“vector space”的概念、定义、习题解题方法及翻译-数学辞典

单词

vector space

释义

vector space

(linear space) A main mathematical structure in linear algebra. A vector space V over a field F consists of a set V with operations of addition +: V × V → V, and scalar multiplication F × V → V, such that V is an abelian group under + and further

where α,β ∈ F and v,w ε V. Elements of F are called scalars and elements of V are called vectors. Examples of real vectors spaces (where F = R) include:

the space of m × n real matrices;

the space of polynomials with real coefficients;

the solution space of homogeneous simultaneous linear equations;

the kernel and image of a linear map;

the space Hom(V,W) of all linear maps between vector spaces V and W;

sequence spaces (see c and l∞);

the space of continuous real-valued functions on a topological space.

A vector space V is finite-dimensional if it has a finite basis, in which case all bases will contain the same number of elements, the space's dimension. Once a basis is chosen, coordinates may be uniquely assigned to vectors identifying V with R^dimV. See affine space, dual space, inner product, module, normed vector space, n-dimensional space, vector subspace.

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