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Chapter 4: Q51E (page 196)

In Exercises 48 through 53, let V be the space spanned by the two functions cos(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.

51.role="math" localid="1660417513133" T(f(t))=f(t-Ï€2)

Short Answer

Expert verified

The solution is B=0-110

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

Let T(f(t))=f(t-Ï€2) be the transformation.

Assume fx=acosx+bsinx be the function.

Hence B=TcosxB   TsinxB 

Then, the transformation be as follows.

Tcosx=cosx-π2                   =sinx  

Similarly further simplification is as follows

Tsinx=sinx-π2                 =-cosx

Therefore the matrix be B as follows

B=0-110

Hence the solution

02

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

Consider B=0-110 be the matrix of the given transformation T.

B=detB        =ad-bc        =0--1       =1≠0

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

Hence the solution.

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