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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 and , 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'': : where 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 be a module defined over the ring (that is, is an object in the category ). Then we define the dual space of , denoted , to be , the module formed of all -linear module homomorphisms from into . Note then that we may define a dual to the dual, referred to as the double dual of , written as , and defined as . To formally construct a basis for the dual space, we shall now restrict our view to the case where is a finite-dimensional free (left) -module, where is a ring of unity. Then, we assume that the set is a basis for . From here, we define the Kronecker Delta function over the basis by if and if . Then the set describes a linearly independent set with each . Since is finite-dimensional, the basis is of finite cardinality. Then, the set is a basis to and is a free (right) -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
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