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.
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