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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク fundamental theorem of linear algebra ： ウィキペディア英語版 fundamental theorem of linear algebra In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces. These may be stated concretely in terms of the rank ''r'' of an matrix ''A'' and its singular value decomposition: :$A=U\backslash Sigma\; V^\backslash mathrm\backslash$ First, each matrix $A\; \backslash in\; \backslash mathbf^$ ($A$ has $m$ rows and $n$ columns) induces four ''fundamental subspaces''. These ''fundamental subspaces'' are: (A) |$\backslash mathbf^m$ |$r$ (rank) |The first $r$ columns of $U$ |- |nullspace or kernel |$\backslash mathrm(A)$ or $\backslash mathrm\; (A)$ |$\backslash mathbf^n$ |$n\; -\; r$ (nullity) |The last $(n\; -\; r)$ columns of $V$ |- |row space or coimage |$\backslash mathrm(A^\backslash mathrm)$ or $\backslash mathrm\; (A^\backslash mathrm)$ |$\backslash mathbf^n$ |$r$ (rank) |The first $r$ columns of $V$ |- |left nullspace or cokernel |$\backslash mathrm(A^\backslash mathrm)$ or $\backslash mathrm\; (A^\backslash mathrm)$ |$\backslash mathbf^m$ |$m\; -\; r$ (corank) |The last $(m\; -\; r)$ columns of $U$ |} Secondly: # In $\backslash mathbf^n$, $\backslash mathrm(A)\; =\; (\backslash mathrm(A^\backslash mathrm))^\backslash perp$, that is, the nullspace is the orthogonal complement of the row space # In $\backslash mathbf^m$, $\backslash mathrm(A^\backslash mathrm)\; =\; (\backslash mathrm(A))^\backslash perp$, that is, the left nullspace is the orthogonal complement of the column space. The dimensions of the subspaces are related by the rank–nullity theorem, and follow from the above theorem. Further, all these spaces are intrinsically defined—they do not require a choice of basis—in which case one rewrites this in terms of abstract vector spaces, operators, and the dual spaces as $A\backslash colon\; V\; \backslash to\; W$ and $A^$ * \colon W^ * \to V^ *: the kernel and image of $A^$ * are the cokernel and coimage of $A$. == See also == * Rank–nullity theorem * Closed range theorem 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「fundamental theorem of linear algebra」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.