Spații vectoriale

De?nition 3.1. A vector space V over a ?eld F (or F vector space) is a set

V with an addition ? (internal composition law) such that pV; ?q is a group

and a scalar multiplication a : FtV t V; p(R); vq t (R) av  (R)v, satisfying the

following properties:

1. (R)pv ? wq  (R)v ? (R)w, @(R) P F; @v;w P F

2. p(R) ? ?qv  (R)v ? ?v; @(R); ? P F; @v P V

3. (R)p?vq  p(R)?qv

4. 1 a v  v; @v P V

The elements of V are called vectors and the elements of F are called

scalars. The scalar multiplication depends upon F. For this reason when

we need to be exact we will say that V is a vector space over F, instead

19

20 3. Vector Spaces

of simply saying that V is a vector space. Usually a vector space over R

is called a real vector space and a vector space over C is called a complex

vector space.

Remark. From the de?nition of a vector space V over F the following

rules for calculus are easily deduced:

2 (R) a 0V  0

2 0F a v  0V

2 (R) a v  0V t (R)  0F or v  0V

Examples. We will list a number of simple examples, which appear

frequently in practice.

2 V  Cn has a structure of R vector space, but it also has a structure

of C vector space.

2 V  FrXs is a F vector space.

2 Mm;npFq is a F vector space.

2 C0

ra;bs is a R vector space.

3.2 Subspaces of a vector space

It is natural to ask about subsets of a vector space V which are conveniently

closed with respect to the operations in the vector space. For this reason

we give the following:

De?nition 3.2. Let V a vector space over F. A subset U ? V is called

subspace of V over F if it is stable with respect to the composition laws (

that is v ? u P U; @v; u P U; and (R)v P U@(R) P F; v P U) and the induced

operations verify the properties form the de?nition of a vector space over F.