{"id":4014,"date":"2026-05-01T07:31:49","date_gmt":"2026-04-30T22:31:49","guid":{"rendered":"https:\/\/anagileway.com\/?p=4014"},"modified":"2026-05-06T09:37:38","modified_gmt":"2026-05-06T00:37:38","slug":"the-4-subspaces","status":"publish","type":"post","link":"https:\/\/anagileway.com\/en\/2026\/05\/01\/the-4-subspaces\/","title":{"rendered":"4\u3064\u306e\u90e8\u5206\u7a7a\u9593"},"content":{"rendered":"

\u3053\u308c\u306f\u4f55?<\/h2>\n\n\n\n

\u66f8\u7c4d\u300e\u56f3\u89e3\u7dda\u5f62\u4ee3\u6570\uff1a\u30b9\u30c8\u30e9\u30f3\u30b0\u6d41\u76f4\u611f\u7684\u7406\u89e3<\/a>\u300f\u304b\u3089\u3001\u30c8\u30d4\u30c3\u30af\u3092\u7d39\u4ecb\u3059\u308b\u77ed\u3044\u8a18\u4e8b\u3067\u3059\u3002\u305d\u306e\u4e2d\u304b\u3089\u3001\u7dda\u5f62\u4ee3\u6570\u3092\u7e26\u65ad\u3057\u305f\u898b\u53d6\u308a\u56f3\u3067\u3042\u308b\u300c4\u3064\u306e\u90e8\u5206\u7a7a\u9593<\/strong>\u300d\uff08The Four Subspaces\uff09\u306b\u3064\u3044\u3066\u89e3\u8aac\u3057\u307e\u3057\u3087\u3046\u3002 <\/p>\n\n\n\n

$n$ \u5909\u6570 $m$ \u5f0f\u304b\u3089\u306a\u308b\u5b9f\u6570\u4fc2\u6570\u306e\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u3092\u884c\u5217\u8868\u73fe\u3059\u308b\u3068\u3001\u3000<\/p>\n\n\n\n

$$
A \\bx = \\bb
$$<\/p>\n\n\n\n

\u3068\u66f8\u3051\u307e\u3059\u304c\u3001\u5b9f\u6570\u6210\u5206\u306e $m \\times n$ \u884c\u5217 $A$ \u3092\u6c7a\u3081\u308b\u3068\u3001$\\bR^n$ \uff08 $\\bx$ \u304c\u5b58\u5728\u3059\u308b\uff09\u3068 $\\bR^m$ \uff08 $\\bb$ \u304c\u5b58\u5728\u3059\u308b\uff09\u306f\u305d\u308c\u305e\u308c\u3001\u4e8c\u3064\u306e\u90e8\u5206\u7a7a\u9593\u306b\u81ea\u7136\u306b\u76f4\u548c\u5206\u89e3\u3055\u308c\u307e\u3059\u3002\u3053\u308c\u3092\u4f7f\u3063\u3066\u3001\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f $A\\bx = \\bb$ \u306e\u89e3\u306e\u6709\u7121\u3001\u81ea\u7531\u5ea6\u3001\u6700\u5c0f\u30ce\u30eb\u30e0\u89e3\u3001\u6700\u5c0f\u8aa4\u5dee\u89e3\u306a\u3069\u3092\u56f3\u7684\u306b\u7406\u89e3\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3055\u3089\u306b\u3001$A = QR$ \u5206\u89e3\u3084 $A = U\\Sigma V\\transp$ \uff08SVD\uff1a\u7279\u7570\u5024\u5206\u89e3\uff09\u3001\u64ec\u4f3c\u9006\u884c\u5217 $A^+$\u3082\u3053\u3053\u306b\u66f8\u304d\u8fbc\u3080\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\u3059\uff08\u3053\u308c\u3089\u306f\u5f8c\u307b\u3069\uff09\u3002<\/p>\n\n\n\n

