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Chapter 4: Q77E (page 186)

If Tis a linear transformation from Vtoand L is a linear transformation from Wto U, is the composite transformation L鈭�Tfrom Vto U linear? How can you tell? IfTand L are isomorphisms, isL鈭�T an isomorphism as well?

Short Answer

Expert verified

If T and L are isomorphisms, thenL鈭�T is also an isomorphism.

Step by step solution

01

Definition of Linear transformation.

Consider two linear spacesand. A functionfromtois called linear transformation if

  1. T(f+g)=T(f)+T(g)T(f+g)=T(f)+T(g)
  2. T(kf)=kT(f)T(kf)=kT(f)

for all elements f and g ofand for all scalars.

Ifis finite dimensional, then

dim(V)=rank(T)+nullity(T)=dim(imT)+dim(kerT)

02

Definition of Isomorphism.

A linear transformation T:V鈫�Wis called isomorphism if its both one-to-one and onto.

03

Proof of L∘T is an isomorphism.

If T and L are linear, then

L鈭�T)f+g=LT(f+g=L(Tf+Tg)=LTf+LTg=L鈭�Tf+L鈭�T(g

Also,

L鈭�T)kf=LT(kf=L(kTf)=kLTf=kL鈭�Tf

Therefore,L鈭�Tis linear.

The composite of invertible function is invertible.

Thus, T and L are isomorphism.

This impliesL鈭�Tis also an isomorphism.

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Most popular questions from this chapter

Consider linear transformation T from Vto Wand from Wto u. If ker T and ker L are both finite dimensional, and if im T = W,show that ker (LoT)is finite dimensional as well and that.

(ker (LoT)) = dim (Ker (T)) + dim (Ker (L))

Find the transformation is linear and determine whether they are isomorphism.

Find the basis of all Pn, and determine its dimension.

Find the image, kernel, rank, and nullity of the transformationT inT(f(t))=f(7) from P2to R.

Find the kernel and nullity of the transformation T inT(f)=f-f fromc鈭� to c鈭�.
See all solutions

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