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Chapter 4: Q64E (page 185)

Define an isomorphism fromP3 to R2脳2.

Short Answer

Expert verified

The solution is the transformation is an isomorphism.

Step by step solution

01

Definition of Linear Transformation

Consider two linear spaces V and W. A function T is said to be linear transformation if the following holds.

T(f+g)=T(f)+T(g)T(kf)=kT(f)

For all elementsf,g of V and is scalar.

A linear transformationT:V鈫�W is said to be an isomorphism if and only ifker(T)={0} andim(T)=W or dim(V)=dim(W).

02

Explanation of the solution

Consider the transformation as follows.

T:P3鈫�R2脳2and it is given as follows.

T(a0+a1t+a2t2+a3t3)=a0a1a2a3

For any non-zero polynomialsa0+a1t+a2t2+a3t3where0鈮�i鈮�3and there is at least one ai鈮�0.

There exist a non-zero matrix as follows.

a0a1a2a3鈭�R2脳2

Therefore, the kernel as follows.

ker(T)={0}

And the dimensions as follows.

dim(P3)=dim(R2脳2)=4

Thus, the linear transformationT:P3鈫�R2脳2 is an isomorphism.

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Most popular questions from this chapter

Which of the subsets Vof鈩�3脳3given in Exercise 6through 11are subspaces of鈩�3脳3. The3脳3matrices in reduced row-echelon form.

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, T(f(t))=f(7) fromP2torole="math" localid="1659412169328" 鈩�.

T denotes the space of infinity sequence of real numbers,T(f(t))=f(t)+f''(t)+sin(t)fromC鈭�to C鈭�.

Question: TRUE OR FALSE?

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

In Exercises 5 through 40, find the matrix of the given linear transformationwith respect to the given basis. If no basis is specified, use standard basis:渭=(1,t,t)forP2,

role="math" localid="1659423247247" 渭=([1000],[0100],[0010],[0001])

for2脳2androle="math" localid="1659421462939" 渭=(1,i)for,.For the spaceU2脳2of upper triangular2脳2matrices, use the basis

渭=([1000],[0100],[0001])

Unless another basis is given. In each case, determine whether Tis an isomorphism. If Tisn鈥檛 an isomorphism, find bases of the kernel and image of Tand thus determine the rank of T.

12. T(M)=M[2003]fromR2脳2toR2脳2.
See all solutions

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