\"\"<\/a><\/figure>\n\n\n\n


<\/p>\n\n\n\n

\u3053\u308c\u306f\u3001Wikipedia \u306b\u3082\u300e\u7dda\u5f62\u4ee3\u6570\u5b66\u306e\u57fa\u672c\u5b9a\u7406<\/a>\u00a0\u300f\u3068\u3057\u3066\u7d39\u4ecb\u3055\u308c\u3066\u3044\u307e\u3059\u3002Strang \u5148\u751f\u306e\u30a2\u30a4\u30c7\u30a3\u30a2\u3068\u3057\u3066\u6559\u80b2\u4e0a\u306e\u529f\u7e3e\u304c\u5927\u304d\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

\u307e\u305f\u3001\u3053\u306e\u7d75\u3092\u5730\u56f3\u3068\u3057\u3066\u77e5\u3063\u3066\u304a\u304f\u3068\u3001\u7dda\u5f62\u4ee3\u6570\u306e\u3055\u307e\u3056\u307e\u306a\u5b9a\u7406\u3092\u3053\u306e\u5730\u56f3\u306b\u66f8\u304d\u8fbc\u3080\u3053\u3068\u304c\u3067\u304d\u3001\u7406\u89e3\u3092\u3057\u3063\u304b\u308a\u3057\u305f\u3082\u306e\u306b\u3067\u304d\u307e\u3059\u3002\u3067\u306f\u3001\u89e3\u8aac\u3057\u3066\u3044\u304d\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\u305d\u308c\u305e\u308c\u306e\u7a7a\u9593\u306e\u540d\u524d\u3068\u610f\u5473<\/h2>\n\n\n\n

$r$ \u3092 $A$ \u306e\u30e9\u30f3\u30af $r = \\rank A$ \u3068\u3057\u3066\u3001<\/p>\n\n\n\n

\u540d\u524d<\/strong><\/td>\u5b9a\u7fa9<\/strong><\/td>\u8a18\u53f7<\/strong><\/td>\u5c5e\u3059\u308b\u7a7a\u9593<\/strong><\/td>\u6b21\u5143<\/strong><\/td><\/tr>
\u5217\u7a7a\u9593
(Column Space)<\/td>		$A$ \u306e\u5217\u30d9\u30af\u30c8\u30eb
\u306e\u5f35\u308b\u7a7a\u9593<\/td>		$\\bCSpace(A)$<\/td>$\\bR^m$\uff08\u51fa\u529b\u5074\uff09<\/td>$r$<\/td><\/tr>
\u884c\u7a7a\u9593
(Row Space)<\/td>		$A$ \u306e\u884c\u30d9\u30af\u30c8\u30eb
\u306e\u5f35\u308b\u7a7a\u9593 <\/td>		$\\bCSpace(A\\transp)$<\/td>$\\bR^n$\uff08\u5165\u529b\u5074\uff09 <\/td>$r$<\/td><\/tr>
\u96f6\u7a7a\u9593
(Null Space)<\/td>		$A\\bx=\\bzero$
\u3068\u306a\u308b $\\bx$ \u5168\u4f53<\/td>		$\\bNSpace(A)$ <\/td>$\\bR^n$\uff08\u5165\u529b\u5074\uff09<\/td>$n – r$<\/td><\/tr>
\u5de6\u96f6\u7a7a\u9593
(Left Null Space)<\/td>		$\\by\\transp A=\\bzero$
\u3068\u306a\u308b $\\by$ \u5168\u4f53<\/td>		$\\bCSpace(A\\transp)$<\/td>$\\bR^m$\uff08\u51fa\u529b\u5074\uff09<\/td>$m – r$<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

