Page 24 - Matrices theory and applications

1.1. Basics
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for every x, y ∈ E and a ∈ K. One then has u(0) = 0. The preimage
−1
u
(0), denoted by ker u,is the kernel of u. It is a linear subspace of E.
The range u(E) is also a linear subspace of F. The set of homomorphisms
of E into F is a K-vector space, denoted by L(E, F). If F = E, one defines
End(E):= L(E, F); its elements are the endomorphisms of E.
n
The identification of M n×1 (K)with K allows us to consider the matri-
n
ces of size n × m as linear maps from K m to K .If M ∈ M n×m (K), one
proceeds as in the following diagram:
n
K m → M m×1 (K) → M n×1(K) → K ,
x → X → Y = MX → y.
Namely, the image of the vector x with coordinates x 1 ,... ,x m is the vector
y with coordinates y 1 ,... ,y n given by
m

y i = m ij x j . (1.1)
j=1
n
m
One thus obtains an isomorphism between M n×m (K)and L(K ; K ),
which we shall use frequently in studying matrix properties.
More generally, if E, F are K-vector spaces of respective dimensions m
and n, in which one chooses bases β = {e 1 ,... ,e m } and γ = {f 1,... ,f n },
one may construct the linear map u : E → F by
u(x 1 e 1 + ··· + x m e m)= y 1 f 1 + ··· + y n f n ,
via the formulas (1.1). One says that M is the matrix of u in the bases β,
γ.
Let E, F, G be three K-vector spaces of dimensions p, m, n.Let us

choose respective bases α, β, γ. Given two matrices M, M of sizes n × m

and m × p, corresponding to linear maps u : F → G and u : E → F,the

product MM is the matrix of the linear map u ◦ u : E → G. Here lies
the origin of the definition of the product of matrices. The associativity
of the product expresses that of the composition of maps. One will note,
however, that the isomorphism between M n×m(K)and L(E, F)is by no
means canonical, since the correspondence M → u always depends on an
arbitrary choice of two bases. One thus cannot reduce the entire theory of
matrices to that of linear maps, and vice versa.
When E = F is a K-vector space of dimension n,it isoften worth
choosing a single basis (γ = β with the previous notation). One then has
an algebra isomorphism M → u between M n(K)and End(E), the algebra
of endomorphisms of E. Again, this isomorphism depends on an arbitrary
choice of basis.
If M is the matrix of u ∈L(E, F) in the bases α, β, the linear subspace
u(E) is spanned by the vectors of F whose representations in the basis β
are the columns M (j) of M. Its dimension thus equals rkM.
If M ∈ M n×m (K), one defines the kernel of M to be the set ker M of
those X ∈ M m×1(K) such that MX =0 n. The image of K m under M is