Page 25 - Matrices theory and applications

1. Elementary Theory
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called the range of M, sometimes denoted by R(M). The kernel and the
m
n
and K , respectively. The range is
range of M are linear subspaces of K
spanned by the columns of M and therefore has dimension rk M.
Proposition 1.1.1 Let K be a field. If M ∈ M n×m (K),then
Proof m =dim ker M + rk M.
Let {f 1 ,... ,f r } be a basis of R(M). By construction, there exist vectors
{e 1,... ,e r } of K m such that Me j = f j .Let E be the linear subspace

spanned by the e j .If e = a j e j ∈ ker M,then a j f j = 0, and thus the
j j
a j vanish. It follows that the restriction M : E → R(M) is an isomorphism,
so that dim E =rk M.
m

If e ∈ K ,then Me ∈ R(M), and there exists e ∈ E such that Me =
Me. Therefore, e = e +(e − e ) ∈ E +ker M,so that K m = E +ker M.

Since E ∩ ker M = {0}, one has m =dim E +dim ker M.
1.2 Change of Basis
Let E be a K-vector space, in which one chooses a basis β = {e 1,... ,e n }.
1

Let P ∈ M n (K)be an invertiblematrix. The set β = {e ,... ,e } defined
n
1
by
n

e =

i p ji e j
j=1
is a basis of E.One says that P is the matrix of the change of basis β → β ,

or the change-of-basis matrix. If x ∈ E has coordinates (x 1 ,... ,x n )inthe

basis β and (x ,... ,x ) in the basis β , one then has the formulas

1
n
n

x j = p ji x .
i
i=1
If u : E → F is a linear map, one may compare the matrices of u for
different choices of the bases of E and F.Let β, β be bases of E and let

γ, γ be bases of F. Let us denote by P, Q the change-of-basis matrices of
β → β and γ → γ . Finally, let M, M be the matrices of u in the bases

β, γ and β ,γ , respectively. Then

MP = QM ,
or M = Q −1 MP,where Q −1 denotes the inverse of Q.One says that M

and M are equivalent. Two equivalent matrices have same rank.
1
See Section 2.2 for the meaning of this notion.