\u4e0a\u8a18\u306b\u306f\u3001\u884c\u7a7a\u9593\u3068\u5217\u7a7a\u9593\u306e\u6b21\u5143\u306f\u7b49\u3057\u3044\u3002 \u3059\u306a\u308f\u3061
$$
r = \\rank A = \\rank A\\transp
$$
\u304c\u542b\u307e\u308c\u3066\u3044\u307e\u3059\u3002\u3053\u306e\u3053\u3068\u306b\u3064\u3044\u3066\u306f\u5225\u8a18\u4e8b<\/a>\u306b\u66f8\u304d\u307e\u3057\u305f\u3002 $\\bR^n$ \u306b\u304a\u3044\u3066\u3001\u96f6\u7a7a\u9593\u306f\u884c\u7a7a\u9593\u306e\u76f4\u4ea4\u88dc\u7a7a\u9593\u3068\u306a\u308a\u3001 $\\bR^m$ \u306b\u304a\u3044\u3066\u3001\u5de6\u96f6\u7a7a\u9593\u306f\u5217\u7a7a\u9593\u306e\u306e\u76f4\u4ea4\u88dc\u7a7a\u9593\u3068\u306a\u308a\u307e\u3059\u3002
$$
\\bR^n = \\bCSpace(A\\transp) \\oplus \\bNSpace(A), \\quad \\bCSpace(A\\transp) \\perp \\bNSpace(A)\\\\
\\bR^m = \\bCSpace(A) \\oplus \\bNSpace(A\\transp), \\quad \\bCSpace(A) \\perp \\bNSpace(A\\transp)
$$
\u3059\u3054\u3044\uff01\uff08\u307e\u3060\u5206\u304b\u3089\u306a\u304f\u3066\u3044\u3044\u3067\u3059\u3002\uff09<\/p>\n\n\n\n

\u5177\u4f53\u4f8b\u3067\u89e3\u8aac<\/h2>\n\n\n\n

Strang\u6d41\u306b\u5247\u3063\u3066\u3001\u5177\u4f53\u7684\u4f8b\u793a\u3067\u9032\u3081\u307e\u3059\u3002\u884c\u5217 $A$ \u306f\u524d\u56de\u306e\u8a18\u4e8b\uff08\u884c\u5217\u306e\u884c\u30e9\u30f3\u30af\u3068\u5217\u30e9\u30f3\u30af\u306f\u306a\u305c\u7b49\u3057\u3044\u304b\uff1f<\/a>\uff09\u3067\u4f7f\u3063\u305f\u4f8b\u3067\u3059\u3002<\/p>\n\n\n\n

$$
\\by = A\\bx = \\begin{bmatrix}
2 & 2 & 4 \\\\
4 & 4 & 8 \\\\
1 & 2 & 3 \\\\
\\end{bmatrix}
\\begin{bmatrix}
x_1 \\\\ x_2 \\\\ x_3
\\end{bmatrix}
$$
\u3068\u3044\u3046\u3001$\\bx $ \u304b\u3089 $\\by$ \u3078\u306e\u7dda\u5f62\u5199\u50cf $f(\\bx) = A\\bx$ \u3092\u8003\u3048\u307e\u3059\u3002\u3053\u3053\u3067\u3001 $\\bx$ \u304c\u56f3\u306e\u5de6\u306b\u3042\u308b $\\bR^n$\u3001$\\by$\u304c\u56f3\u306e\u53f3\u306b\u3042\u308b $\\bR^m$ \u306e\u4e2d\u306b\u5b58\u5728\u3057\u307e\u3059\u3002\u305d\u3057\u3066\u3001$A$ \u304c\u305d\u306e\u9593\u306e\u5909\u63db $f$ \u306e\u5909\u63db\u884c\u5217\u3068\u8003\u3048\u307e\u3059\u3002\u3053\u3053\u3067\u3001\u308f\u3056\u3068\u30011\u5217\u76ee\u30682\u5217\u76ee\u3092\u8db3\u3057\u30663\u5217\u76ee\u306b\u306a\u308b\u3088\u3046\u306b\u7d30\u5de5\u3057\u3066\u3042\u308a\u307e\u3059\u3002

\u56f3\u306b\u306f\u3001\u53f3\u306b2\u3064\u3001\u5de6\u306b2\u3064\u306e\u90e8\u5206\u7a7a\u9593\u304c\u305d\u308c\u305e\u308c\u3042\u308a\u307e\u3059\u3002\u53f3\u4e0a\u306e\u5217\u7a7a\u9593 $\\bCSpace(A)$ \u304b\u3089\u521d\u3081\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n

