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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 Fredholm alternative ： ウィキペディア英語版 Fredholm alternative In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue. ==Linear algebra== If ''V'' is an ''n''-dimensional vector space and $T:V\backslash to\; V$ is a linear transformation, then exactly one of the following holds: #For each vector ''v'' in ''V'' there is a vector ''u'' in ''V'' so that $T(u)\; =\; v$. In other words: ''T'' is surjective (and so also bijective, since ''V'' is finite-dimensional). #$\backslash dim(\backslash ker(T))\; >\; 0$. A more elementary formulation, in terms of matrices, is as follows. Given an ''m''×''n'' matrix ''A'' and a ''m''×1 column vector b, exactly one of the following must hold: #''Either:'' ''A'' x = b has a solution x #''Or:'' ''A''T y = 0 has a solution y with yTb ≠ 0. In other words, ''A'' x = b has a solution $(\backslash mathbf\; \backslash in\; \backslash operatorname(A))$ if and only if for any y s.t. ''A''T y = 0, yTb = 0 $(\backslash mathbf\; \backslash in\; \backslash ker(A^T)^)$. 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「Fredholm alternative」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.