翻訳と辞書 Words near each other ・ Dual wavelet ・ Dual wield ・ Dual Work Exchanger Energy Recovery ・ Dual! Parallel Trouble Adventure ・ Dual-Axis Radiographic Hydrodynamic Test Facility ・ Dual-band blade antenna ・ Dual-beta ・ Dual-career commuter couples ・ Dual-clutch transmission ・ Dual-coding theory ・ Dual-control modes of ventilation ・ Dual-covenant theology ・ Dual-energy X-ray absorptiometry ・ Dual Analog Controller ・ Dual ball joint suspension ・ Dual basis ・ Dual basis in a field extension ・ Dual Blades ・ Dual board ・ Dual Box ・ Dual brain theory ・ Dual bundle ・ Dual carbon battery ・ Dual carriageway ・ Dual chartering ・ Dual City Sessions ・ Dual code ・ Dual Committee ・ Dual cone and polar cone ・ Dual consciousness Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Dual basis ： ウィキペディア英語版 Dual basis In linear algebra, given a vector space ''V'' with a basis ''B'' of vectors indexed by an index set ''I'' (the cardinality of ''I'' is the dimensionality of ''V''), its dual set is a set ''B''∗ of vectors in the dual space ''V''∗ with the same index set ''I'' such that ''B'' and ''B''∗ form a biorthogonal system. The dual set is always linearly independent but does not necessarily span ''V''∗. If it does span ''V''∗, then ''B''∗ is called the dual basis for the basis ''B''. Denoting the indexed vector sets as $B\; =\; \backslash \_$ and $B^\; =\; \backslash \_$, being biorthogonal means that the elements pair to 1 if the indexes are equal, and to zero otherwise. Symbolically, evaluating a dual vector in ''V''∗ on a vector in the original space ''V'': :$$ v^i(v_j) = \delta^i_j = \begin 1 & \text i = j\\ 0 & \text i \ne j\text \end where $\backslash delta^i\_j$ is the Kronecker delta symbol. ==A categorical and algebraic construction of the dual space== Another way to introduce the dual space of a vector space (module) is by introducing it in a categorical sense. To do this, let $A$ be a module defined over the ring $R$ (that is, $A$ is an object in the category $R\backslash text\backslash mathbf$). Then we define the dual space of $A$, denoted $A^$, to be $\backslash text\_R(A,R)$, the module formed of all $R$-linear module homomorphisms from $A$ into $R$. Note then that we may define a dual to the dual, referred to as the double dual of $A$, written as $A^$, and defined as $\backslash text\_R(A^,R)$. To formally construct a basis for the dual space, we shall now restrict our view to the case where $F$ is a finite-dimensional free (left) $R$-module, where $R$ is a ring of unity. Then, we assume that the set $X$ is a basis for $F$. From here, we define the Kronecker Delta function $\backslash delta\_$ over the basis $X$ by $\backslash delta\_=1$ if $x=y$ and $\backslash delta\_=0$ if $x\backslash ne\; y$. Then the set $S\; =\; \backslash lbrace\; f\_x:F\; \backslash to\; R\; \backslash ;\; |\; \backslash ;\; f\_x(y)=\backslash delta\_\; \backslash rbrace$ describes a linearly independent set with each $f\_x\; \backslash in\; \backslash text\_R(F,R)$. Since $F$ is finite-dimensional, the basis $X$ is of finite cardinality. Then, the set $S$ is a basis to $F^\backslash ast$ and $F^\backslash ast$ is a free (right) $R$-module. 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「Dual 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.