\u5217\u7a7a\u9593<\/h3>\n\n\n\n

$$
A\\bx = \\begin{bmatrix}
2 & 2 & 4 \\\\
4 & 4 & 8 \\\\
1 & 2 & 3 \\\\
\\end{bmatrix}
\\begin{bmatrix}
x_1 \\\\ x_2 \\\\ x_3
\\end{bmatrix} =
(x_1)\\begin{bmatrix} 2\\\\ 4\\\\ 1\\\\ \\end{bmatrix}
+ (x_2)\\begin{bmatrix} 2 \\\\ 4 \\\\ 2 \\\\ \\end{bmatrix}
+ (x_3)\\begin{bmatrix} 4 \\\\ 8 \\\\ 3 \\\\ \\end{bmatrix}
$$
\u3068\u3044\u3046\u98a8\u306b\u30d0\u30e9\u3057\u3066\u898b\u307e\u3059\u3002\u3053\u306e\u30d0\u30e9\u3057\u65b9\u3092\u3053\u306e\u3088\u3046\u306b\u76f4\u611f\u7684\u306b\u898b\u3048\u308b\u3088\u3046\u306b\u6163\u308c\u3066\u304a\u304f\u3068\u3088\u3044\u3067\u3059\u3002<\/p>\n\n\n

\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

\u3059\u308b\u3068\u3001 $\\bx$ \u304c $\\bR^n$ \u5168\u4f53\uff08\u5de6\u5074\uff09\u3092\u52d5\u304f\u3068\u304d\u3001 $A\\bx$ \u306f\u30013\u3064\u306e\u5217\u30d9\u30af\u30c8\u30eb\u306e\u7dda\u5f62\u7d50\u5408\u5168\u4f53\u3068\u306a\u308a\u3001 $A$ \u306e\u5217\u30d9\u30af\u30c8\u30eb\u304c\u5f35\u308b\u7a7a\u9593<\/strong>\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u3092\u3001$A$ \u306e\u5217\u7a7a\u9593<\/strong>\u3068\u547c\u3073\u3001 $\\bCSpace(A)$ \u3068\u66f8\u304d\u307e\u3059\u3002\u3053\u308c\u306f\u3001 $\\bR^m$ \u306e\u90e8\u5206\u7a7a\u9593\u3068\u306a\u308a\u307e\u3059\u3002 \u5199\u50cf\u306e\u8a00\u8449\u3067 $f$ \u306e\u50cf(Image)\u3068\u3082\u547c\u3070\u308c\u307e\u3059\u3002\u8a18\u53f7\u3092\u4f7f\u3063\u3066\u3088\u308a\u4e00\u822c\u7684\u306b\u66f8\u304f\u3068\u3001
$$
A = \\begin{bmatrix}
| & | & | \\\\
\\ba_1 & \\ba_2 & \\ba_3 \\\\
| & | & |
\\end{bmatrix}, \\;
\\bx = \\begin{bmatrix}
x_1 \\\\ x_2 \\\\ x_3
\\end{bmatrix}
$$
\u3068\u3057\u3066\u3001
$$
A\\bx=
x_1\\begin{bmatrix} | \\\\ \\ba_1 \\\\ | \\\\ \\end{bmatrix}
+ x_2 \\begin{bmatrix} | \\\\ \\ba_2 \\\\ | \\end{bmatrix}
+ x_3 \\begin{bmatrix} | \\\\ \\ba_3 \\\\ | \\end{bmatrix}\\\\
$$
\u3068 $\\ba_1, \\ba_2, \\ba_3$\u306e\u7dda\u5f62\u7d50\u5408\u3068\u3057\u3066
$$
\\bCSpace(A) = \\{y \\in \\mathbb{R}^m \\mid y = A\\bx, \\bx \\in \\bR^n \\}
$$
\u3068\u8868\u73fe\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u96f6\u7a7a\u9593<\/h3>\n\n\n\n

