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Chapter 4: Q48E (page 196)

In Exercises 48 through 53, let V be the space spanned by the two functionscos(t)and sin(t). In each exercise, find the matrix of the given transformation T with respect to the basis cos(t),sin(t), and determine whether T is an isomorphism. T(f)=f'

Short Answer

Expert verified

The solution isB=(0鈭�110)

Step by step solution

01

Step 1:Solution for the matrix of the given transformation T

Consider the matrix of the given transformation be T

LetT(f)=f'be the transformation.

Assumef(x)=acos(x)+bsin(x)be the function.

HenceB=[[T(cosx)]B鈥夆赌夆赌�[T(sinx)]B鈥�]

Then, the transformation be as follows.

Tcos(x)=鈭�sin(x)T(sinx)=cos(x)

Therefore the matrix be B as follows

B=(0鈭�110)

Hence the solution.

02

Step 2:Solution for the isomorphism of the given transformation T

Consider B=(0鈭�110)be the matrix of the given transformation T.

[B]=det(B)鈥夆赌夆赌夆�夆赌夆赌夆�夆��=ad鈭�bc鈥夆赌夆赌夆�夆赌夆赌夆�夆��=0鈭�(鈭�1)鈥夆赌夆赌夆�夆赌夆赌夆��=1鈮�0

Thus the transformation T is invertible as well.So T is an isomorphism.

Hence the solution.

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Most popular questions from this chapter

TRUE OR FALSE?

38. There exists a subspace of鈩�3脳4that is isomorphic toP9.

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism,T(x+iy)=x2+y2 from 鈩�to 鈩�.

Find the image, kernel, rank, and nullity of the transformation T in T(ft)=(f'')(t)+4f'(t)fromP2toP2

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism,T(ft)=鈭�-23f(t)dt from toP2to鈩�.

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.
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