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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 Column vector ： ウィキペディア英語版 Row and column vectors In linear algebra, a column vector or column matrix is an ''m'' × 1 matrix, that is, a matrix consisting of a single column of ''m'' elements, :$\backslash mathbf\; =\; \backslash begin\; x\_1\; \backslash \backslash \; x\_2\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; x\_m\; \backslash end\; \backslash ,.$ Similarly, a row vector or row matrix is a 1 × ''m'' matrix, that is, a matrix consisting of a single row of ''m'' elements〔, p. 8〕 : Throughout, boldface is used for the row and column vectors. The transpose (indicated by T) of a row vector is a column vector :$\backslash begin\; x\_1\; \backslash ;\; x\_2\; \backslash ;\; \backslash dots\; \backslash ;\; x\_m\; \backslash end^\; =\; \backslash begin\; x\_1\; \backslash \backslash \; x\_2\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; x\_m\; \backslash end\; \backslash ,,$ and the transpose of a column vector is a row vector :$\backslash begin\; x\_1\; \backslash \backslash \; x\_2\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; x\_m\; \backslash end^\; =\; \backslash begin\; x\_1\; \backslash ;\; x\_2\; \backslash ;\; \backslash dots\; \backslash ;\; x\_m\; \backslash end\; \backslash ,.$ The set of all row vectors forms a vector space called row space, similarly the set of all column vectors forms a vector space called column space. The dimensions of the row and column spaces equals the number of entries in the row or column vector. The column space can be viewed as the dual space to the row space, since any linear functional on the space of column vectors can be represented uniquely as an inner product with a specific row vector. == Notation == To simplify writing column vectors in-line with other text, sometimes they are written as row vectors with the transpose operation applied to them. :$\backslash mathbf\; =\; \backslash begin\; x\_1\; \backslash ;\; x\_2\; \backslash ;\; \backslash dots\; \backslash ;\; x\_m\; \backslash end^$ or :$\backslash mathbf\; =\; \backslash begin\; x\_1,\; x\_2,\; \backslash dots,\; x\_m\; \backslash end^$ Some authors also use the convention of writing both column vectors and row vectors as rows, but separating row vector elements with commas and column vector elements with semicolons (see alternative notation 2 in the table below). | align=center| $\backslash begin\; x\_1\; \backslash \backslash \; x\_2\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; x\_m\; \backslash end\; \backslash text\; \backslash begin\; x\_1\; \backslash ;\; x\_2\; \backslash ;\; \backslash dots\; \backslash ;\; x\_m\; \backslash end^$ |- | Alternative notation 1 (commas, transpose signs) | align=center| $\backslash begin\; x\_1,\; x\_2,\; \backslash dots,\; x\_m\; \backslash end$ | align=center| $\backslash begin\; x\_1,\; x\_2,\; \backslash dots,\; x\_m\; \backslash end^$ |- | Alternative notation 2 (commas and semicolons, no transpose signs) | align=center| $\backslash begin\; x\_1,\; x\_2,\; \backslash dots,\; x\_m\; \backslash end$ | align=center| $\backslash begin\; x\_1;\; x\_2;\; \backslash dots;\; x\_m\; \backslash end$ |} 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「Row and column vectors」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.