\u6b21\u306b \u5de6\u4e0b\u306e $\\bNSpace(A)$ \u3067\u3059\u3002\u3053\u308c\u306f\u3001 $A\\bx=\\bzero$ \u3068\u306a\u308b $x$ \u306e\u96c6\u5408\u3067\u3059\u3002 $A$ \u306e\u96f6\u7a7a\u9593<\/strong>\u3068\u304b\u5199\u50cf\u306e\u8a00\u8449\u3067\u5909\u63db $f$ \u306e\u6838\u7a7a\u9593(Kernel)\u3001 $\\bzero \\in \\bR^m$ \u306e\u9006\u50cf $f^{-1}(\\bzero)$ \uff08preimage\uff09\u3000\u3068\u3082\u547c\u3070\u308c\u3001$\\bR^n$ \u306e\u90e8\u5206\u7a7a\u9593\u3067\u3059\u3002
$$
A\\bx = \\begin{bmatrix}
2 & 2 & 4 \\\\
4 & 4 & 8 \\\\
1 & 2 & 3 \\\\
\\end{bmatrix}
\\begin{bmatrix}
x_1 \\\\ x_2 \\\\ x_3
\\end{bmatrix} = \\begin{bmatrix}0 \\\\ 0\\\\ 0\\end{bmatrix} \\quad \\text{\u3068\u306a\u308b $\\bx$ \u5168\u4f53}
$$
\u4eca\u5ea6\u306f\u3053\u308c\u3092\u3001\u3053\u3046\u8aad\u307f\u307e\u3059\u3002\u6a2a\u30d9\u30af\u30c8\u30eb\u3068\u7e26\u30d9\u30af\u30c8\u30eb\u306e\u7a4d\u3067\u3059\u3002
$$
A\\bx = \\begin{bmatrix}
\\begin{bmatrix}2 & 2 & 4\\end{bmatrix} \\begin{bmatrix} x_1\\\\ x_2\\\\ x_3 \\end{bmatrix} \\\\
\\begin{bmatrix}4 & 4 & 8\\end{bmatrix} \\begin{bmatrix} x_1\\\\ x_2\\\\ x_3 \\end{bmatrix} \\\\
\\begin{bmatrix}1 & 2 & 3\\end{bmatrix} \\begin{bmatrix} x_1\\\\ x_2\\\\ x_3 \\end{bmatrix}
\\end{bmatrix} = \\begin{bmatrix}
2x_1+2x_2+4x_3 \\\\
4x_1+4x_2+8x_3 \\\\
x_1+2x_2+3x_3 \\\\
\\end{bmatrix} = \\begin{bmatrix}
0\\\\
0 \\\\
0 \\\\
\\end{bmatrix}
$$<\/p>\n\n\n\n

\u3053\u306e\u898b\u65b9\u3092\u8996\u899a\u7684\u306b\u66f8\u304f\u3068\u3053\u3093\u306a\u611f\u3058\u3067\u3059\u3002\u30af\u30ed\u30b9\uff08\uff0b\uff09\u3067\u5185\u7a4d\u3092\u8868\u73fe\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n

\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n

\u96c6\u5408\u3068\u3057\u3066\u66f8\u304f\u3068\u3001
$$
\\bNSpace(A) = \\{\\bx \\in \\bR^n \\mid A\\bx=\\bzero \\}
$$
\u3068\u8868\u73fe\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n

\u884c\u7a7a\u9593<\/h3>\n\n\n\n

\u6b21\u306b\u5de6\u4e0a $\\bCSpace(A\\transp)$ \u306f\u3001 $A$\u306e\u884c\u7a7a\u9593<\/strong>\u3067\u3001 $A$ \u306e\u884c\u3079\u30af\u30c8\u30eb( $A\\transp$ \u306e\u5217\u30d9\u30af\u30c8\u30eb)\u304c\u5f35\u308b\u7a7a\u9593\u3067\u3059\u3002$\\by$ \u304c $\\bR^m$ \uff08\u53f3\u5074\uff09\u5168\u4f53\u3092\u52d5\u304f\u3068\u304d\u3001$A\\transp \\by$ \u306f\u3001$A$ \u306e\u884c\u30d9\u30af\u30c8\u30eb(\u3092\u5217\u30d9\u30af\u30c8\u30eb\u3068\u3057\u305f\u3082\u306e)\u306e\u7dda\u5f62\u7d50\u5408\u5168\u4f53\u3067\u3059\u3002\u307e\u305f\u3001\u3053\u308c\u306f $\\bR^n$ \u306e\u90e8\u5206\u7a7a\u9593\u3068\u306a\u308a\u307e\u3059\u3002\u8a18\u53f7\u3092\u7bc0\u7d04\u3057\u3066\u8ee2\u7f6e\u3092\u4f7f\u3063\u3066\u3044\u307e\u3059\u3002
$$
\\bCSpace(A\\transp) = \\{\\bx \\in \\bR^n \\mid \\bx = A\\transp \\by, \\by \\in \\bR^m \\}
$$<\/p>\n\n\n\n

