Finite-dimensional vector spaces

We are concerned with real vector spaces, but the results extend readily to complex vector spaces, as well. We describe briefly the ideas and results that we need.

Let K denote either the field R of real numbers or the field C of complex numbers. A vector space E over K is an abelian additive group (E, +), together with a mapping (scalar multiplication) (λ, x) → λx of K × E into E which satisfies

1. • 1.x = x,

2. • (λ + μ)x = λx + μx,

3. • λ(μx) = (λμ)x,

4. • λ(x + y) = λx + λy,

for λ, μ ∈ K and x, y ∈ E. The elements of E are called vectors and the elements of K are called scalars.

It then follows that 0.x = 0 and λ.0 = 0 for x ∈ E and λ ∈ K. (Note that the same symbol 0 is used for the additive identity element in E and the zero element in K.)

A non-empty subset F of a vector space E is a linear subspace if it is a subgroup of E and if λx ∈ F whenever λ ∈ K and x ∈ F. A linear subspace is then a vector space, with the operations inherited from E.