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Chapter 4: Q47E (page 200)

Question: There exist invertible 2脳2matrices P and Q such that the transformationT(f)=PM-QMis an isomorphism.

Short Answer

Expert verified

The solution is the statement is true.

Step by step solution

01

Determine the isomorphism

There exists invertible 2脳2matrices P and Q such that the transformation TM=PM-QMis an isomorphism.

Consider the matrices P and Q as follows.

P=1001andQ=-100-1

Simplify as follows.

T(M)=I(M)-(-I)MT(M)=M+MT(M)=2M

This implies that (T) = 0

02

Determine if statement is True or False

From the rank nullity theorem as follows.

dimT=rankT+nullityTdimV=rankT+0dimV=lmT

The dimension of domain of T is equal to range of T.

Hence, T is an isomorphism.

Thus, the given statement 鈥淭here exists invertible 2脳2matrices P and Q such that the transformation TM=PM-QMis an isomorphism鈥� is true

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Most popular questions from this chapter

In Exercise 72through 74, letZnbe the set of all polynomials of degree鈮�nsuch that f(0) = 0.

74. Define an isomorphism fromZntoPn-1(think calculus!).

Show that if 0 is the neutral element of a linear space V then k0=0, for all scalars k.

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

Find the basis of all nxn diagonal matrix, and determine its dimension.

Question: TRUE OR FALSE?

4. The kernel of a linear transformation is a subspace of the domain.
See all solutions

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