\u5de6\u96f6\u7a7a\u9593<\/h3>\n\n\n\n

\u6700\u5f8c\u306b\u3001\u53f3\u4e0b $\\bNSpace(A\\transp)$ \u306f\u5de6\u96f6\u7a7a\u9593<\/strong>\u3068\u540d\u4ed8\u3051\u3066\u3044\u307e\u3059\u3002$A\\transp$ \u306e\u96f6\u7a7a\u9593\u3067\u3001 $A\\transp=\\bzero$ \u3068\u306a\u308b $\\by$ \u306e\u96c6\u5408\u3067\u3001 $\\bR^m$ \u306e\u90e8\u5206\u7a7a\u9593\u3068\u306a\u308a\u307e\u3059\u3002
$$
\\bNSpace(A\\transp) = \\{\\by \\in \\bR^m \\mid A\\transp \\by= \\bzero \\}
$$<\/p>\n\n\n\n

\u76f4\u4ea4\u3059\u308b\u90e8\u5206\u7a7a\u9593\u305f\u3061<\/h2>\n\n\n\n

\u96f6\u7a7a\u9593\u306b\u542b\u307e\u308c\u308b\u3059\u3079\u3066\u306e\u30d9\u30af\u30c8\u30eb $\\bx$ \u306f\u3001\u884c\u7a7a\u9593\u306b\u542b\u307e\u308c\u308b\u3059\u3079\u3066\u306e\u30d9\u30af\u30c8\u30eb\u306b\u76f4\u4ea4\u3057\u307e\u3059(\u3053\u308c\u304c\u3001\u56f3\u3067\u76f4\u4ea4\u8a18\u53f7\u304c\u5165\u3063\u3066\u3044\u308b\u8a33\u3067\u3059)\u3002\u306a\u305c\u3063\u3066\u3001 $A\\bx=\\bzero$ \u3067\u3059\u304b\u3089\u306d\u3002 $A$ \u306e\u5404\u884c $ \\ba^*_i$\u3068 $\\bx$ \u306e\u5185\u7a4d\u306f\u3059\u3079\u3066 0 \u3067\u3059\u3002
$$
A\\bx = \\begin{bmatrix}
– & \\ba^*_1 & – \\\\
– & \\ba^*_2 & – \\\\
– & \\ba^*_3 & –
\\end{bmatrix} x =
\\begin{bmatrix}
\\ba^*_1 \\bx \\\\ \\ba^*_2 \\bx \\\\ \\ba^*_3 \\bx
\\end{bmatrix} =
\\begin{bmatrix}
0 \\\\ 0 \\\\ 0
\\end{bmatrix}
$$<\/p>\n\n\n

