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In linear algebra and related fields of mathematics, a linear subspace (or vector subspace) is a vector space that is a subset of some other (higher-dimension) vector space. A linear subspace is usually called simply a ''subspace'' when the context serves to distinguish it from other kinds of subspaces. == Definition and useful characterization of subspace == Let ''K'' be a field (such as the real numbers), and let ''V'' be a vector space over ''K''. As usual, we call elements of ''V'' ''vectors'' and call elements of ''K'' ''scalars''. Ignoring the full extent of mathematical generalization, scalars can be understood simply as numbers. Suppose that ''W'' is a subset of ''V''. If ''W'' is a vector space itself (which means that it is closed under operations of addition and scalar multiplication), with the same vector space operations as ''V'' has, then ''W'' is a subspace of ''V''. To use this definition, we don't have to prove that all the properties of a vector space hold for ''W''. Instead, we can prove a theorem that gives us an easier way to show that a subset of a vector space is a subspace. Theorem: Let ''V'' be a vector space over the field ''K'', and let ''W'' be a subset of ''V''. Then ''W'' is a subspace if and only if ''W'' satisfies the following three conditions: #The zero vector, 0, is in ''W''. #If u and v are elements of ''W'', then the sum is an element of ''W''; #If u is an element of ''W'' and ''c'' is a scalar from ''K'', then the product ''c''u is an element of ''W''. Proof: Firstly, property 1 ensures ''W'' is nonempty. Looking at the definition of a vector space, we see that properties 2 and 3 above assure closure of ''W'' under addition and scalar multiplication, so the vector space operations are well defined. Since elements of ''W'' are necessarily elements of ''V'', axioms 1, 2 and 5–8 of a vector space are satisfied. By the closure of ''W'' under scalar multiplication (specifically by 0 and −1), the vector space's definitional axiom identity element of addition and axiom inverse element of addition are satisfied. Conversely, if ''W'' is a subspace of ''V'', then ''W'' is itself a vector space under the operations induced by ''V'', so properties 2 and 3 are satisfied. By property 3, −w is in ''W'' whenever w is, and it follows that ''W'' is closed under subtraction as well. Since ''W'' is nonempty, there is an element x in ''W'', and is in ''W'', so property 1 is satisfied. One can also argue that since ''W'' is nonempty, there is an element x in ''W'', and 0 is in the field ''K'' so and therefore property 1 is satisfied. 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 subspace」の詳細全文を読む スポンサード リンク
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