Page 20 - Matrices theory and applications

3
1.1. Basics
If G⊂ E, one often identifies G and the associated family (g) g∈G .The set
G of linear combinations of elements of G is a linear subspace E, called the
linear subspace spanned by G. It is the smallest linear subspace E containing
G, equal to the intersection of all linear subspaces containing G. The subset
G is generating when G = E.
One can prove that every K-vector space admits at least one basis. In
the most general setting, this is a consequence of the axiom of choice.
All the bases of E have the same cardinality, which is therefore called the
dimension of E, denoted by dim E. The dimension is an upper (respectively
a lower) bound for the cardinality of free (respectively generating) families.
In this book we shall only use finite-dimensional vector spaces. If F, G are
two linear subspaces of E, the following formula holds:
dim F +dim G =dim F ∩ G +dim(F + G).
If F ∩ G = {0},one writes F ⊕ G instead of F + G, and one says that F
and G are in direct sum. One has then
dim F ⊕ G =dim F +dim G.
i
Given a set I, the family (e ) i∈I , defined by

0,j = i,
i
(e ) j =
1,j = i,
I
I
is a basis of K , called the canonical basis. The dimension of K is therefore
equal to the cardinality of I.
In a vector space, every generating family contains at least one basis of
E. Similarly, given a free family, it is contained in at least one basis of E.
This is the incomplete basis theorem.
Let L be a field and K a subfield of L.If F is an L-vector space, then F
is also a K-vector space. As a matter of fact, L is itself a K-vector space,
and one has

dim K F =dim L F · dim K L.
The most common example (the only one that we shall consider) is K = IR,
L = CC,for whichwe have
dim IR F =2 dim CC F.
Conversely, if G is an IR-vector space, one builds its complexification G CC
as follows:
G CC = G × G,

with the induced structure of an additive group. An element (x, y)of G CC
is also denoted x + iy. One defines multiplication by a complex number by

(λ = a + ib, z = x + iy) → λz := (ax − by, ay + bx).