\n
\"\"<\/a><\/figure>\n<\/div>\n\n\n


$\\bx$ \u306f $A$ \u306e\u3059\u3079\u3066\u306e\u884c\u306b\u76f4\u4ea4\u3057\u3066\u3044\u308b\u306e\u3067\u3059\u3002\u96f6\u7a7a\u9593\u306e\u5143 $\\bx$ \u3068\u884c\u7a7a\u9593\u306e\u5143 $A\\transp y$ \u306e\u5185\u7a4d\u3092\u3068\u308b\u3068\u3001
$$
\\bx \\cdot (A\\transp \\by) = \\bx\\transp (A\\transp \\by) = (A\\bx)\\transp \\by = (\\bzero)\\transp \\by = 0
$$
\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u3082\u5206\u304b\u308b\u3067\u3057\u3087\u3046\u3002\u3053\u306e\u3053\u3068\u304b\u3089\u3001 \u96f6\u7a7a\u9593 $\\bNSpace(A)$ \u3068\u884c\u7a7a\u9593 $\\bCSpace(A\\transp)$ \u306f\u76f4\u4ea4\u3057\u307e\u3059\u3002\u3088\u3063\u3066\u3001\u4e21\u65b9\u306e\u5171\u901a\u90e8\u5206\u306f $\\{\\bzero\\}$ \u3067\u3059\uff08\u5171\u901a\u90e8\u5206\u306e $\\bx$ \u306f\u76f4\u4ea4\u95a2\u4fc2\u304b\u3089\u3001 $\\| \\bx \\|^2 = \\bx \\cdot \\bx = 0 \\implies \\bx = \\bzero$\uff09\u3002\u3055\u3089\u306b\u3001(\u884c\u57fa\u672c\u5909\u5f62\u306b\u3088\u308b\u884c\u968e\u6bb5\u884c\u5217\u3067\u660e\u3089\u304b\u306b\u3055\u308c\u308b\u30d4\u30dc\u30c3\u30c8\u3068\u81ea\u7531\u5909\u6570\u306e\u6570\u306e\u95a2\u4fc2\u304b\u3089)\u4e21\u7a7a\u9593\u306e\u6b21\u5143\u3092\u8db3\u3059\u3068 $n$ \u3067 $\\bR^n$ \u3092\u5145\u6e80\u3059\u308b\u306e\u3067\u3059\u3002
$$
\\begin{eqnarray}
\\quad \\bCSpace(A\\transp) \\perp \\bNSpace(A)& \\\\
\\dim \\bCSpace(A\\transp) + \\dim \\bNSpace(A) &= n \\\\
\\bCSpace(A\\transp) \\oplus \\bNSpace(A) &= \\bR^n
\\end{eqnarray}
$$
\u307e\u3063\u305f\u304f\u540c\u69d8\u306e\u76f4\u4ea4\u6027\u3068\u6b21\u5143\u95a2\u4fc2\u304c $\\mathbb{R}^m$\u5074\u306e\u5217\u7a7a\u9593\u3068\u5de6\u96f6\u7a7a\u9593\u306b\u3064\u3044\u3066\u3082\u8a00\u3048\u307e\u3059\u3002
$$
\\begin{eqnarray}
\\quad \\bCSpace(A) \\perp \\bNSpace(A\\transp)& \\\\
\\dim \\bCSpace(A) + \\dim \\bNSpace(A\\transp) &= m \\\\
\\bCSpace(A) \\oplus \\bNSpace(A\\transp)&= \\bR^m
\\end{eqnarray}
$$
\u7f8e\u3057\u3044\uff01
<\/p>\n\n\n\n

\u4f8b\u3067\u78ba\u8a8d<\/h2>\n\n\n\n

\u4eca\u56de\u306e\u4f8b\u3067\u3001
$$
A = \\begin{bmatrix}
2 & 2 & 4 \\\\
4 & 4 & 8 \\\\
1 & 2 & 3 \\\\
\\end{bmatrix}
$$
\u306b\u5bfe\u5fdc\u3059\u308b\u7a7a\u9593\u3092\u56f3\u306e\u4e2d\u306b\u66f8\u3044\u3066\u304a\u304d\u307e\u3059\u3002\u3082\u3057 $A$ \u304c\u6b63\u5247\uff08\u53ef\u9006\uff09\u3067\u3042\u308c\u3070\u3001\u5de6\u53f3\u306e\u96f6\u7a7a\u9593\u306f ${\\bzero}$ \u3068\u306a\u3063\u3066\u3064\u3076\u308c\u3066\u3044\u307e\u3059\u3002\u3064\u307e\u308a\u3001 $A$ \u306b\u306f $A^{-1}$ \u304c\u5b58\u5728\u3057\u3001\u5168\u5358\u5c04\u3068\u306a\u308b\u308f\u3051\u3067\u3059\u3002\u4eca\u56de\u306e\u4f8b\u306f\u30013\u3064\u76ee\u306e\u5217\u30d9\u30af\u30c8\u30eb\u304c\u3000$\\ba_3 = \\ba_1 + \\ba_2$ \u3068\u306a\u3063\u3066\u3044\u3066 $A$ \u306e\u30e9\u30f3\u30af\u306f\uff12\u3067\u3059\u3002\u6b8b\u308a\u306e1\u6b21\u5143\u304c\u3001\u96f6\u7a7a\u9593\u306b\u914d\u7f6e\u3055\u308c\u307e\u3059\u3002\u3000<\/p>\n\n\n\n

