428 CHAPTER 18. LINEAR TRANSFORMATIONS

Of course this result holds for any finite product of linear transformations by induc-tion. One way this is quite useful is in the case where you have a finite product of lineartransformations ∏

li=1 Li all in L (V,V ) . Then

dim

(ker

l

∏i=1

Li

)≤

l

∑i=1

dim(kerLi)

and so if you can find a linearly independent set of vectors in ker(∏

li=1 Li

)of size

l

∑i=1

dim(kerLi) ,

then it must be a basis for ker(∏

li=1 Li

).

Definition 18.3.2 Let {Vi}ri=1 be subspaces of V. Then

r

∑i=1

Vi

denotes all sums of the form ∑ri=1 vi where vi ∈Vi. If whenever

r

∑i=1

vi = 0, vi ∈Vi, (18.1)

it follows that vi = 0 for each i, then a special notation is used to denote ∑ri=1 Vi. This

notation isV1⊕·· ·⊕Vr

and it is called a direct sum of subspaces.

Lemma 18.3.3 If V =V1⊕·· ·⊕Vr and if β i ={

vi1, · · · ,vi

mi

}is a basis for Vi, then a basis

for V is {β 1, · · · ,β r}.

Proof: Suppose ∑ri=1 ∑

mij=1 ci jvi

j = 0. then since it is a direct sum, it follows for each i,

mi

∑j=1

ci jvij = 0

and now since{

vi1, · · · ,vi

mi

}is a basis, each ci j = 0. ■

Here is a useful lemma.

Lemma 18.3.4 Let Li be in L (V,V ) and suppose for i ̸= j,LiL j = L jLi and also Li is oneto one on ker(L j) whenever i ̸= j. Then

ker

(p

∏i=1

Li

)= ker(L1)⊕+ · · ·+⊕ker(Lp)

Here ∏pi=1 Li is the product of all the linear transformations. A symbol like ∏ j ̸=i L j is the

product of all of them but Li.