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In linear algebra, the quotient of a vector space ''V'' by a subspace ''N'' is a vector space obtained by "collapsing" ''N'' to zero. The space obtained is called a quotient space and is denoted ''V''/''N'' (read ''V'' mod ''N'' or ''V'' by ''N''). == Definition == Formally, the construction is as follows . Let ''V'' be a vector space over a field ''K'', and let ''N'' be a subspace of ''V''. We define an equivalence relation ~ on ''V'' by stating that ''x'' ~ ''y'' if ''x'' − ''y'' ∈ ''N''. That is, ''x'' is related to ''y'' if one can be obtained from the other by adding an element of ''N''. From this definition, one can deduce that any element of ''N'' is related to the zero vector; more precisely all the vectors in ''N'' get mapped into the equivalence class of the zero vector. The equivalence class of ''x'' is often denoted :() = ''x'' + ''N'' since it is given by :() = . The quotient space ''V''/''N'' is then defined as ''V''/~, the set of all equivalence classes over ''V'' by ~. Scalar multiplication and addition are defined on the equivalence classes by *α() = () for all α ∈ ''K'', and *() + () = (). It is not hard to check that these operations are well-defined (i.e. do not depend on the choice of representative). These operations turn the quotient space ''V''/''N'' into a vector space over ''K'' with ''N'' being the zero class, (). The mapping that associates to ''v'' ∈ ''V'' the equivalence class () is known as the quotient map. 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「Quotient space (linear algebra)」の詳細全文を読む スポンサード リンク
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