\"\"<\/a><\/figure>\n\n\n\n


\u96f6\u7a7a\u9593\u306e\u30d9\u30af\u30c8\u30eb\u304c\u884c\u7a7a\u9593\u3068\u3001\u5de6\u96f6\u7a7a\u9593\u306e\u30d9\u30af\u30c8\u30eb\u304c\u5217\u7a7a\u9593\u3068\u76f4\u4ea4\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u8a08\u7b97\u3067\u78ba\u304b\u3081\u3066\u304f\u3060\u3055\u3044\u3002
$$
A\\bx = \\begin{bmatrix}
\\begin{bmatrix}2 & 2 & 4\\end{bmatrix} \\begin{bmatrix} x_1\\\\ x_2\\\\ x_3 \\end{bmatrix} \\\\
\\begin{bmatrix}4 & 4 & 8\\end{bmatrix} \\begin{bmatrix} x_1\\\\ x_2\\\\ x_3 \\end{bmatrix} \\\\
\\begin{bmatrix}1 & 2 & 3\\end{bmatrix} \\begin{bmatrix} x_1\\\\ x_2\\\\ x_3 \\end{bmatrix}
\\end{bmatrix} = \\begin{bmatrix}
2x_1+2x_2+4x_3 \\\\
4x_1+4x_2+8x_3 \\\\
x_1+2x_2+3x_3 \\\\
\\end{bmatrix} = \\begin{bmatrix}
0\\\\
0 \\\\
0 \\\\
\\end{bmatrix}
$$
\u304b\u3089\u3001
$$
x=\\begin{bmatrix}
1 \\\\
1 \\\\
-1 \\\\
\\end{bmatrix}
$$
\u304c\u96f6\u7a7a\u9593\u3000$\\bNSpace(A)$ \u306e\u57fa\u5e95\u306b\u3068\u308c\u3001
$$
A^T \\by=\\begin{bmatrix}
2y_1+4y_2+y_3 \\\\
2y_1+4y_2+2y_3 \\\\
4y_1+8y_2+3y_3 \\\\
\\end{bmatrix} = \\begin{bmatrix}
0\\\\
0 \\\\
0 \\\\
\\end{bmatrix}
$$
\u304b\u3089\u3001
$$
y=\\begin{bmatrix}
2 \\\\
-1 \\\\
0 \\\\
\\end{bmatrix}
$$
\u304c\u5de6\u96f6\u7a7a\u9593 $\\bNSpace(A\\transp)$ \u306e\u57fa\u5e95\u306b\u3068\u308c\u307e\u3059\u3002<\/p>\n\n\n\n

\u3053\u3053\u304b\u3089\u4f55\u304c\u8a00\u3048\u308b?\uff08\u6b21\u5143\u5b9a\u7406\uff09<\/h2>\n\n\n\n

\u3053\u306e\u8a18\u4e8b\u3068\u306f\u5225\u306b\u3001\u884c\u5217\u306e\u5217\u30e9\u30f3\u30af\u3068\u884c\u30e9\u30f3\u30af\u304c\u7b49\u3057\u3044\u3053\u3068\u3001\u3059\u306a\u308f\u3061\u3001
$$
\\dim \\bCSpace(A) = \\dim \\bCSpace(A\\transp) = \\rank A \\tag{*1}
$$
\u304c\u8a00\u3048\u307e\u3059\u3002\u8a73\u3057\u304f\u306f\u4ee5\u4e0b\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\n
CR\u5206\u89e3 – \u884c\u3068\u5217\u306e\u30e9\u30f3\u30af\u306f\u306a\u305c\u7b49\u3057\u3044\u304b<\/a><\/blockquote>