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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 Matroid representation ： ウィキペディア英語版 Matroid representation In the mathematical theory of matroids, a matroid representation is a family of vectors whose linear independence relation is the same as that of a given matroid. Matroid representations are analogous to group representations; both types of representation provide abstract algebraic structures (matroids and groups respectively) with concrete descriptions in terms of linear algebra. A linear matroid is a matroid that has a representation, and an ''F''-linear matroid (for a field ''F'') is a matroid that has a representation using a vector space over ''F''. Matroid representation theory studies the existence of representations and the properties of linear matroids. ==Definitions== A (finite) matroid $(E,\backslash mathcal)$ is defined by a finite set $E$ (the elements of the matroid) and a family $\backslash mathcal$ of the subsets of $E$, called the independent sets of the matroid. It is required to satisfy the properties that every subset of an independent set is itself independent, and that if one independent set $A$ is larger than a second independent set $B$ then there exists an element $x\backslash in\; A\backslash setminus\; B$ that can be added to $B$ to form a larger independent set. One of the key motivating examples in the formulation of matroids was the notion of linear independence of vectors in a vector space: if $E$ is a finite set or multiset of vectors, and $\backslash mathcal$ is the family of linearly independent subsets of $E$, then $(E,\backslash mathcal)$ is a matroid.〔. For the rank function, see p. 26.〕〔.〕 More generally, if $(E,\backslash mathcal)$ is any matroid, then a representation of $(E,\backslash mathcal)$ may be defined as a function $f$ that maps $E$ to a vector space $V$, with the property that a subset $A$ of $E$ is independent if and only if $f(A)$ is linearly independent. A matroid with a representation is called a linear matroid, and if $V$ is a vector space over field ''F'' then the matroid is called an ''F''-linear matroid. Thus, the linear matroids are exactly the matroids that are isomorphic to the matroids defined from sets or multisets of vectors. The function $f$ will be one-to-one if and only if the underlying matroid is simple (having no two-element dependent sets). Matroid representations may also be described more concretely using matrices over a field ''F'', with one column per matroid element and with a set of elements being independent in the matroid if and only if the corresponding set of matrix columns is linearly independent. The rank function of a linear matroid is given by the matrix rank of submatrices of this matrix, or equivalently by the dimension of the linear span of subsets of vectors.〔, p. 12.〕 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「Matroid representation」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.