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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 Standard basis ： ウィキペディア英語版 Standard basis In mathematics, the standard basis (also called natural basis or canonical basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system. For example, the standard basis for the Euclidean plane is formed by vectors :$\backslash mathbf\_x\; =\; (1,0),\backslash quad\; \backslash mathbf\_y\; =\; (0,1),$ and the standard basis for three-dimensional space is formed by vectors :$\backslash mathbf\_x\; =\; (1,0,0),\backslash quad\; \backslash mathbf\_y\; =\; (0,1,0),\backslash quad\; \backslash mathbf\_z=(0,0,1).$ Here the vector e''x'' points in the ''x'' direction, the vector e''y'' points in the ''y'' direction, and the vector e''z'' points in the ''z'' direction. There are several common notations for these vectors, including , , , and . These vectors are sometimes written with a hat to emphasize their status as unit vectors. Each of these vectors is sometimes referred to as the versor of the corresponding Cartesian axis. These vectors are a basis in the sense that any other vector can be expressed uniquely as a linear combination of these. For example, every vector v in three-dimensional space can be written uniquely as :$v\_x\backslash ,\backslash mathbf\_x\; +\; v\_y\backslash ,\backslash mathbf\_y\; +\; v\_z\backslash ,\backslash mathbf\_z,$ the scalars ''v''''x'', ''v''''y'', ''v''''z'' being the scalar components of the vector v. In $n$-dimensional Euclidean space, the standard basis consists of ''n'' distinct vectors :$\backslash ,$ where e''i'' denotes the vector with a 1 in the $i$th coordinate and 0's elsewhere. Standard bases can be defined for other vector spaces, such as polynomials and matrices. In both cases, the standard basis consists of the elements of the vector space such that all coefficients but one are 0 and the non-zero one is 1. For polynomials, the standard basis thus consists of the monomials and is commonly called monomial basis. For matrices $\backslash mathcal\_$, the standard basis consists of the ''m''×''n''-matrices with exactly one non-zero entry, which is 1. For example, the standard basis for 2×2 matrices is formed by the 4 matrices : \mathbf_ = \begin 0 & 1 \\ 0 & 0 \end,\quad \mathbf_ = \begin 0 & 0 \\ 1 & 0 \end,\quad \mathbf_ = \begin 0 & 0 \\ 0 & 1 \end. == Properties == By definition, the standard basis is a sequence of orthogonal unit vectors. In other words, it is an ordered and orthonormal basis. However, an ordered orthonormal basis is not necessarily a standard basis. For instance the two vectors representing a 30° rotation of the 2D standard basis described above, i.e. :$v\_1\; =\; \backslash left(\; ,\; \backslash right)\; \backslash ,$ :$v\_2\; =\; \backslash left(\; ,\; \backslash right)\; \backslash ,$ are also orthogonal unit vectors, but they are not aligned with the axes of the Cartesian coordinate system, so the basis with these vectors does not meet the definition of standard basis. 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「Standard basis」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.