vector space

[edit] English

Wikipedia has an article on:

Vector space

Wikipedia

(mathematics) A type of set of vectors that satisfies a specific group of constraints.

A vector space is a set of vectors which can be linearly combined.

Each vector space has a basis and dimension.

vector space over the field F

(linear algebra) A set V, whose elements are called "vectors", together with a binary operation + forming a module (V,+), and a set F^* of bilinear unary functions f^*:V→V, each of which corresponds to a "scalar" element f of a field F, such that the composition of elements of F^* corresponds isomorphically to multiplication of elements of F, and such that for any vector v, 1^*(v) = v.
- Any field $\mathbb{F}$ is a one-dimensional vector space over itself.
- If $\mathbb{V}$ is a vector space over $\mathbb{F}$ and S is any set, then $\mathbb{V}^S=\{f|f:S\rightarrow \mathbb{V} \}$ is a vector space over $\mathbb{F}$ , and $\mbox{dim} ( \mathbb{V}^S ) = \mbox{card}(S) \, \mbox{dim} (\mathbb{V})$ .
- If $\mathbb{V}$ is a vector space over $\mathbb{F}$ then any closed subset of $\mathbb{V}$ is also a vector space over $\mathbb{F}$ .
- The above three rules suffice to construct all vector spaces.