In this chapter we consider vector spaces over a field which is either ℝ or ℂ. We shall start from the most general situation of scalar products. We then consider the situations when scalar products are non-degenerate and positive definite, respectively.

Scalar products and basic properties

In this section, we use F to denote the field ℝ or ℂ.

Definition 4.1 Let U be a vector space over F. A scalar product over U is defined to be a bilinear symmetric function f : U × U → F, written simply as (u, v) ≡ f(u, v), u, v ∈ U. In other words the following properties hold.

1. (Symmetry) (u, υ) = (υ, u) ∈ F for u, υ ∈ U.

2. (Additivity) (u + υ, w) = (u, w) + (υ, w) for u, υ, w ∈ U.

3. (Homogeneity) (au, υ) = a(u, υ) for a ∈ F and u, υ ∈ U.

We say that u, υ ∈ U are mutually perpendicular or orthogonal to each other, written as u ⊥ υ, if (u, υ) = 0. More generally for any non-empty subset S of U we use the notation

S⊥ = {u ∈ U | (u, υ) = 0 for any υ ∈ S}.

For u ∈ U we say that u is a null vector if (u, u) = 0.

It is obvious that S⊥ is a subspace of U for any nonempty subset S of U. Moreover {0}⊥ = U. Furthermore it is easy to show that if the vectors u1, …, uk are mutually perpendicular and not null then they are linearly independent.

Let u, υ ∈ U so that u is not null. Then we can resolve υ into the sum of two mutually perpendicular vectors, one in Span {u}, say cu for some scalar c, and one in Span{u}⊥, say w.