00\u5ea7\u8fde\u63a5\u4e24\u4e2a\u4e0d\u540c\u7ef4\u5ea6\u4e16\u754c\u7684\u6865\u6881\uff1a\u5c06**\u8f93\u5165\u7a7a\u95f4 $\\mathbb{R}^n$** \u6620\u5c04\u5230**\u8f93\u51fa\u7a7a\u95f4 $\\mathbb{R}^m$**\u3002\u5728\u8fd9\u4e2a\u6620\u5c04\u8fc7\u7a0b\u4e2d\uff0c\u4e09\u5927\u5b50\u7a7a\u95f4\u7684\u4ee3\u6570\u7ed3\u6784\u51b3\u5b9a\u4e86\u53d8\u6362\u7684\u51e0\u4f55\u672c\u8d28\u3002<\/p>\n<p>**1. \u5217\u7a7a\u95f4 $\\text{col}(A)$\uff1a\u8f93\u51fa\u7a7a\u95f4\u7684\u7586\u754c $\\text{(Output Space and Range)}$**<\/p>\n<p>**\u5b9a\u4e49\uff1a** \u77e9\u9635 $A$ \u7684\u6240\u6709\u5217\u5411\u91cf\u5f20\u6210\u7684 $\\mathbb{R}^m$ \u7684\u5b50\u7a7a\u95f4\uff0c\u8bb0\u4e3a $\\text{col}(A) = \\text{span}(\\vec{a}_1, \\vec{a}_2, \\dots, \\vec{a}_n)$\u3002<\/p>\n<p>**\u6570\u5b66\u63a8\u5bfc\uff1a**<br \/>\n\u8bbe $A = [\\vec{a}_1, \\vec{a}_2, \\dots, \\vec{a}_n]$\uff0c\u5bf9\u4e8e\u8f93\u5165\u7a7a\u95f4\u4e2d\u7684\u4efb\u610f\u5411\u91cf $\\vec{x} = [x_1, x_2, \\dots, x_n]^T \\in \\mathbb{R}^n$\uff0c\u77e9\u9635\u4e58\u6cd5\u53ef\u5c55\u5f00\u4e3a\u5217\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\uff1a<\/p>\n<p>$$A\\vec{x} = x_1\\vec{a}_1 + x_2\\vec{a}_2 + \\dots + x_n\\vec{a}_n$$<\/p>\n<p>**\u7ed3\u8bba\uff1a** \u5217\u7a7a\u95f4\u5b8c\u5168\u5b58\u5728\u4e8e**\u8f93\u51fa\u4e16\u754c $\\mathbb{R}^m$** \u4e2d\u3002\u7ebf\u6027\u53d8\u6362 $T(\\vec{x}) = A\\vec{x}$ \u7684**\u503c\u57df $\\text{(Range)}$** \u7cbe\u786e\u7b49\u4e8e\u5217\u7a7a\u95f4\u3002\u65e0\u8bba\u8f93\u5165\u5411\u91cf $\\vec{x}$ \u5982\u4f55\u53d6\u503c\uff0c\u6620\u5c04\u540e\u7684\u7ed3\u679c\u5411\u91cf\u5fc5\u7136\u88ab\u4e25\u683c\u9501\u6b7b\u5728 $\\text{col}(A)$ \u5185\u3002<\/p>\n<p>**\u7a7a\u95f4\u5305\u542b\u5173\u7cfb\u63cf\u8ff0\uff1a**<br \/>\n\u5728\u6570\u5b66\u8bed\u8a00\u4e2d\uff0c\u6211\u4eec\u5c06\u5217\u7a7a\u95f4\u4e0e**\u76ee\u6807\u7a7a\u95f4 $\\text{(Codomain, \u5373 } \\mathbb{R}^m)$** \u7684\u5173\u7cfb\u5199\u4f5c\uff1a<br \/>\n$$\\text{col}(A) = T(\\mathbb{R}^n) \\subseteq \\mathbb{R}^m$$<\/p>\n<p>- **\u5f53 $\\text{rank}(A) < m$ \u65f6**\uff1a\u53d8\u6362\u540e\u7684\u7ed3\u679c\u53ea\u80fd\u586b\u5145 $\\mathbb{R}^m$ \u4e2d\u7684\u4e00\u4e2a\u4f4e\u7ef4\u7247\u6bb5\uff08\u5982\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u4e8c\u7ef4\u5e73\u9762\uff09\u3002\u6b64\u65f6 $T(\\mathbb{R}^n)$ \u662f $\\mathbb{R}^m$ \u7684**\u771f\u5b50\u7a7a\u95f4**\u3002\n- **\u5f53 $\\text{rank}(A) = m$ \u65f6\uff08\u884c\u6ee1\u79e9\uff09**\uff1a\u6b64\u65f6\u53d8\u6362\u662f**\u6ee1\u5c04 $\\text{(Onto \/ Surjective)}$** \u7684\uff0c\u610f\u5473\u7740\u8f93\u51fa\u53ef\u4ee5\u8986\u76d6\u6574\u4e2a\u76ee\u6807\u7a7a\u95f4\u3002\u53ea\u6709\u5728\u8fd9\u79cd\u7279\u6b8a\u60c5\u51b5\u4e0b\uff0c\u624d\u6210\u7acb\uff1a\n$$T(\\mathbb{R}^n) = \\mathbb{R}^m$$\n\n**\u5373\u4e3a\uff1a**\n$\\mathbb{R}^m$ \u662f\u53d8\u6362\u9884\u8bbe\u7684\u201c\u7740\u9646\u573a\u201d**\uff0c\u800c $\\text{col}(A)$ \u662f\u53d8\u6362\u771f\u6b63\u80fd**\u201c\u89e6\u8fbe\u7684\u8303\u56f4\u201d\u3002\n\n\u9664\u975e\u77e9\u9635\u7684**\u79e9 $\\text{(Rank)}$** \u8db3\u591f\u9ad8\uff0c\u9ad8\u5230\u80fd\u586b\u6ee1\u6574\u4e2a\u8f93\u51fa\u7ef4\u5ea6\u7684\u4e16\u754c\uff0c\u5426\u5219\u8fd9\u53f0\u673a\u5668\u751f\u6210\u7684\u5411\u91cf\u6c38\u8fdc\u53ea\u80fd\u5728 $\\mathbb{R}^m$ \u7684\u67d0\u4e2a\u5c40\u90e8\u5207\u9762\uff08\u5b50\u7a7a\u95f4\uff09\u5185\u6d3b\u52a8\u3002\n\n**2. \u96f6\u7a7a\u95f4 $\\text{null}(A)$\uff1a\u8f93\u5165\u7a7a\u95f4\u4e2d\u7684\u201c\u9ed1\u6d1e\u201d $\\text{(Kernel in Input Space)}$**\n\n**\u5b9a\u4e49\uff1a** \u6ee1\u8db3\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4 $A\\vec{x} = \\vec{0}$ \u7684\u6240\u6709\u89e3\u5411\u91cf $\\vec{x}$ \u6784\u6210\u7684 $\\mathbb{R}^n$ \u7684\u5b50\u7a7a\u95f4\u3002\n\n**\u5b50\u7a7a\u95f4\u8bc1\u660e\uff1a**\n\u8bbe $\\vec{u}, \\vec{v} \\in \\text{null}(A)$\uff0c\u4e14 $c, d$ \u4e3a\u4efb\u610f\u5b9e\u6570\u6807\u91cf\u3002\u6839\u636e\u77e9\u9635\u4e58\u6cd5\u7684\u7ebf\u6027\u6027\u8d28\uff1a\n$$A(c\\vec{u} + d\\vec{v}) = c(A\\vec{u}) + d(A\\vec{v}) = c\\vec{0} + d\\vec{0} = \\vec{0}$$\n\n\u56e0\u4e3a\u7ebf\u6027\u7ec4\u5408\u5411\u91cf $c\\vec{u} + d\\vec{v}$ \u4ecd\u6ee1\u8db3\u65b9\u7a0b\uff0c\u6545 $\\text{null}(A)$ \u5bf9\u7ebf\u6027\u7ec4\u5408\u5c01\u95ed\uff0c\u662f\u4e00\u4e2a\u5408\u6cd5\u7684\u5b50\u7a7a\u95f4\u3002\u5b83\u5b8c\u5168\u5b58\u5728\u4e8e**\u8f93\u5165\u4e16\u754c $\\mathbb{R}^n$** \u4e2d\uff0c\u4ee3\u8868\u4e86\u90a3\u4e9b\u5728\u53d8\u6362\u4e2d\u4e27\u5931\u6240\u6709\u4fe1\u606f\u3001\u88ab\u4e0d\u53ef\u9006\u5730\u574d\u7f29\u81f3\u96f6\u5411\u91cf $\\vec{0}$ \u7684\u201c\u65e0\u6548\u8f93\u5165\u201d\u3002\n\n**3. \u884c\u7a7a\u95f4 $\\text{row}(A)$\uff1a\u5b58\u6d3b\u7684\u6709\u6548\u8f93\u5165\u7a7a\u95f4 $\\text{(Effective Input Space)}$**\n\n**\u5b9a\u4e49\uff1a** \u77e9\u9635 $A$ \u7684\u6240\u6709\u884c\u5411\u91cf\u5f20\u6210\u7684 $\\mathbb{R}^n$ \u7684\u5b50\u7a7a\u95f4\uff0c\u8bb0\u4e3a $\\text{row}(A)$\u3002\u5b83\u7b49\u4ef7\u4e8e\u8f6c\u7f6e\u77e9\u9635\u7684\u5217\u7a7a\u95f4 $\\text{col}(A^T)$\u3002\n\n**\u6b63\u4ea4\u6027\u63a8\u5bfc\uff08\u6838\u5fc3\uff09\uff1a**\n\u8bbe\u77e9\u9635 $A$ \u7684\u884c\u5411\u91cf\u4e3a $\\vec{r}_1^T, \\vec{r}_2^T, \\dots, \\vec{r}_m^T$\u3002\u8003\u8651 $A\\vec{x} = \\vec{0}$ \u8fd9\u4e00\u65b9\u7a0b\u7ec4\uff0c\u6309\u77e9\u9635\u5206\u5757\u4e58\u6cd5\u5c55\u5f00\uff1a\n$$A\\vec{x} = \\begin{bmatrix} \\vec{r}_1^T \\\\ \\vec{r}_2^T \\\\ \\vdots \\\\ \\vec{r}_m^T \\end{bmatrix} \\vec{x} = \\begin{bmatrix} \\vec{r}_1 \\cdot \\vec{x} \\\\ \\vec{r}_2 \\cdot \\vec{x} \\\\ \\vdots \\\\ \\vec{r}_m \\cdot \\vec{x} \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{bmatrix} = \\vec{0}$$\n\n\u4ece\u4e0a\u5f0f\u53ef\u4ee5\u6e05\u6670\u770b\u51fa\uff0c\u5982\u679c\u5411\u91cf $\\vec{x} \\in \\text{null}(A)$\uff0c\u5219 $\\vec{x}$ \u5fc5\u987b\u4e0e $A$ \u7684**\u6bcf\u4e00\u4e2a\u884c\u5411\u91cf $\\vec{r}_i$ \u7684\u5185\u79ef\uff08\u70b9\u79ef\uff09\u4e3a\u96f6**\u3002\n\n\u63a8\u800c\u5e7f\u4e4b\uff0c$\\vec{x}$ \u5fc5\u7136\u4e0e\u884c\u5411\u91cf\u7684\u4efb\u610f\u7ebf\u6027\u7ec4\u5408\u6b63\u4ea4\u3002\u56e0\u6b64\uff0c**\u96f6\u7a7a\u95f4\u4e0e\u884c\u7a7a\u95f4\u5728\u8f93\u5165\u7a7a\u95f4 $\\mathbb{R}^n$ \u4e2d\u4e92\u4e3a\u6b63\u4ea4\u8865**\uff0c\u8bb0\u4e3a\uff1a\n$$\\text{null}(A) = \\text{row}(A)^\\perp$$\n\n**4. \u6620\u5c04\u7684\u6838\u5fc3\u903b\u8f91\uff1a\u4ece\u8f93\u5165\u5230\u8f93\u51fa\u7684\u6b63\u4ea4\u5206\u89e3\u4e0e\u540c\u6784**\n\n\u6839\u636e\u6b63\u4ea4\u8865\u5b9a\u7406\uff0c**\u8f93\u5165\u7a7a\u95f4 $\\mathbb{R}^n$** \u4e2d\u7684\u4efb\u610f\u5411\u91cf $\\vec{x}$ \u90fd\u53ef\u4ee5\u88ab\u552f\u4e00\u5206\u89e3\u4e3a\u201c\u6709\u6548\u8f93\u5165\uff08\u884c\u7a7a\u95f4\u5206\u91cf\uff09\u201d\u4e0e\u201c\u65e0\u6548\u8f93\u5165\uff08\u96f6\u7a7a\u95f4\u5206\u91cf\uff09\u201d\u4e4b\u548c\uff1a\n$$\\vec{x} = \\vec{x}_{\\text{row}} + \\vec{x}_{\\text{null}} \\quad (\\text{\u5176\u4e2d } \\vec{x}_{\\text{row}} \\in \\text{row}(A), \\vec{x}_{\\text{null}} \\in \\text{null}(A))$$\n\n\u5f53\u7ebf\u6027\u53d8\u6362 $A$ \u4f5c\u7528\u4e8e\u8f93\u5165\u5411\u91cf $\\vec{x}$ \u65f6\uff1a\n$$A\\vec{x} = A(\\vec{x}_{\\text{row}} + \\vec{x}_{\\text{null}}) = A\\vec{x}_{\\text{row}} + A\\vec{x}_{\\text{null}} = A\\vec{x}_{\\text{row}} + \\vec{0} = A\\vec{x}_{\\text{row}}$$\n\u8fd9\u8bf4\u660e\uff0c\u51b3\u5b9a**\u8f93\u51fa\u4e16\u754c**\u4e2d $A\\vec{x}$ \u6700\u7ec8\u4f4d\u7f6e\u7684\uff0c\u5b8c\u5168\u4e14\u4ec5\u4ec5\u662f\u8f93\u5165\u5411\u91cf\u4e2d\u7684**\u884c\u7a7a\u95f4\u5206\u91cf $\\vec{x}_{\\text{row}}$**\u3002\n\n### \u2b50\u5b9e\u6218\u4f53\u4f1a\u8f93\u5165\u7a7a\u95f4\u7684\u6b63\u4ea4\u5206\u89e3\u6620\u5c04\n\n\u4e3a\u4e86\u76f4\u89c2\u4f53\u4f1a $\\vec{x} = \\vec{x}_{\\text{row}} + \\vec{x}_{\\text{null}}$ \u7684\u5a01\u529b\uff0c\u6211\u4eec\u6765\u770b\u4e00\u4e2a\u5177\u4f53\u7684\u7269\u7406\u6620\u5c04\u6848\u4f8b\u3002\n\n```ad-example\ntitle: \u4f8b\u9898 \u4e09\u7ef4\u8f93\u5165\u7a7a\u95f4\u7684\u6b63\u4ea4\u5256\u5206\u4e0e\u6295\u5f71\n\u5df2\u77e5\u7ebf\u6027\u53d8\u6362 $T: \\mathbb{R}^3 \\to \\mathbb{R}^2$ \u7531\u77e9\u9635 $A$ \u5b9a\u4e49\uff1a\n$$A = \\begin{bmatrix} 1 & 1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix}$$\n\u8bbe\u8f93\u5165\u5411\u91cf\u4e3a $\\vec{x} = \\begin{bmatrix} 5 \\\\ 1 \\\\ 4 \\end{bmatrix}$\u3002\u8bf7\u8ba1\u7b97\u5176\u5728\u884c\u7a7a\u95f4\u4e0e\u96f6\u7a7a\u95f4\u4e0a\u7684\u6b63\u4ea4\u5206\u89e3\uff0c\u5e76\u9a8c\u8bc1\u6620\u5c04\u7ed3\u679c\u3002\n\n**\u89e3\uff1a**\n**1. \u786e\u5b9a\u5b50\u7a7a\u95f4\u57fa\u5e95**\n* **\u96f6\u7a7a\u95f4 $\\text{null}(A)$**\uff1a\u89e3 $A\\vec{x} = \\vec{0}$ \u5f97 $x_1 = -x_2, x_3 = 0$\u3002\u57fa\u5e95\u65b9\u5411\u5411\u91cf\u4e3a $\\vec{v} = \\begin{bmatrix} 1 \\\\ -1 \\\\ 0 \\end{bmatrix}$\u3002\u8fd9\u662f\u4e00\u6761\u76f4\u7ebf\u3002\n* **\u884c\u7a7a\u95f4 $\\text{row}(A)$**\uff1a\u57fa\u5e95\u4e3a $\\{[1, 1, 0]^T, [0, 0, 1]^T\\}$\u3002\u8fd9\u662f\u4e00\u4e2a\u5e73\u9762\u3002\n\n**2. \u8fdb\u884c\u6b63\u4ea4\u5206\u89e3**\n\u6211\u4eec\u9700\u8981\u5c06 $\\vec{x} = [5, 1, 4]^T$ \u62c6\u89e3\u4e3a $\\vec{x}_{\\text{row}} + \\vec{x}_{\\text{null}}$\u3002\n\u6700\u7b80\u5355\u7684\u65b9\u6cd5\u662f\uff1a\u5148\u5229\u7528\u4e00\u7ef4\u6295\u5f71\u516c\u5f0f\uff0c\u6c42\u51fa $\\vec{x}$ \u5728\u96f6\u7a7a\u95f4\u76f4\u7ebf\u4e0a\u7684\u6295\u5f71 $\\vec{x}_{\\text{null}}$\u3002\n$$\\vec{x}_{\\text{null}} = \\text{proj}_{\\vec{v}}(\\vec{x}) = \\frac{\\vec{x} \\cdot \\vec{v}}{\\vec{v} \\cdot \\vec{v}} \\vec{v}$$\n\u4ee3\u5165\u6570\u636e\u8ba1\u7b97\u5185\u79ef\uff1a\n* $\\vec{x} \\cdot \\vec{v} = (5)(1) + (1)(-1) + (4)(0) = 4$\n* $\\vec{v} \\cdot \\vec{v} = (1)^2 + (-1)^2 + 0^2 = 2$\n$$\\vec{x}_{\\text{null}} = \\frac{4}{2} \\begin{bmatrix} 1 \\\\ -1 \\\\ 0 \\end{bmatrix} = 2 \\begin{bmatrix} 1 \\\\ -1 \\\\ 0 \\end{bmatrix} = \\begin{bmatrix} 2 \\\\ -2 \\\\ 0 \\end{bmatrix}$$\n\u6839\u636e\u5411\u91cf\u51cf\u6cd5\uff0c\u5269\u4e0b\u7684\u90e8\u5206\u5fc5\u7136\u5b8c\u5168\u843d\u5728\u884c\u7a7a\u95f4\u4e2d\uff1a\n$$\\vec{x}_{\\text{row}} = \\vec{x} - \\vec{x}_{\\text{null}} = \\begin{bmatrix} 5 \\\\ 1 \\\\ 4 \\end{bmatrix} - \\begin{bmatrix} 2 \\\\ -2 \\\\ 0 \\end{bmatrix} = \\begin{bmatrix} 3 \\\\ 3 \\\\ 4 \\end{bmatrix}$$\n*(\u9a8c\u8bc1\uff1a$\\vec{x}_{\\text{row}}$ \u786e\u5b9e\u7b49\u4e8e $3[1, 1, 0]^T + 4[0, 0, 1]^T$\uff0c\u5b8c\u7f8e\u5b58\u5728\u4e8e\u884c\u7a7a\u95f4\u5185)*\n\n**3. \u6620\u5c04\u9a8c\u8bc1**\n\u73b0\u5728\u8ba9\u77e9\u9635 $A$ \u5206\u522b\u4f5c\u7528\u4e8e\u8fd9\u4e09\u4e2a\u5411\u91cf\uff1a\n* **\u539f\u5411\u91cf\u6620\u5c04**\uff1a$$A\\vec{x} = \\begin{bmatrix} 1 & 1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} \\begin{bmatrix} 5 \\\\ 1 \\\\ 4 \\end{bmatrix} = \\begin{bmatrix} 6 \\\\ 4 \\end{bmatrix}$$\n* **\u884c\u7a7a\u95f4\u5206\u91cf\u6620\u5c04**\uff1a$$A\\vec{x}_{\\text{row}} = \\begin{bmatrix} 1 & 1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} \\begin{bmatrix} 3 \\\\ 3 \\\\ 4 \\end{bmatrix} = \\begin{bmatrix} 6 \\\\ 4 \\end{bmatrix}$$\n* **\u96f6\u7a7a\u95f4\u5206\u91cf\u6620\u5c04**\uff1a$$A\\vec{x}_{\\text{null}} = \\begin{bmatrix} 1 & 1 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} \\begin{bmatrix} 2 \\\\ -2 \\\\ 0 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}$$\n\n**\u7ed3\u8bba\uff1a**\n\u8ba1\u7b97\u7ed3\u679c\u5b8c\u7f8e\u8bc1\u5b9e\u4e86\u540c\u6784\u6620\u5c04\u7684\u903b\u8f91\uff1a$$A\\vec{x} = A\\vec{x}_{\\text{row}} + A\\vec{x}_{\\text{null}} = \\begin{bmatrix} 6 \\\\ 4 \\end{bmatrix} + \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix}$$\n\u8f93\u5165\u5411\u91cf\u4e2d $\\vec{x}_{\\text{null}}$ \u7684\u90e8\u5206\u5f7b\u5e95\u574d\u7f29\uff0c\u8361\u7136\u65e0\u5b58\uff1b\u800c $\\vec{x}_{\\text{row}}$ \u7684\u90e8\u5206\u5219\u6beb\u65e0\u6298\u635f\u5730\u6784\u6210\u4e86\u6700\u7ec8\u7684\u8f93\u51fa\u7ed3\u679c\u3002\n```\n\n\u901a\u8fc7\u4e0b\u65b9\u7684\u4ea4\u4e92\u5f0f 3D \u6a21\u578b $\\text{(Interactive 3D Model)}$\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u89c2\u5730\u89c2\u5bdf\u5230\u77e9\u9635 $A$ \u5bf9\u8f93\u5165\u7a7a\u95f4\u7684\u201c\u4fee\u526a\u201d\u8fc7\u7a0b $\\text{(Pruning Process)}$\u3002\u8be5\u6a21\u578b\u6f14\u793a\u4e86\u5f53 $\\vec{x} = [5, 1, 4]^T$ \u4f5c\u7528\u4e8e\u79e9\u4e3a 2 \u7684 $2 \\times 3$ \u77e9\u9635\u65f6\uff0c\u51e0\u4f55\u5143\u7d20\u7684\u5bf9\u5e94\u5173\u7cfb\uff1a\n\n**1. \u9ed1\u8272\u5411\u91cf**\uff1a\u8f93\u5165\u7684\u539f\u59cb\u8f93\u5165\u5411\u91cf $\\vec{x}$ $\\text{(Input