Linear Algebra, Theory And Applications, 2012a

Linear Transformations
Canonical Forms
10.1 A Theorem Of Sylvester, Direct Sums
The notation is defined as follows.
Definition 10.1.1 Let L ∈L(V,W) . Then ker (L) ≡{v ∈ V : Lv =0} .
Lemma 10.1.2 Whenever L ∈L(V,W) , ker (L) is a subspace.
Proof: If a, b are scalars and v,w are in ker (L) , then
L (av + bw) =aL (v)+bL (w) =0+0=0
Suppose now that A ∈L(V,W) andB ∈L(W, U)whereV,W,U are all finite dimensional
vector spaces. Then it is interesting to consider ker (BA). The following theorem of
Sylvester is a very useful and important result.
Theorem 10.1.3 Let A ∈L(V,W) and B ∈L(W, U) where V,W,U are all vector spaces
over a field F. Suppose also that ker (A) and A (ker (BA)) are finite dimensional subspaces.
Then
dim (ker (BA)) ≤ dim (ker (B)) + dim (ker (A)) .
Proof: If x ∈ ker (BA) , then Ax ∈ ker (B)andsoA (ker (BA)) ⊆ ker (B) . The following
picture may help.
ker(BA)
ker(A)
A
✲
ker(B)
A(ker(BA))
Now let {x 1 , ··· ,x n } be a basis of ker (A) and let {Ay 1 , ··· ,Ay m } be a basis for
A (ker (BA)) . Take any z ∈ ker (BA) . Then Az = ∑ m
i=1 a iAy i and so
(
)
m∑
A z − a i y i = 0
which means z − ∑ m
i=1 a iy i ∈ ker (A) and so there are scalars b i such that
m∑
n∑
z − a i y i = b i x i .
i=1
i=1 j=1
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