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In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. Linear maps can generally be represented as matrices, and simple examples include rotation and reflection linear transformations. An important special case is when , in which case the map is called a linear operator, or an endomorphism of . Sometimes the term ''linear function'' has the same meaning as ''linear map'', while in analytic geometry it does not. A linear map always maps linear subspaces onto linear subspaces (possibly of a lower dimension); for instance it maps a plane through the origin to a plane, straight line or point. In the language of abstract algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring. == Definition and first consequences == Let ''V'' and ''W'' be vector spaces over the same field ''K''. A function is said to be a ''linear map'' if for any two vectors x and y in ''V'' and any scalar ''α'' in ''K'', the following two conditions are satisfied: ) = f(\mathbf)+f(\mathbf) \! | additivity |- | style="padding:0 20pt"| | homogeneity of degree 1 |} This is equivalent to requiring the same for any linear combination of vectors, i.e. that for any vectors and scalars , the following equality holds: : Denoting the zero elements of the vector spaces ''V'' and ''W'' by 0''V'' and 0''W'' respectively, it follows that because letting in the equation for homogeneity of degree 1, : Occasionally, ''V'' and ''W'' can be considered to be vector spaces over different fields. It is then necessary to specify which of these ground fields is being used in the definition of "linear". If ''V'' and ''W'' are considered as spaces over the field ''K'' as above, we talk about ''K''-linear maps. For example, the conjugation of complex numbers is an R-linear map , but it is not C-linear. A linear map from ''V'' to ''K'' (with ''K'' viewed as a vector space over itself) is called a linear functional. These statements generalize to any left-module ''R''''M'' over a ring ''R'' without modification, and to any right-module upon reversing of the scalar multiplication. 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「linear map」の詳細全文を読む スポンサード リンク
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