Signal)}$\u3002\n**2. \u84dd\u8272\u5411\u91cf**\uff1a\u884c\u7a7a\u95f4\u6295\u5f71\u5206\u91cf $\\vec{x}_{\\text{row}}$ $\\text{(Effective Component)}$\uff0c\u4ee3\u8868\u8f93\u5165\u4e2d\u80fd\u88ab\u4f20\u9012\u7684\u6709\u6548\u4fe1\u53f7\u3002\n**3. \u7ea2\u8272\u5411\u91cf**\uff1a\u96f6\u7a7a\u95f4\u6295\u5f71\u5206\u91cf $\\vec{x}_{\\text{null}}$ $\\text{(Null\/Noise Component)}$\uff0c\u4ee3\u8868\u6620\u5c04\u4e2d\u4f1a\u88ab\u6ee4\u9664\u7684\u5197\u4f59\u4fe1\u606f\u3002\n**4. \u7d2b\u8272\u5411\u91cf**\uff1a\u7ecf\u8fc7\u77e9\u9635\u53d8\u6362\u8fd0\u7b97\u540e\u5f97\u5230\u7684\u8f93\u51fa\u5411\u91cf $A\\vec{x} \\space \\text{(Mapping Result)}$\u3002\n**5. \u9752\u8272\u5e73\u9762**\uff1a\u77e9\u9635\u7684\u884c\u7a7a\u95f4 $\\text{row}(A)$\uff0c\u5373\u4fe1\u606f\u7684\u201c\u901a\u5e26\u201d\u3002\n**6. \u7c89\u8272\u76f4\u7ebf**\uff1a\u77e9\u9635\u7684\u96f6\u7a7a\u95f4 $\\text{null}(A)$\uff0c\u5373\u4fe1\u606f\u7684\u201c\u963b\u5e26\u201d\u3002\n\n<iframe loading=\"lazy\" src=\"https:\/\/www.geogebra.org\/calculator\/qt7hfpyh?embed\" width=\"100%\" height=\"600\" allowfullscreen style=\"border: 1px solid #e4e4e4;border-radius: 4px;\" frameborder=\"0\"><\/iframe><\/p>\n<p>**\u7b80\u8981\u53d9\u8ff0\uff1a**<br \/>\n\u77e9\u9635 $A$ \u7684\u6620\u5c04\u672c\u8d28\u4e0a\u662f\u4e00\u4e2a**\u8fc7\u6ee4\u5668 $\\text{(Filter)}$**\u3002\u7531\u4e8e\u7c89\u8272\u76f4\u7ebf\u6240\u5728\u7684\u96f6\u7a7a\u95f4\u662f\u77e9\u9635\u7684\u201c\u76f2\u533a\u201d\uff0c\u7ea2\u8272\u5206\u91cf $\\vec{x}_{\\text{null}}$ \u5728\u53d8\u6362\u540e\u4f1a\u5f7b\u5e95\u201c\u574d\u7f29\u201d\u5f52\u96f6\uff1b\u800c\u9752\u8272\u5e73\u9762\uff08\u884c\u7a7a\u95f4\uff09\u5219\u662f\u901a\u5f80\u8f93\u51fa\u7aef\u7684\u552f\u4e00\u901a\u9053\u3002\u6700\u7ec8\u770b\u5230\u7684\u7d2b\u8272\u8f93\u51fa\u5411\u91cf $A\\vec{x}$\uff0c\u5176\u6570\u503c\u548c\u65b9\u5411\u5b8c\u5168\u7531\u84dd\u8272\u5206\u91cf $\\vec{x}_{\\text{row}}$ \u51b3\u5b9a\u3002\u8fd9\u610f\u5473\u7740\uff0c\u53ea\u8981\u84dd\u8272\u5206\u91cf\u4fdd\u6301\u4e0d\u53d8\uff0c\u65e0\u8bba\u4f60\u5982\u4f55\u6cbf\u7740\u96f6\u7a7a\u95f4\u65b9\u5411\u62c9\u957f\u7ea2\u8272\u5411\u91cf\uff0c\u7d2b\u8272\u8f93\u51fa\u7ed3\u679c\u90fd\u4f1a\u7a33\u5982\u6cf0\u5c71\u3001\u7eb9\u4e1d\u4e0d\u52a8\u3002\u8fd9\u79cd\u5206\u89e3\u63ed\u793a\u4e86\u77e9\u9635\u5982\u4f55\u4ece\u6742\u4e71\u7684\u8f93\u5165\u4e2d\u7cbe\u51c6\u63d0\u53d6\u51fa\u6838\u5fc3\u7279\u5f81\u5e76\u5c06\u5176\u8de8\u7a7a\u95f4\u642c\u8fd0\u3002<\/p>\n<p>### \u2b50\u6b63\u4ea4\u5206\u89e3\u53ef\u89c6\u5316<\/p>\n<p>\u4f60\u53ef\u4ee5\u901a\u8fc7\u62d6\u52a8\u4e0b\u65b9 3D \u6a21\u578b\uff0c\u89c2\u5bdf\u5411\u91cf $\\vec{x}$ \u5982\u4f55\u88ab\u62c6\u89e3\u4e3a\u4e24\u4e2a\u5782\u76f4\u7684\u5206\u91cf\uff0c\u5e76\u7406\u89e3\u4e3a\u4ec0\u4e48\u53ea\u6709\u884c\u7a7a\u95f4\u5206\u91cf\u80fd\u901a\u8fc7\u201c\u6620\u5c04\u5927\u95e8\u201d\u3002<\/p>\n<p>**\u5373\u4e3a\uff1a**<\/p>\n<p>\u901a\u8fc7\u8fd9\u4e2a\u4f8b\u9898\u6211\u4eec\u53ef\u4ee5\u53d1\u73b0\uff0c\u77e9\u9635 $A$ \u5c31\u50cf\u662f\u4e00\u4e2a\u5177\u6709**\u65b9\u5411\u9009\u62e9\u6027**\u7684\u8fc7\u6ee4\u5668\u3002<\/p>\n<p>\u5b83\u5728\u8f93\u5165\u7a7a\u95f4\u4e2d\u5efa\u7acb\u4e86\u4e00\u5957\u201c\u5782\u76f4\u201d\u7684\u5ba1\u67e5\u673a\u5236\uff1a\u51e1\u662f\u843d\u5728**\u96f6\u7a7a\u95f4 $\\text{(Null Space)}$** \u91cc\u7684\u5206\u91cf\uff0c\u65e0\u8bba\u591a\u5927\uff0c\u90fd\u4f1a\u88ab\u7edf\u7edf\u62b9\u9664\uff1b\u53ea\u6709\u843d\u5728**\u884c\u7a7a\u95f4 $\\text{(Row Space)}$** \u91cc\u7684\u5206\u91cf\uff0c\u624d\u80fd\u5e26\u7740\u5b83\u7684\u4fe1\u606f\uff0c\u8de8\u8d8a\u7a7a\u95f4\u7684\u9e3f\u6c9f\uff0c\u6295\u5c04\u5230\u8f93\u51fa\u7a7a\u95f4\u7684\u5217\u7a7a\u95f4\u4e2d\u3002<\/p>\n<p>\u8fd9\u79cd**\u6b63\u4ea4\u5206\u89e3 $\\text{(Orthogonal Decomposition)}$** \u7684\u89c6\u89d2\uff0c\u6b63\u662f\u6211\u4eec\u540e\u7eed\u7406\u89e3\u201c\u6700\u5c0f\u4e8c\u4e58\u6cd5\u201d\u548c\u201c\u5947\u5f02\u503c\u5206\u89e3 (SVD)\u201d\u7684\u51e0\u4f55\u57fa\u77f3\u3002<\/p>\n<p>**\u540c\u6784\u6620\u5c04 $\\text{(Isomorphism)}$ \u7684\u4e25\u683c\u8bc1\u660e\uff1a**<br \/>\n\u6211\u4eec\u4e0d\u4ec5\u77e5\u9053\u8f93\u51fa\u7531 $\\vec{x}_{\\text{row}}$ \u51b3\u5b9a\uff0c\u8fd8\u80fd\u8bc1\u660e\u77e9\u9635 $A$ \u5728\u201c\u884c\u7a7a\u95f4\uff08\u6709\u6548\u8f93\u5165\uff09\u201d\u548c\u201c\u5217\u7a7a\u95f4\uff08\u5168\u90e8\u8f93\u51fa\uff09\u201d\u4e4b\u95f4\u5efa\u7acb\u4e86\u4e00\u4e2a**\u5b8c\u7f8e\u7684 1:1 \u6620\u5c04\uff08\u53cc\u5c04\uff09**\u3002<br \/>\n\u5047\u8bbe\u884c\u7a7a\u95f4\u4e2d\u6709\u4e24\u4e2a\u4e0d\u540c\u7684\u5411\u91cf $\\vec{x}_{\\text{row}, 1}$ \u548c $\\vec{x}_{\\text{row}, 2}$\uff0c\u7ecf\u8fc7 $A$ \u6620\u5c04\u540e\u5f97\u5230\u4e86\u76f8\u540c\u7684\u8f93\u51fa\u5411\u91cf\uff0c\u5373\uff1a<br \/>\n$$A\\vec{x}_{\\text{row}, 1} = A\\vec{x}_{\\text{row}, 2}$$<\/p>\n<p>\u5219\u6709\uff1a<br \/>\n$$A(\\vec{x}_{\\text{row}, 1} - \\vec{x}_{\\text{row}, 2}) = \\vec{0}$$<\/p>\n<p>\u8fd9\u610f\u5473\u7740\u5dee\u5411\u91cf $(\\vec{x}_{\\text{row}, 1} - \\vec{x}_{\\text{row}, 2})$ \u5fc5\u987b\u5c5e\u4e8e\u96f6\u7a7a\u95f4 $\\text{null}(A)$\u3002<br \/>\n\u4f46\u540c\u65f6\uff0c\u56e0\u4e3a $\\vec{x}_{\\text{row}, 1}$ \u548c $\\vec{x}_{\\text{row}, 2}$ \u90fd\u5728\u884c\u7a7a\u95f4 $\\text{row}(A)$ \u4e2d\uff0c\u6839\u636e\u5b50\u7a7a\u95f4\u7684\u5c01\u95ed\u6027\uff0c\u5b83\u4eec\u7684\u5dee\u5411\u91cf\u4e5f\u5fc5\u987b\u5c5e\u4e8e\u884c\u7a7a\u95f4\u3002\u552f\u4e00\u80fd\u540c\u65f6\u5c5e\u4e8e\u4e24\u4e2a\u6b63\u4ea4\u8865\u7a7a\u95f4\u7684\u5411\u91cf\uff0c\u53ea\u6709\u96f6\u5411\u91cf $\\vec{0}$\u3002<br \/>\n\u56e0\u6b64\uff1a<br \/>\n$$\\vec{x}_{\\text{row}, 1} - \\vec{x}_{\\text{row}, 2} = \\vec{0} \\implies \\vec{x}_{\\text{row}, 1} = \\vec{x}_{\\text{row}, 2}$$<\/p>\n<p>**\u6df1\u523b\u7ed3\u8bba**\uff1a<br \/>\n\u884c\u7a7a\u95f4\u4e2d\u7684\u4e0d\u540c\u5411\u91cf\uff0c\u7edd\u4e0d\u53ef\u80fd\u6620\u5c04\u5230\u5217\u7a7a\u95f4\u4e2d\u7684\u540c\u4e00\u70b9\u3002\u77e9\u9635 $A$ \u7684\u53d8\u6362\u8fc7\u7a0b\uff0c\u672c\u8d28\u4e0a\u662f**\u5c06\u8f93\u5165\u4e16\u754c $\\mathbb{R}^n$ \u4e2d\u7684\u96f6\u7a7a\u95f4\u5206\u91cf\u201c\u65e0\u60c5\u5f52\u96f6\u201d\uff0c\u5e76\u5c06\u8f93\u5165\u4e16\u754c\u4e2d\u7684\u884c\u7a7a\u95f4\uff08\u6709\u6548\u8f93\u5165\uff09\uff0c\u4ee5\u6781\u5176\u7cbe\u786e\u7684 