**Appendix B**

Linear algebra

**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 need^{1}.

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 ...

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