Page 29 - Matrices theory and applications

1. Elementary Theory
12
m
Completing the basis of F as a basis of K , one sees that l is the restric-
m
m
by its
tion of a linear form L on K . Let us define the vector x ∈ K
j
coordinates in the canonical basis: x j = L(e ). One has L(y)= x, y
for
m
every y ∈ K ;that is, l = T (x). Finally, we obtain
m =dim kerT +rk T =dim F
⊥
∗
+dim F .
The dual formulas between kernels and ranges are frequently used. If
M ∈ M n×m(K), one has
T
T
n
K m =ker M ⊕ R(M ), K =ker(M ) ⊕ R(M),
⊥
⊥
⊥
where ⊕ means a direct sum of orthogonal subspaces. We conclude that
T
T ⊥
rk M T =dim R(M )= m − dim R(M ) = m − dim ker M,
and finally, that
rk M T =rk M.
1.2.3 Matrices and Bilinear Forms
Let E, F be two K-vector spaces. One chooses two respective bases β =
{e 1,... ,e n} and γ = {f 1 ,... ,f m }.If B : E × F → K is a bilinear form,
then

B(x, y)= B(e i ,f j )x i y j ,
i,j
where the x i ,y j are the coordinates of x, y. One can define a matrix M ∈
M n×m(K)by m ij = B(e i ,f j ). Conversely, if M ∈ M n×m(K)is given, one
can construct a bilinear form on E × F by the formula

T
B(x, y):= x My = m ij x i y j .
i,j
Therefore, there is an isomorphism between M n×m (K) and the set of bi-
linear forms on E × F.One says that M is the matrix of B with respect
to the bases β, γ. This isomorphism depends on the choice of the bases.
A particular case arises when E = K n and F = K m are endowed with
canonical bases.
If M is associated to B, it is clear that M T is associated to the bilinear
form defined on F × E by
(y, x) → B(x, y).
When M is a square matrix, one may take F = E and γ = β.In that
case, M is symmetric if and only if B is symmetric: B(x, y)= B(y, x).
Likewise, one says that B is alternate if B(x, x) ≡ 0, that is if M itself is
an alternate matrix.