B.1 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 need1.
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.x = x,
•(λ + μ)x = λx + μx,
•λ(μx) = (λμ) x,
•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 ...