1:1 \u540c\u6784\u6620\u5c04\uff0c\u5b8c\u5168\u5e73\u79fb\u5e76\u62c9\u4f38\u6210\u4e86\u8f93\u51fa\u4e16\u754c $\\mathbb{R}^m$ \u4e2d\u7684\u5217\u7a7a\u95f4\uff08\u5168\u90e8\u8f93\u51fa\uff09**\u3002<\/p>\n<p>```ad-example<br \/>\ntitle: \u4f8b\u9898 \u7efc\u5408\u4f53\u4f1a\u77e9\u9635\u7684\u4e09\u5927\u5b50\u7a7a\u95f4\u6620\u5c04<\/p>\n<p>\u5df2\u77e5 $3 \\times 3$ \u77e9\u9635 $A = \\begin{bmatrix} 1 & 2 & 3 \\\\ 0 & 1 & 1 \\\\ 2 & 5 & 7 \\end{bmatrix}$\uff0c\u5b83\u4ee3\u8868\u4e86\u4e00\u4e2a\u4ece $\\mathbb{R}^3$ \u5230 $\\mathbb{R}^3$ \u7684\u7ebf\u6027\u53d8\u6362 $T(\\vec{x}) = A\\vec{x}$\u3002<\/p>\n<p>**\u4efb\u52a1\uff1a** \u8ba1\u7b97\u5e76\u63cf\u8ff0\u8be5\u77e9\u9635\u7684\u96f6\u7a7a\u95f4 $\\text{null}(A)$\u3001\u884c\u7a7a\u95f4 $\\text{row}(A)$ \u548c\u5217\u7a7a\u95f4 $\\text{col}(A)$ \u7684\u57fa\u5e95\u3001\u7ef4\u5ea6\u53ca\u51e0\u4f55\u5f62\u6001\u3002<\/p>\n<p>---<\/p>\n<p>**\u7b2c\u4e00\u6b65\uff1a\u77e9\u9635\u7684\u884c\u5316\u7b80 (Row Reduction)**<br \/>\n\u4e3a\u4e86\u63ed\u793a\u7a7a\u95f4\u7ed3\u6784\uff0c\u6211\u4eec\u9996\u5148\u5c06 $A$ \u5316\u7b80\u4e3a\u7b80\u5316\u884c\u9636\u68af\u5f62\u77e9\u9635 $\\text{(RREF, Reduced Row Echelon Form)}$\uff1a<br \/>\n$$A = \\begin{bmatrix} 1 & 2 & 3 \\\\ 0 & 1 & 1 \\\\ 2 & 5 & 7 \\end{bmatrix} \\xrightarrow{R_3 - 2R_1} \\begin{bmatrix} 1 & 2 & 3 \\\\ 0 & 1 & 1 \\\\ 0 & 1 & 1 \\end{bmatrix} \\xrightarrow{R_3 - R_2} \\begin{bmatrix} 1 & 2 & 3 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 0 \\end{bmatrix} \\xrightarrow{R_1 - 2R_2} \\begin{bmatrix} 1 & 0 & 1 \\\\ 0 & 1 & 1 \\\\ 0 & 0 & 0 \\end{bmatrix} = R$$<br \/>\n\u89c2\u5bdf $R$ \u53ef\u77e5\uff1a\u4e3b\u5143\u5728\u7b2c 1\u30012 \u5217\uff0c\u7b2c 3 \u5217\u5bf9\u5e94\u81ea\u7531\u53d8\u91cf\u3002\u56e0\u6b64\uff0c\u77e9\u9635\u7684\u79e9 $\\text{rank}(A) = 2$\u3002<\/p>\n<p>---<\/p>\n<p>**1. \u96f6\u7a7a\u95f4 $\\text{null}(A)$\uff1a\u88ab\u574d\u7f29\u7684\u7ef4\u5ea6**<br \/>\n**\u8ba1\u7b97\uff1a** \u89e3\u9f50\u6b21\u65b9\u7a0b\u7ec4 $R\\vec{x} = \\vec{0}$\u3002<br \/>\n$$\\begin{cases} x_1 + x_3 = 0 \\implies x_1 = -x_3 \\\\ x_2 + x_3 = 0 \\implies x_2 = -x_3 \\end{cases}$$<br \/>\n\u4ee4\u81ea\u7531\u53d8\u91cf $x_3 = t$\uff08$t$ \u4e3a\u4efb\u610f\u5b9e\u6570\uff09\uff0c\u5219\u89e3\u96c6\u4e3a $\\vec{x} = t \\begin{bmatrix} -1 \\\\ -1 \\\\ 1 \\end{bmatrix}$\u3002<br \/>\n**\u57fa\u5e95\uff1a** $\\left\\{ \\begin{bmatrix} -1 \\\\ -1 \\\\ 1 \\end{bmatrix} \\right\\}$<br \/>\n**\u7a7a\u95f4\u8868\u793a\uff1a** $\\text{null}(A) = \\text{span}\\left\\{ \\begin{bmatrix} -1 \\\\ -1 \\\\ 1 \\end{bmatrix} \\right\\}$<br \/>\n**\u7ef4\u5ea6\uff1a** $\\dim(\\text{null}(A)) = 1$\uff08\u4e5f\u79f0\u4e3a\u96f6\u5316\u5ea6 $\\text{nullity} = 1$\uff09\u3002<br \/>\n**\u51e0\u4f55\u63cf\u8ff0\uff1a** \u5728\u8f93\u5165\u7a7a\u95f4 $\\mathbb{R}^3$ \u4e2d\uff0c\u8fd9\u662f\u4e00\u6761\u7a7f\u8fc7\u539f\u70b9\u3001\u65b9\u5411\u5411\u91cf\u4e3a $[-1, -1, 1]^T$ \u7684**\u76f4\u7ebf**\u3002\u5728\u8fd9\u6761\u76f4\u7ebf\u4e0a\u7684\u6240\u6709\u5411\u91cf\uff0c\u5728\u7ecf\u8fc7 $A$ \u7684\u53d8\u6362\u540e\uff0c\u90fd\u4f1a\u88ab\u201c\u7c89\u788e\u201d\u6210\u8f93\u51fa\u7a7a\u95f4\u7684\u539f\u70b9 $\\vec{0}$\u3002<\/p>\n<p>---<\/p>\n<p>**2. \u884c\u7a7a\u95f4 $\\text{row}(A)$\uff1a\u5b58\u6d3b\u7684\u6709\u6548\u8f93\u5165**<br \/>\n**\u8ba1\u7b97\uff1a** \u884c\u7a7a\u95f4\u4e0d\u4f1a\u56e0\u4e3a\u521d\u7b49\u884c\u53d8\u6362\u800c\u6539\u53d8\uff0c\u56e0\u6b64 $\\text{RREF}$ \u77e9\u9635 $R$ \u7684\u975e\u96f6\u884c\u5c31\u662f $\\text{row}(A)$ \u7684\u4e00\u7ec4\u6700\u7b80\u57fa\u5e95\u3002<br \/>\n**\u57fa\u5e95\uff1a** $\\left\\{ \\begin{bmatrix} 1 \\\\ 0 \\\\ 1 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 1 \\\\ 1 \\end{bmatrix} \\right\\}$<br \/>\n**\u7a7a\u95f4\u8868\u793a\uff1a** $\\text{row}(A) = \\text{span}\\left\\{ \\begin{bmatrix} 1 \\\\ 0 \\\\ 1 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 1 \\\\ 1 \\end{bmatrix} \\right\\}$<br \/>\n**\u7ef4\u5ea6\uff1a** $\\dim(\\text{row}(A)) = 2$\u3002<br \/>\n**\u51e0\u4f55\u63cf\u8ff0\uff1a** \u5728\u8f93\u5165\u7a7a\u95f4 $\\mathbb{R}^3$ \u4e2d\uff0c\u8fd9\u662f\u7531\u4e24\u4e2a\u57fa\u5411\u91cf\u5f20\u6210\u7684\u4e00\u4e2a**\u4e8c\u7ef4\u5e73\u9762**\uff08\u5176\u6cd5\u5411\u91cf\u6070\u597d\u662f\u96f6\u7a7a\u95f4\u7684\u57fa\u5e95 $[-1, -1, 1]^T$\uff0c\u8fd9\u8bc1\u660e\u4e86\u884c\u7a7a\u95f4\u4e0e\u96f6\u7a7a\u95f4\u5728\u8f93\u5165\u4e16\u754c\u91cc\u662f\u4e92\u76f8\u5782\u76f4\u7684\u201c\u6b63\u4ea4\u8865\u201d\uff09\u3002\u8fd9\u4e2a\u5e73\u9762\u4e0a\u7684\u5411\u91cf\u662f\u201c\u6709\u6548\u8f93\u5165\u201d\uff0c\u5b83\u4eec\u5728\u53d8\u6362\u4e2d\u5b58\u6d3b\u4e86\u4e0b\u6765\u3002<\/p>\n<p>---<\/p>\n<p>**3. \u5217\u7a7a\u95f4 $\\text{col}(A)$\uff1a\u53d8\u6362\u7684\u503c\u57df\u6295\u5f71**<br \/>\n**\u8ba1\u7b97\uff1a** \u6839\u636e\u5b9a\u7406\uff0c\u4e3b\u5143\u5217\uff08\u7b2c 1\u30012 \u5217\uff09\u5bf9\u5e94\u7684**\u539f\u77e9\u9635 $A$** \u4e2d\u7684\u5217\u6784\u6210\u4e86\u5217\u7a7a\u95f4\u7684\u57fa\u5e95\u3002\uff08\u6ce8\u610f\uff1a\u5343\u4e07\u4e0d\u80fd\u7528 $R$ \u7684\u5217\uff0c\u884c\u53d8\u6362\u7834\u574f\u4e86\u5217\u7a7a\u95f4\uff01\uff09<br \/>\n**\u57fa\u5e95\uff1a** $\\left\\{ \\begin{bmatrix} 1 \\\\ 0 \\\\ 2 \\end{bmatrix}, \\begin{bmatrix} 2 \\\\ 1 \\\\ 5 \\end{bmatrix} \\right\\}$<br \/>\n**\u7a7a\u95f4\u8868\u793a\uff1a** $\\text{col}(A) = \\text{span}\\left\\{ \\begin{bmatrix} 1 \\\\ 0 \\\\ 2 \\end{bmatrix}, \\begin{bmatrix} 2 \\\\ 1 \\\\ 5 \\end{bmatrix} \\right\\}$<br \/>\n**\u7ef4\u5ea6\uff1a** $\\dim(\\text{col}(A)) = 2$\u3002<br \/>\n**\u51e0\u4f55\u63cf\u8ff0\uff1a** \u5728\u8f93\u51fa\u7a7a\u95f4 $\\mathbb{R}^3$ \u4e2d\uff0c\u8fd9\u662f\u4e00\u4e2a\u7ecf\u8fc7\u539f\u70b9\u7684**\u4e8c\u7ef4\u5e73\u9762**\u3002<\/p>\n<p>---<\/p>\n<p>**\u7ed3\u8bba\uff1a\u51e0\u4f55\u6620\u5c04\u7684\u5b8f\u89c2\u89c6\u89d2 $T(\\vec{x}) = A\\vec{x}$**<br \/>\n\u5728\u8fd9\u4e2a $\\mathbb{R}^3 \\to \\mathbb{R}^3$ \u7684\u53d8\u6362\u4e2d\uff1a<br \/>\n\u8f93\u5165\u7a7a\u95f4\uff08\u4e09\u7ef4\uff09\u88ab\u77e9\u9635 $A$ \u5288\u6210\u4e86\u4e24\u534a\uff1a\u4e00\u6761\u4e00\u7ef4\u7684\u76f4\u7ebf\uff08\u96f6\u7a7a\u95f4\uff09\u548c\u4e00\u4e2a\u4e8c\u7ef4\u7684\u5e73\u9762\uff08\u884c\u7a7a\u95f4\uff09\u3002<br \/>\n\u5f53\u53d8\u6362\u53d1\u751f\u65f6\uff0c\u90a3\u6761\u76f4\u7ebf\u4e0a\u7684\u6240\u6709\u70b9\u90fd\u88ab\u538b\u7f29\u6210\u4e86\u5bf9\u9762\u4e16\u754c\u7684\u4e00\u4e2a\u9ed1\u6d1e\uff08\u539f\u70b9 $\\vec{0}$\uff09\uff1b\u800c\u90a3\u4e2a\u4e8c\u7ef4\u5e73\u9762\uff08\u884c\u7a7a\u95f4\uff09\u5219\u88ab\u77e9\u9635 $A$ \u50cf\u8f6c\u52a8\u4e00\u5757\u7eb8\u677f\u4e00\u6837\uff0c\u4ee5 1:1 \u7684\u65e0\u635f\u6bd4\u4f8b\uff0c\u7cbe\u51c6\u5730\u8d34\u5408\u5230\u4e86\u8f93\u51fa\u7a7a\u95f4\u7684\u53e6\u4e00\u4e2a\u4e8c\u7ef4\u5e73\u9762\uff08\u5217\u7a7a\u95f4\uff09\u4e0a\u3002<\/p>\n<p>\u8fd9\u4e5f\u5b8c\u7f8e\u5370\u8bc1\u4e86\u4e0a\u4e00\u8282\u7684**\u79e9\u5b9a\u7406**\uff1a$\\text{rank}(A) + \\text{nullity}(A) = 2 + 1 = 3$\uff08\u8f93\u5165\u7a7a\u95f4\u7684\u7ef4\u5ea6\uff09\u3002<br \/>\n```<\/p>\n<p>```ad-example<br \/>\ntitle:\u4f8b\u98983.41 \u5224\u5b9a\u5411\u91cf\u4e0e\u5b50\u7a7a\u95f4\u7684\u4ece\u5c5e\u5173\u7cfb\u53ca\u7a7a\u95f4\u63cf\u8ff0<\/p>\n<p>\u5df2\u77e5\u77e9\u9635 $A = \\begin{bmatrix} 1 & -1 \\\\ 0 & 1 \\\\ 3 & -3 \\end{bmatrix}$\u3002<\/p>\n<p>---<\/p>\n<p>**\u95ee\u9898 $\\text{(a)}$\uff1a\u5224\u65ad\u5411\u91cf $\\vec{b} = [1, 2, 3]^T$ \u662f\u5426\u5728 $A$ \u7684\u5217\u7a7a\u95f4 $\\text{col}(A)$ \u4e2d\uff1f**<\/p>\n<p>**\u89e3\u9898\u601d\u8def**\uff1a\u6839\u636e\u5b9a\u4e49\uff0c$\\vec{b}$ \u5728 $\\text{col}(A)$ \u4e2d\u5f53\u4e14\u4ec5\u5f53\u7ebf\u6027\u65b9\u7a0b\u7ec4 $A\\vec{x} = \\vec{b}$ \u6709\u89e3\uff08\u4e00\u81f4\u6027\uff09\u3002<br \/>\n**\u8ba1\u7b97\u8fc7\u7a0b**\uff1a\u6784\u9020\u589e\u5e7f\u77e9\u9635\u5e76\u8fdb\u884c\u884c\u5316\u7b80\uff1a<br \/>\n$$[A \\mid \\vec{b}] = \\left[ \\begin{array}{cc|c} 1 & -1 & 1 \\\\ 0 & 1 & 2 \\\\ 3 & -3 & 3 \\end{array} \\right] \\xrightarrow{R_3 - 3R_1} \\left[ \\begin{array}{cc|c} 1 & -1 & 1 \\\\ 0 & 1 & 2 \\\\ 0 & 0 & 0 \\end{array} \\right] \\xrightarrow{R_1 + R_2} \\left[ \\begin{array}{cc|c} 1 & 0 & 3 \\\\ 0 & 1 & 2 \\\\ 0 & 0 & 0 \\end{array} \\right]$$<br \/>\n**\u7ed3\u8bba**\uff1a\u6700\u540e\u4e00\u884c\u5168\u4e3a 0\uff0c\u65b9\u7a0b\u7ec4\u6709\u89e3\uff08\u4e14\u6709\u552f\u4e00\u89e3\uff09\u3002\u56e0\u6b64\uff0c$\\vec{b} \\in \\text{col}(A)$\u3002<\/p>\n<p>---<\/p>\n<p>**\u95ee\u9898 $\\text{(b)}$\uff1a\u5224\u65ad\u5411\u91cf $\\vec{w} = [4, 5]$ \u662f\u5426\u5728 $A$ \u7684\u884c\u7a7a\u95f4 $\\text{row}(A)$ \u4e2d\uff1f**<\/p>\n<p>**\u89e3\u9898\u601d\u8def**\uff1a\u82e5 $\\vec{w}$ \u662f $A$ \u884c\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u5219\u5c06 $\\vec{w}$ \u62fc\u5728 $A$ \u4e0b\u65b9\u5f62\u6210\u589e\u5e7f\u77e9\u9635 $\\begin{bmatrix} A \\\\ \\vec{w} \\end{bmatrix}$\uff0c\u5229\u7528 $A$ \u7684\u884c\u53ef\u4ee5\u5c06 $\\vec{w}$ \u5b8c\u5168\u6d88\u5143\u3002<br \/>\n**\u8ba1\u7b97\u8fc7\u7a0b**\uff1a\u6784\u9020\u4e0a\u4e0b\u589e\u5e7f\u77e9\u9635\u5e76\u8fdb\u884c\u884c\u6d88\u5143\uff1a<br \/>\n$$\\begin{bmatrix} A \\\\ \\hline \\vec{w} \\end{bmatrix} = \\left[ \\begin{array}{cc} 1 & -1 \\\\ 0 & 1 \\\\ 3 & -3 \\\\ \\hline 4 & 5 \\end{array} \\right] \\xrightarrow{R_3 - 3R_1, R_4 - 4R_1} \\left[ \\begin{array}{cc} 1 & -1 \\\\ 0 & 1 \\\\ 0 & 0 \\\\ \\hline 0 & 9 \\end{array} \\right] \\xrightarrow{R_4 - 9R_2} \\left[ \\begin{array}{cc} 1 & -1 \\\\ 0 & 1 \\\\ 0 & 0 \\\\ \\hline 0 & 0 \\end{array} \\right]$$<br \/>\n**\u7ed3\u8bba**\uff1a$\\vec{w}$ \u6210\u529f\u5316\u4e3a\u5168 0 \u884c\uff0c\u8bf4\u660e $\\vec{w}$ \u662f $A$ \u884c\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\u3002\u56e0\u6b64\uff0c$\\vec{w} \\in \\text{row}(A)$\u3002<\/p>\n<p>---<\/p>\n<p>**\u95ee\u9898 $\\text{(c)}$\uff1a\u63cf\u8ff0 $\\text{row}(A)$ \u548c $\\text{col}(A)$ \u7684\u51e0\u4f55\u5f62\u6001\u3002**<\/p>\n<p>**\u5bf9\u4e8e $\\text{row}(A)$**\uff1a<br \/>\n\u53ef\u4ee5\u9a8c\u8bc1\u5bf9\u4e8e\u4efb\u610f\u5411\u91cf $\\vec{w} = [x, y]$\uff0c\u589e\u5e7f\u77e9\u9635 $\\begin{bmatrix} A \\\\ \\vec{w} \\end{bmatrix}$ \u90fd\u80fd\u88ab\u5316\u7b80\u4e3a\u57ab\u5e95\u5168 0 \u884c\u7684\u5f62\u5f0f\u3002\u8fd9\u610f\u5473\u7740 $\\mathbb{R}^2$ \u4e2d\u7684\u6bcf\u4e00\u4e2a\u5411\u91cf\u90fd\u5728 $\\text{row}(A)$ \u4e2d\u3002<br \/>\n**\u7ed3\u8bba**\uff1a$\\text{row}(A) = \\mathbb{R}^2$\u3002<\/p>\n<p>**\u5bf9\u4e8e $\\text{col}(A)$**\uff1a<br \/>\n$\\text{col}(A)$ \u662f $\\mathbb{R}^3$ \u7684\u5b50\u7a7a\u95f4\u3002\u901a\u8fc7\u89e3\u65b9\u7a0b\u53ef\u77e5\uff0c\u5176\u5217\u5411\u91cf\u5f20\u6210\u7684\u7a7a\u95f4\u662f $\\mathbb{R}^3$ \u4e2d\u6ee1\u8db3\u65b9\u7a0b $3x - z = 0$ \u7684\u4e00\u4e2a\u8fc7\u539f\u70b9\u7684\u5e73\u9762\u3002<\/p>\n<p>**\u5907\u6ce8:**<br \/>\n\u5224\u5b9a\u884c\u7a7a\u95f4\u96b6\u5c5e\u5173\u7cfb\u8fd8\u6709\u53e6\u4e00\u79cd\u65b9\u6cd5\uff1a\u7531\u4e8e $A$ \u7684\u884c\u5411\u91cf\u5c31\u662f $A^T$ \u7684\u5217\u5411\u91cf\uff0c\u56e0\u6b64\u201c$\\vec{w}$ \u662f\u5426\u5728 $\\text{row}(A)$ \u4e2d\u201d\u7b49\u4ef7\u4e8e\u201c$\\vec{w}^T$ \u662f\u5426\u5728 $\\text{col}(A^T)$ \u4e2d\u201d\u3002\u8fd9\u53ef\u4ee5\u5c06\u95ee\u9898 $\\text{(b)}$ \u8f6c\u5316\u56de\u95ee\u9898 $\\text{(a)}$ \u4e2d\u7684 $A^T\\vec{x} = \\vec{w}^T$ \u65b9\u7a0b\u6c42\u89e3\u95ee\u9898\u3002<br \/>\n```<\/p>\n<p>```ad-theorem<br \/>\ntitle: \u5b9a\u7406 3.20\uff1a\u884c\u7b49\u4ef7\u77e9\u9635\u5177\u6709\u76f8\u540c\u7684\u884c\u7a7a\u95f4<br \/>\n\u8bbe $B$ \u662f\u4efb\u610f\u4e00\u4e2a\u4e0e\u77e9\u9635 $A$ \u884c\u7b49\u4ef7\u7684\u77e9\u9635\u3002\u5219 $\\text{row}(B) = \\text{row}(A)$\u3002<br \/>\n```<\/p>\n<p>**\u8bc1\u660e\uff1a**<br \/>\n\u77e9\u9635 $A$ \u53ef\u4ee5\u901a\u8fc7\u4e00\u7cfb\u5217\u521d\u7b49\u884c\u53d8\u6362\u8f6c\u5316\u4e3a $B$\u3002\u56e0\u6b64\uff0c$B$ \u7684\u5404\u4e2a\u884c\u5411\u91cf\u90fd\u662f $A$ \u7684\u884c\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\uff1b\u4ece\u800c\uff0c$B$ \u7684\u884c\u5411\u91cf\u7684\u4efb\u4f55\u7ebf\u6027\u7ec4\u5408\u4e5f\u5fc5\u7136\u662f $A$ \u7684\u884c\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\u3002\u7531\u6b64\u53ef\u5f97 $\\text{row}(B) \\subseteq \\text{row}(A)$\u3002<\/p>\n<p>\u53e6\u4e00\u65b9\u9762\uff0c\u7531\u4e8e\u521d\u7b49\u884c\u53d8\u6362\u662f\u53ef\u9006\u7684\uff0c\u9006\u8f6c\u8fd9\u4e9b\u884c\u64cd\u4f5c\u5373\u53ef\u5c06 $B$ \u8f6c\u5316\u4e3a $A$\u3002\u56e0\u6b64\uff0c\u540c\u6837\u7684\u903b\u8f91\u8868\u660e $\\text{row}(A) \\subseteq \\text{row}(B)$\u3002<\/p>\n<p>\u7efc\u5408\u8fd9\u4e24\u4e2a\u5305\u542b\u5173\u7cfb\uff0c\u6211\u4eec\u5f97\u5230\u7ed3\u8bba\uff1a$\\text{row}(A) = \\text{row}(B)$\u3002<\/p>\n<p>**\u5373\u4e3a\uff1a**<br \/>\n\u5bf9\u4e00\u77e9\u9635\u53ea\u8fdb\u884c\u82e5\u5e72\u6b21**\u521d\u7b49\u884c\u53d8\u6362 $\\text{(Elementary Row Operations)}$**\uff0c\u8fdb\u884c\u884c\u53d8\u6362\u524d\u540e\u7684\u77e9\u9635\u662f**\u884c\u7b49\u4ef7 $\\text{(Row Equivalent)}$** \u7684\uff0c\u53d8\u6362\u524d\u540e\u4e24\u77e9\u9635\u5bf9\u5e94\u7684**\u884c\u7a7a\u95f4 $\\text{(Row Space)}$** \u4e5f\u662f\u76f8\u540c\u7684\u3002<\/p>\n<p>```ad-theorem<br \/>\ntitle: \u5b9a\u7406 3.21\uff1a\u96f6\u7a7a\u95f4\u662f\u4e00\u4e2a\u5b50\u7a7a\u95f4<br \/>\n\u8bbe $A$ \u662f\u4e00\u4e2a $m \\times n$ \u77e9\u9635\uff0c\u4ee4 $N$ \u4e3a\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4 $A\\vec{x} = \\vec{0}$ \u7684\u89e3\u96c6\u3002\u5219 $N$ \u662f $\\mathbb{R}^n$ \u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4\u3002<br \/>\n```<\/p>\n<p>**\u8bc1\u660e\uff1a**<br \/>\n*\u6ce8\uff1a\u4e3a\u4e86\u4f7f\u77e9\u9635\u4e58\u6cd5 $A\\vec{x}$ \u6709\u5b9a\u4e49\uff0c\u5411\u91cf $\\vec{x}$ \u5fc5\u987b\u662f $\\mathbb{R}^n$ \u4e2d\u7684\u5217\u5411\u91cf\uff1b\u4e14\u65b9\u7a0b\u53f3\u4fa7\u7684 $\\vec{0}$ \u4ee3\u8868 $\\mathbb{R}^m$ \u4e2d\u7684\u96f6\u5411\u91cf\u3002*<\/p>\n<p>\u9996\u5148\uff0c\u56e0\u4e3a $A\\vec{0} = \\vec{0}$\uff0c\u6240\u4ee5**\u96f6\u5411\u91cf $\\vec{0}$ \u5c5e\u4e8e $N$**\u3002<\/p>\n<p>\u5176\u6b21\uff0c\u8bbe $\\vec{u}$ \u548c $\\vec{v}$ \u662f $N$ \u4e2d\u7684\u4efb\u610f\u4e24\u4e2a\u5411\u91cf\u3002\u6839\u636e\u5b9a\u4e49\uff0c\u6709 $A\\vec{u} = \\vec{0}$ \u4e14 $A\\vec{v} = \\vec{0}$\u3002\u6839\u636e\u77e9\u9635\u4e58\u6cd5\u7684\u5206\u914d\u5f8b\uff0c\u53ef\u5f97\uff1a<br \/>\n$$A(\\vec{u} + \\vec{v}) = A\\vec{u} + A\\vec{v} = \\vec{0} + \\vec{0} = \\vec{0}$$<\/p>\n<p>\u56e0\u6b64\uff0c\u5411\u91cf\u548c $\\vec{u} + \\vec{v}$ \u4e5f\u5728 $N$ \u4e2d\uff0c**\u52a0\u6cd5\u5c01\u95ed\u6027\u6210\u7acb**\u3002<\/p>\n<p>\u6700\u540e\uff0c\u5bf9\u4e8e\u4efb\u610f\u5b9e\u6570\u6807\u91cf $c$\uff0c\u6839\u636e\u77e9\u9635\u4e58\u6cd5\u7684\u9f50\u6b21\u6027\uff1a<br \/>\n$$A(c\\vec{u}) = c(A\\vec{u}) = c\\vec{0} = \\vec{0}$$<\/p>\n<p>\u56e0\u6b64\uff0c\u6807\u91cf\u4e58\u79ef $c\\vec{u}$ \u4e5f\u5c5e\u4e8e $N$\uff0c**\u6807\u91cf\u4e58\u6cd5\u5c01\u95ed\u6027\u6210\u7acb**\u3002<\/p>\n<p>\u7531\u4e8e\u6ee1\u8db3\u5b50\u7a7a\u95f4\u7684\u6240\u6709\u4e09\u4e2a\u5224\u5b9a\u6761\u4ef6\uff0c\u7531\u6b64\u5f97\u51fa $N$ \uff08\u5373 $\\text{null}(A)$\uff09\u662f $\\mathbb{R}^n$ \u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4\u3002<\/p>\n<p>### \u7ebf\u6027\u65b9\u7a0b\u7ec4\u89e3\u7684\u7ed3\u6784\u5206\u6790<\/p>\n<p>\u7ebf\u6027\u65b9\u7a0b\u7ec4 $A\\vec{x} = \\vec{b}$ \u7684\u89e3\u7684\u6570\u91cf\u5e76\u975e\u5076\u7136\uff0c\u5b83\u6df1\u523b\u5730\u4f9d\u8d56\u4e8e\u9f50\u6b21\u65b9\u7a0b\u7ec4\u89e3\u7a7a\u95f4\uff08\u5373\u96f6\u7a7a\u95f4\uff09\u7684\u6027\u8d28\u3002\u6839\u636e **\u5b9a\u7406 3.21** \u8bc1\u660e\u7684\u96f6\u7a7a\u95f4\u5b50\u7a7a\u95f4\u5c5e\u6027\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u7ebf\u6027\u65b9\u7a0b\u7ec4\u89e3\u7684\u79bb\u6563\u6027\u89c4\u5f8b\u3002<\/p>\n<p>```ad-theorem<br \/>\ntitle: \u5b9a\u7406 3.22\uff1a\u7ebf\u6027\u65b9\u7a0b\u7ec4\u89e3\u7684\u6570\u91cf\u6cd5\u5219<br \/>\n\u5bf9\u4e8e\u4efb\u610f**\u7ebf\u6027\u65b9\u7a0b\u7ec4 $\\text{(System of Linear Equations)}$** $A\\vec{x} = \\vec{b}$\uff0c\u5176\u89e3\u7684\u72b6\u51b5**\u5fc5\u4e14\u4ec5\u6709**\u4ee5\u4e0b\u4e09\u79cd\u4e4b\u4e00\uff1a<br \/>\n1. **\u65e0\u89e3 $\\text{(No solution)}$**\uff1a\u5411\u91cf $\\vec{b}$ \u4e0d\u5728\u77e9\u9635 $A$ \u7684\u5217\u7a7a\u95f4 $\\text{col}(A)$ \u5185\u3002<br \/>\n2. **\u552f\u4e00\u89e3 $\\text{(Unique solution)}$**\uff1a$\\vec{b} \\in \\text{col}(A)$ \u4e14\u96f6\u7a7a\u95f4\u4e3a\u5e73\u51e1\u5b50\u7a7a\u95f4\uff08\u5373 $\\text{null}(A) = \\{\\vec{0}\\}$\uff09\u3002<br \/>\n3. **\u65e0\u7a77\u591a\u89e3 $\\text{(Infinitely many solutions)}$**\uff1a$\\vec{b} \\in \\text{col}(A)$ \u4e14\u96f6\u7a7a\u95f4\u5305\u542b\u975e\u96f6\u5411\u91cf\u3002<br \/>\n```<\/p>\n<p>**\u8bc1\u660e\uff1a**<br \/>\n\u6211\u4eec\u53ea\u9700\u8bc1\u660e\u4e00\u4e2a\u6838\u5fc3\u547d\u9898\uff1a**\u5982\u679c\u65b9\u7a0b\u7ec4\u6709\u81f3\u5c11\u4e24\u4e2a\u4e0d\u540c\u7684\u89e3\uff0c\u90a3\u4e48\u5b83\u4e00\u5b9a\u6709\u65e0\u7a77\u591a\u4e2a\u89e3\u3002**<\/p>\n<p>**1. \u5bfb\u627e\u65b9\u5411\u5411\u91cf**\uff1a<br \/>\n\u5047\u8bbe $\\vec{x}_1$ \u548c $\\vec{x}_2$ \u662f $A\\vec{x} = \\vec{b}$ \u7684\u4e24\u4e2a\u4e0d\u540c\u89e3\uff08$\\vec{x}_1 \\neq \\vec{x}_2$\uff09\uff0c\u5219\u6709 $A\\vec{x}_1 = \\vec{b}$ \u548c $A\\vec{x}_2 = \\vec{b}$\u3002<\/p>\n<p>\u4e24\u5f0f\u76f8\u51cf\uff1a<br \/>\n$$A\\vec{x}_1 - A\\vec{x}_2 = \\vec{b} - \\vec{b} = \\vec{0}$$<\/p>\n<p>\u5229\u7528\u77e9\u9635\u5206\u914d\u5f8b\uff1a<br \/>\n$$A(\\vec{x}_1 - \\vec{x}_2) = \\vec{0}$$<\/p>\n<p>\u8fd9\u610f\u5473\u7740\u5dee\u5411\u91cf $\\vec{v} = \\vec{x}_1 - \\vec{x}_2$ \u662f**\u96f6\u7a7a\u95f4 $\\text{null}(A)$** \u4e2d\u7684\u4e00\u4e2a**\u975e\u96f6\u5411\u91cf**\u3002<\/p>\n<p>**2. \u5229\u7528\u5b50\u7a7a\u95f4\u7684\u5c01\u95ed\u6027\u63a8\u5bfc**\uff1a<br \/>\n\u6839\u636e **\u5b9a\u7406 3.21**\uff0c\u96f6\u7a7a\u95f4\u662f\u4e00\u4e2a\u5b50\u7a7a\u95f4\uff0c\u56e0\u6b64\u5b83\u5bf9**\u6807\u91cf\u4e58\u6cd5 $\\text{(Scalar Multiplication)}$** \u5c01\u95ed\u3002<\/p>\n<p>\u8fd9\u610f\u5473\u7740\u5bf9\u4e8e\u4efb\u4f55\u5b9e\u6570\u6807\u91cf $c$\uff0c\u5411\u91cf $c\\vec{v}$ \u4ecd\u7136\u5728\u96f6\u7a7a\u95f4\u5185\uff0c\u5373\u6ee1\u8db3 $A(c\\vec{v}) = \\vec{0}$\u3002<\/p>\n<p>**3. \u6784\u9020\u65e0\u7a77\u89e3\u96c6**\uff1a<br \/>\n\u6784\u9020\u65b0\u7684\u5411\u91cf\u5e8f\u5217 $\\vec{x}_{new} = \\vec{x}_1 + c\\vec{v}$\u3002\u5c06\u5176\u4ee3\u5165\u539f\u65b9\u7a0b\u68c0\u9a8c\uff1a<br \/>\n$$A(\\vec{x}_1 + c\\vec{v}) = A\\vec{x}_1 + A(c\\vec{v}) = \\vec{b} + \\vec{0} = \\vec{b}$$<\/p>\n<p>\u7531\u4e8e\u5b9e\u6570 $c$ \u5177\u6709\u8fde\u7eed\u6027\uff08\u65e0\u7a77\u591a\u4e2a\u53d6\u503c\uff09\uff0c\u56e0\u6b64 $A\\vec{x} = \\vec{b}$ \u5bf9\u5e94\u7684\u89e3\u96c6\u4e5f\u5fc5\u7136\u662f\u65e0\u7a77\u5927\u7684\u3002<\/p>\n<p>**\u5373\u4e3a\uff1a**<br \/>\n\u7ebf\u6027\u65b9\u7a0b\u7ec4**\u4e0d\u53ef\u80fd\u53ea\u6709\u201c\u4e24\u4e2a\u89e3\u201d\u6216\u201c\u4e09\u4e2a\u89e3\u201d\u3002**<\/p>\n<p>\u56e0\u4e3a\u4e00\u65e6\u4f60\u627e\u5230\u4e86\u4e24\u4e2a\u4e0d\u540c\u7684\u89e3 $\\vec{x}_1$ \u548c $\\vec{x}_2$\uff0c\u5b83\u4eec\u4e4b\u95f4\u7684\u5dee\u503c\u5c31\u843d\u5728\u4e86**\u96f6\u7a7a\u95f4 $\\text{(Null Space)}$** \u91cc\u3002\u7531\u4e8e\u96f6\u7a7a\u95f4\u662f\u5b50\u7a7a\u95f4\uff0c\u5b83\u5fc5\u7136\u5305\u542b\u8fd9\u4e2a\u5dee\u503c\u5411\u91cf\u6240\u5ef6\u4f38\u51fa\u7684\u6574\u6761\u76f4\u7ebf\uff08\u751a\u81f3\u662f\u5e73\u9762\uff09\u3002<\/p>\n<p>\u4f60\u53ef\u4ee5\u5c06\u975e\u9f50\u6b21\u65b9\u7a0b\u7684\u89e3\u96c6\u770b\u4f5c\uff1a**\u4e00\u4e2a\u56fa\u5b9a\u4f4d\u7f6e\uff08\u7279\u89e3\uff09+ \u96f6\u7a7a\u95f4\uff08\u81ea\u7531\u4f38\u7f29\u7684\u65b9\u5411\uff09**\u3002<\/p>\n<p>\u5982\u679c\u96f6\u7a7a\u95f4\u4e0d\u4e3a\u96f6\uff0c\u5b83\u5c31\u50cf\u4e00\u6839\u53ef\u4ee5\u65e0\u9650\u62c9\u957f\u7684\u201c\u6a61\u76ae\u7b4b\u201d\uff0c\u628a\u89e3\u7684\u6570\u91cf\u76f4\u63a5\u4ece\u201c\u4e24\u4e2a\u70b9\u201d\u62c9\u6210\u201c\u4e00\u6761\u7ebf\u201d\u751a\u81f3\u201c\u4e00\u4e2a\u9762\u201d\uff0c\u4e2d\u95f4\u6ca1\u6709\u4efb\u4f55\u505c\u987f\u3002<\/p>\n<p>### \u2b50\u5b9a\u7406\u8054\u7cfb\u603b\u7ed3<\/p>\n<p>**\u5b9a\u7406 3.20**\uff1a**\u521d\u7b49\u884c\u53d8\u6362 $\\text{(Elementary Row Operations)}$** \u4e0d\u6539\u53d8**\u884c\u7a7a\u95f4 $\\text{(Row Space)}$**\uff0c\u8fd9\u786e\u4fdd\u4e86\u5728\u65b9\u7a0b\u5316\u7b80\u8fc7\u7a0b\u4e2d\uff0c\u63cf\u8ff0\u7cfb\u7edf\u7279\u6027\u7684**\u81ea\u7531\u5ea6 $\\text{(Degrees of Freedom)}$** \u662f\u5b88\u6052\u7684\u3002<\/p>\n<p>**\u5b9a\u7406 3.21**\uff1a**\u96f6\u7a7a\u95f4 $\\text{(Null Space)}$** \u5177\u5907**\u5b50\u7a7a\u95f4 $\\text{(Subspace)}$** \u7684\u5b8c\u5907\u5c5e\u6027\uff0c\u5373\u5bf9\u52a0\u6cd5\u4e0e\u6807\u91cf\u4e58\u6cd5\u5177\u6709**\u5c01\u95ed\u6027 $\\text{(Closure)}$**\u3002<\/p>\n<p>**\u5b9a\u7406 3.22**\uff1a\u6b63\u662f\u7531\u4e8e\u96f6\u7a7a\u95f4\u7684\u5c01\u95ed\u6027\uff0c\u7ebf\u6027\u65b9\u7a0b\u7ec4\u89e3\u7684\u6570\u91cf\u624d\u5b8c\u6210\u4e86\u4ece\u201c\u5b64\u7acb\u70b9\u201d $\\text{(Isolated Point)}$ \u5230\u201c\u8fde\u7eed\u65e0\u7a77\u7a7a\u95f4\u201d $\\text{(Continuous Infinite Space)}$ \u7684\u8dc3\u8fc1\u3002<\/p>\n<p>### 3.5.2 \u57fa\u5e95 $\\text{(Basis)}$<\/p>\n<p>\u6211\u4eec\u53ef\u4ee5\u4ece\u51e0\u4f55\u76f4\u89c9\u51fa\u53d1\uff1a\u5728 $\\mathbb{R}^3$ \u4e2d\uff0c\u4e00\u4e2a\u8fc7\u539f\u70b9\u7684\u5e73\u9762\u662f\u7531\u4efb\u610f\u4e24\u4e2a\u4e0d\u5e73\u884c\u7684\u5411\u91cf**\u5f20\u6210 $\\text{(Spanned)}$** \u7684\u3002\u5728\u4ee3\u6570\u8bed\u8a00\u4e2d\uff0c\u8fd9\u610f\u5473\u7740\u8fd9\u4e24\u4e2a\u5411\u91cf\u5f20\u6210\u4e86\u5e73\u9762\uff0c\u4e14\u5b83\u4eec\u662f**\u7ebf\u6027\u65e0\u5173 $\\text{(Linearly Independent)}$** \u7684\u3002<\/p>\n<p>\u5c11\u4e8e\u4e24\u4e2a\u5411\u91cf\u65e0\u6cd5\u5f20\u6210\u5e73\u9762\uff0c\u591a\u4e8e\u4e24\u4e2a\u5411\u91cf\u5219\u4f1a\u9020\u6210\u5197\u4f59\u3002\u8fd9\u79cd\u201c\u4e0d\u591a\u4e0d\u5c11\u3001\u6070\u597d\u80fd\u652f\u6491\u8d77\u6574\u4e2a\u7a7a\u95f4\u201d\u7684\u5411\u91cf\u96c6\u5408\uff0c\u5c31\u662f\u5b50\u7a7a\u95f4\u7684**\u57fa\u5e95 $\\text{(Basis)}$**\u3002<\/p>\n<p>```ad-definition<br \/>\ntitle: \u5b9a\u4e49\uff1a\u57fa\u5e95 $\\text{(Basis)}$<br \/>\n$\\mathbb{R}^n$ \u7684\u5b50\u7a7a\u95f4 $S$ \u7684\u4e00\u4e2a**\u57fa\u5e95 $\\text{(Basis)}$** \u662f\u6307 $S$ \u4e2d\u7684\u4e00\u4e2a\u5411\u91cf\u96c6\u5408\uff0c\u5b83\u5fc5\u987b\u540c\u65f6\u6ee1\u8db3\uff1a<br \/>\n1. **\u5f20\u6210 $S$**\uff1a$\\text{span}\\{\\vec{b}_1, \\vec{b}_2, \\dots, \\vec{b}_k\\} = S$\u3002<br \/>\n2. **\u7ebf\u6027\u65e0\u5173**\uff1a\u96c6\u5408 $\\{\\vec{b}_1, \\vec{b}_2, \\dots, \\vec{b}_k\\}$ \u662f\u7ebf\u6027\u65e0\u5173\u7684\u3002<br \/>\n```<\/p>\n<p>```ad-example<br \/>\ntitle: \u4f8b\u9898 3.42\uff1a$\\mathbb{R}^n$ \u7684\u6807\u51c6\u57fa\u5e95<br \/>\n\u5728 $\\mathbb{R}^n$ \u4e2d\uff0c\u6807\u51c6\u5355\u4f4d\u5411\u91cf $\\vec{e}_1, \\vec{e}_2, \\dots, \\vec{e}_n$ \u662f\u7ebf\u6027\u65e0\u5173\u7684\uff0c\u5e76\u4e14\u80fd\u591f\u5f20\u6210 $\\mathbb{R}^n$\u3002\u56e0\u6b64\uff0c\u5b83\u4eec\u6784\u6210\u4e86 $\\mathbb{R}^n$ \u7684\u4e00\u4e2a\u57fa\u5e95\uff0c\u79f0\u4e3a**\u6807\u51c6\u57fa\u5e95 $\\text{(Standard Basis)}$**\u3002<br \/>\n```<\/p>\n<p>```ad-example<br \/>\ntitle: \u4f8b\u9898 3.43\uff1a\u57fa\u5e95\u7684\u975e\u552f\u4e00\u6027<br \/>\n\u4e00\u4e2a\u5b50\u7a7a\u95f4\u53ef\u4ee5\u62e5\u6709\u591a\u7ec4\u4e0d\u540c\u7684\u57fa\u5e95\u3002\u4f8b\u5982 $\\mathbb{R}^2$ \u65e2\u6709\u6807\u51c6\u57fa\u5e95 $\\left\\{ \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix}, \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} \\right\\}$\uff0c\u4e5f\u53ef\u4ee5\u7531 $\\left\\{ \\begin{bmatrix} 2 \\\\ -1 \\end{bmatrix}, \\begin{bmatrix} 1 \\\\ 3 \\end{bmatrix} \\right\\}$ \u4f5c\u4e3a\u57fa\u5e95\u3002<\/p>\n<p>\u867d\u7136\u57fa\u5e95\u7684\u5f62\u6001\u4e0d\u540c\uff0c\u4f46\u540c\u4e00\u4e2a\u5b50\u7a7a\u95f4\u7684\u4efb\u4f55\u57fa\u5e95\u6240\u5305\u542b\u7684**\u5411\u91cf\u4e2a\u6570**\u6c38\u8fdc\u662f\u76f8\u540c\u7684\u3002<br \/>\n```<\/p>\n<p>```ad-example<br \/>\ntitle: \u4f8b\u9898 3.44\uff1a\u5bfb\u627e\u5f20\u6210\u7a7a\u95f4\u7684\u57fa\u5e95\uff08\u5254\u9664\u5197\u4f59\u6cd5\uff09<br \/>\n\u5df2\u77e5\u5b50\u7a7a\u95f4 $S = \\text{span}(\\vec{u}, \\vec{v}, \\vec{w})$\uff0c\u5176\u4e2d\uff1a<br \/>\n$$\\vec{u} = \\begin{bmatrix} 3 \\\\ -1 \\\\ 5 \\end{bmatrix}, \\vec{v} = \\begin{bmatrix} 2 \\\\ 1 \\\\ 3 \\end{bmatrix}, \\vec{w} = \\begin{bmatrix} 0 \\\\ -5 \\\\ 1 \\end{bmatrix}$$<br \/>\n\u6c42 $S$ \u7684\u4e00\u4e2a\u57fa\u5e95\u3002<\/p>\n<p>**\u89e3\uff1a**<br \/>\n\u8fd9\u4e09\u4e2a\u5411\u91cf\u5df2\u7ecf\u5f20\u6210\u4e86 $S$\uff0c\u56e0\u6b64\u53ea\u8981\u5b83\u4eec\u7ebf\u6027\u65e0\u5173\uff0c\u5c31\u80fd\u6210\u4e3a\u57fa\u5e95\u3002<br \/>\n\u7ecf\u89c2\u5bdf\uff08\u6216\u901a\u8fc7\u9ad8\u65af\u6d88\u5143\u6cd5\uff09\u53ef\u4ee5\u53d1\u73b0 $\\vec{w} = 2\\vec{u} - 3\\vec{v}$\u3002\u8fd9\u610f\u5473\u7740 $\\vec{w}$ \u662f\u591a\u4f59\u7684\u3002<br \/>\n\u7531\u4e8e $\\vec{w}$ \u662f $\\vec{u}$ \u548c $\\vec{v}$ \u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5254\u9664\u3002<br \/>\n\u800c $\\vec{u}$ \u4e0e $\\vec{v}$ \u4e92\u4e0d\u4e3a\u500d\u6570\uff0c\u663e\u7136\u7ebf\u6027\u65e0\u5173\u3002<\/p>\n<p>**\u7ed3\u8bba\uff1a**<br \/>\n$S$ \u7684\u4e00\u4e2a\u57fa\u5e95\u4e3a $\\{\\vec{u}, \\vec{v}\\}$\u3002\u5728\u51e0\u4f55\u4e0a\uff0c\u8fd9\u610f\u5473\u7740 $\\vec{u}, \\vec{v}, \\vec{w}$ \u4e09\u4e2a\u5411\u91cf\u5171\u9762\uff0c\u800c $\\vec{u}$ \u548c $\\vec{v}$ \u8db3\u4ee5\u4f5c\u4e3a\u8fd9\u4e2a\u5e73\u9762\u7684\u65b9\u5411\u5411\u91cf\u3002<br \/>\n```<\/p>\n<p>**\u5373\u4e3a\uff1a**<br \/>\n**\u57fa\u5e95 $\\text{(Basis)}$** \u5c31\u662f\u5b50\u7a7a\u95f4\u91cc\u4e00\u7ec4\u6700\u7cbe\u7b80\u7684\u201c\u5efa\u7b51\u6750\u6599\u201d\u3002<\/p>\n<p>\u5b83\u5fc5\u987b\u8db3\u591f\u591a\uff0c\u591a\u5230\u80fd\u901a\u8fc7**\u7ebf\u6027\u7ec4\u5408 $\\text{(Linear Combination)}$** \u9020\u51fa\u7a7a\u95f4\u91cc\u7684\u6bcf\u4e00\u4e2a\u5411\u91cf\uff0c\u57fa\u5e95\u4e2d\u7684\u6bcf\u4e00\u4e2a\u5411\u91cf\u90fd\u5404\u81ea\u72ec\u7acb\u5730\u8d21\u732e\u4e00\u4e2a**\u7a7a\u95f4\u7ef4\u5ea6 $\\text{(Spatial Dimension)}$**\uff1b\u5b83\u53c8\u5fc5\u987b\u8db3\u591f\u5c11\uff0c\u5c11\u5230\u91cc\u9762\u6ca1\u6709\u4efb\u4f55\u4e00\u4e2a\u5411\u91cf\u662f\u53ef\u4ee5\u88ab\u522b\u4eba\u66ff\u4ee3\u7684\uff0c\u5373\u6ee1\u8db3**\u7ebf\u6027\u65e0\u5173 $\\text{(Linearly Independent)}$**\u3002<\/p>\n<p>### 3.5.3 \u57fa\u5e95\uff1a\u7a7a\u95f4\u7684\u9aa8\u67b6 (Basis)<\/p>\n<p>\u5b50\u7a7a\u95f4\u901a\u5e38\u5305\u542b\u65e0\u6570\u4e2a\u5411\u91cf\uff0c\u6211\u4eec\u9700\u8981\u627e\u5230\u4e00\u7ec4\u6700\u7cbe\u7b80\u7684\u5411\u91cf\u6765\u552f\u4e00\u5730\u786e\u5b9a\u8be5\u7a7a\u95f4\u3002<\/p>\n<p>```ad-definition<br \/>\ntitle: \u5b9a\u4e49\uff1a\u57fa\u5e95 (Basis)<br \/>\n\u5b50\u7a7a\u95f4 $S$ \u7684\u4e00\u4e2a**\u57fa\u5e95** $\\mathcal{B} = \\{\\vec{b}_1, \\vec{b}_2, \\dots, \\vec{b}_k\\}$ \u5fc5\u987b\u6ee1\u8db3\uff1a<br \/>\n1. $\\mathcal{B}$ \u7ebf\u6027\u65e0\u5173 ($\\text{Linearly Independent}$)\u3002<br \/>\n2. $\\mathcal{B}$ \u5f20\u6210 $S$ ($\\text{Spans } S$)\u3002<br \/>\n```<\/p>\n<p>### \u2b50\u7b97\u6cd5\u5b9e\u73b0\uff1a\u5982\u4f55\u6c42\u53d6\u57fa\u5e95\uff1f<\/p>\n<p>```ad-theorem<br \/>\ntitle: \u5b9a\u7406\uff1a\u77e9\u9635\u5b50\u7a7a\u95f4\u57fa\u5e95\u7684\u6c42\u6cd5<br \/>\n\u8bbe\u77e9\u9635 $A$ \u7ecf\u521d\u7b49\u884c\u53d8\u6362\u5316\u4e3a\u884c\u9636\u68af\u5f62\u77e9\u9635 $R$\uff1a<br \/>\n1. **$\\text{row}(A)$ \u7684\u57fa\u5e95**\uff1a$R$ \u7684\u6240\u6709**\u975e\u96f6\u884c**\u3002<br \/>\n2. **$\\text{col}(A)$ \u7684\u57fa\u5e95**\uff1a$R$ \u4e2d\u4e3b\u5143\u5217 ($\\text{Pivot Columns}$) \u5bf9\u5e94\u7684**\u539f\u77e9\u9635 $A$ \u4e2d\u7684\u5217**\u3002<br \/>\n3. **$\\text{null}(A)$ \u7684\u57fa\u5e95**\uff1a\u89e3\u9f50\u6b21\u65b9\u7a0b $A\\vec{x} = \\vec{0}$\uff0c\u63d0\u53d6\u81ea\u7531\u53d8\u91cf\u5bf9\u5e94\u7684\u7279\u89e3\u5411\u91cf\u3002<br \/>\n```<\/p>\n<p>```ad-warning<br \/>\ntitle: \u6838\u5fc3\u7981\u5fcc<br \/>\n\u6c42\u5217\u7a7a\u95f4\u57fa\u5e95\u65f6\uff0c**\u7981\u6b62**\u76f4\u63a5\u4f7f\u7528\u884c\u9636\u68af\u9635 $R$ \u7684\u5217\u3002\u884c\u53d8\u6362\u867d\u7136\u4fdd\u6301\u4e86\u884c\u7a7a\u95f4\uff0c\u4f46\u6781\u5927\u5730\u6539\u53d8\u4e86\u5217\u7a7a\u95f4\u7684\u65b9\u5411\u3002\u5fc5\u987b\u6620\u5c04\u56de\u539f\u77e9\u9635 $A$ \u63d0\u53d6\u4e3b\u5143\u4f4d\u7f6e\u5bf9\u5e94\u7684\u5217\u3002<br \/>\n```<\/p>\n<p>### 3.5.4 \u7ef4\u5ea6\u4e0e\u79e9 (Dimension and Rank)<\/p>\n<p>\u7ef4\u5ea6\u662f\u5bf9\u5b50\u7a7a\u95f4\u201c\u5927\u5c0f\u201d\u7684\u5ea6\u91cf\uff0c\u5176\u672c\u8d28\u662f\u57fa\u5e95\u4e2d\u5411\u91cf\u7684\u4e2a\u6570\u3002<\/p>\n<p>```ad-definition<br \/>\ntitle: \u5b9a\u4e49\uff1a\u7ef4\u5ea6\u3001\u79e9\u4e0e\u96f6\u5316\u5ea6<br \/>\n1. **\u7ef4\u5ea6 ($\\text{Dimension}$)**\uff1a\u5b50\u7a7a\u95f4 $S$ \u57fa\u5e95\u4e2d\u7684\u5411\u91cf\u4e2a\u6570\uff0c\u8bb0\u4f5c $\\dim(S)$\u3002<br \/>\n2. **\u79e9 ($\\text{Rank}$)**\uff1a\u77e9\u9635 $A$ \u5217\u7a7a\u95f4\u7684\u7ef4\u5ea6\uff08\u7b49\u4e8e\u5176\u884c\u7a7a\u95f4\u7684\u7ef4\u5ea6\uff09\uff0c\u8bb0\u4f5c $\\text{rank}(A)$\u3002<br \/>\n3. **\u96f6\u5316\u5ea6 ($\\text{Nullity}$)**\uff1a\u77e9\u9635 $A$ \u96f6\u7a7a\u95f4\u7684\u7ef4\u5ea6\uff0c\u8bb0\u4f5c $\\text{nullity}(A)$\u3002<br \/>\n```<\/p>\n<p>### 3.5.5 \ud83d\udc51 \u79e9\u5b9a\u7406\uff1a\u7a7a\u95f4\u7684\u5b88\u6052\u5f8b (The Rank Theorem)<\/p>\n<p>\u8fd9\u662f\u7ebf\u6027\u4ee3\u6570\u4e2d\u6700\u5177\u529b\u91cf\u7684\u5b9a\u7406\u4e4b\u4e00\uff0c\u5b83\u63ed\u793a\u4e86\u77e9\u9635\u53d8\u6362\u4e2d\u201c\u4fe1\u606f\u4e22\u5931\u4e0e\u4fdd\u7559\u201d\u7684\u5e73\u8861\u3002<\/p>\n<p>```ad-theorem<br \/>\ntitle: \u79e9\u5b9a\u7406 (Rank-Nullity Theorem)<br \/>\n\u5bf9\u4e8e\u4efb\u610f $m \\times n$ \u77e9\u9635 $A$\uff0c\u59cb\u7ec8\u6709\uff1a<br \/>\n$$\\text{rank}(A) + \\text{nullity}(A) = n$$<br \/>\n\u5176\u4e2d $n$ \u4e3a\u77e9\u9635\u7684\u5217\u6570\uff08\u5373\u8f93\u5165\u7a7a\u95f4\u7684\u7ef4\u5ea6\uff09\u3002<br \/>\n```<\/p>\n","protected":false},"excerpt":{"rendered":"<p>3.5 \u5b50\u7a7a\u95f4\u3001\u57fa\u5e95\u3001\u7ef4\u5ea6\u548c\u79e9 (Subspaces, Basis, Dimension, and Rank)<\/p>\n<p>\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","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-260","post","type-post","status-publish","format-standard","hentry","category-article-cn"],"_links":{"self":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/posts\/260","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=260"}],"version-history":[{"count":18,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/posts\/260\/revisions"}],"predecessor-version":[{"id":280,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/posts\/260\/revisions\/280"}],"wp:attachment":[{"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/media?parent=260"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/categories?post=260"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wuhanqing.cn\/wordpress\/wp-json\/wp\/v2\/tags?post=260"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}