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For the following let V = P2(R), the real vector space of polynomials with degree at most 2 and W = R1, the real vector space of 3x1 matrices (column vectors). a) Explain why V is isomorphic to W. Using the standard bases B, : 1, æ, x² for V and C1 : E1, E2, E3 (where E is a column vector of O's except for a 1 in the ith row) for W, define an isomorphism T from V to W. What is T-1? For these bases find M(T) and M(T 1). How does T compare to the linear map from V to W given by [p(x)]B- for any p(x) E V? b) Consider another basis of V given by B2 : a – 1, x² – x – 1,1. Let T be the linear map from V to W given by T (a(x – 1) + b(x² – x – 1) + c) = [a(x – 1) + b(x² – x – 1) + c]B,. What is the matrix for Î with respect to B2 and C1? What is the matrix for (T) 1 with respect to C1 and B2? Explain how to use your answer in part (a) and these matrices to obtain [p(z)], from [p(x)]B, for any p(x) E V. c) Suppose that S :V → W is a linear map whose matrix with respect to the bases B1 and C1 is given by: [1 1 1] M(S) = |1 1 0 |20 1 For p(x) = a + bx + cx² e V, what is S(p(x))? Using part (a), what is TS(p(x))? d) Given another basis C2 : E1 – E2, E3, E1 + E2 + E3 for W, find the matrix for S from (b) with respect to B, and C2.

For the following let V = P2(R), the real vector space of polynomials with degree at most 2 and W = R1, the real vector space of 3x1 matrices (column vectors). a) Explain why V is isomorphic to W. Using the standard bases B, : 1, æ, x² for V and C1 : E1, E2, E3 (where E is a column vector of O's except for a 1 in the ith row) for W, define an isomorphism T from V to W. What is T-1? For these bases find M(T) and M(T 1). How does T compare to the linear map from V to W given by [p(x)]B- for any p(x) E V? b) Consider another basis of V given by B2 : a – 1, x² – x – 1,1. Let T be the linear map from V to W given by T (a(x – 1) + b(x² – x – 1) + c) = [a(x – 1) + b(x² – x – 1) + c]B,. What is the matrix for Î with respect to B2 and C1? What is the matrix for (T) 1 with respect to C1 and B2? Explain how to use your answer in part (a) and these matrices to obtain [p(z)], from [p(x)]B, for any p(x) E V. c) Suppose that S :V → W is a linear map whose matrix with respect to the bases B1 and C1 is given by: [1 1 1] M(S) = |1 1 0 |20 1 For p(x) = a + bx + cx² e V, what is S(p(x))? Using part (a), what is TS(p(x))? d) Given another basis C2 : E1 – E2, E3, E1 + E2 + E3 for W, find the matrix for S from (b) with respect to B, and C2.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CM: Cumulative Review
Problem 5CM: Take this test to review the material in Chapters 4 and 5. After you are finished, check your work...
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For the following let V = P2(R), the real vector space of polynomials with degree at most 2 and W = R1, the real vector space of 3x1 matrices (column vectors). a) Explain why V is isomorphic to W. Using the standard bases B, : 1, æ, x² for V and C1 : E1, E2, E3 (where E is a column vector of O's except for a 1 in the ith row) for W, define an isomorphism T from V to W. What is T-1? For these bases find M(T) and M(T 1). How does T compare to the linear map from V to W given by [p(x)]B- for any p(x) E V? b) Consider another basis of V given by B2 : a – 1, x² – x – 1,1. Let T be the linear map from V to W given by T (a(x – 1) + b(x² – x – 1) + c) = [a(x – 1) + b(x² – x – 1) + c]B,. What is the matrix for Î with respect to B2 and C1? What is the matrix for (T) 1 with respect to C1 and B2? Explain how to use your answer in part (a) and these matrices to obtain [p(z)], from [p(x)]B, for any p(x) E V. c) Suppose that S :V → W is a linear map whose matrix with respect to the bases B1 and C1 is given by: [1 1 1] M(S) = |1 1 0 |20 1 For p(x) = a + bx + cx² e V, what is S(p(x))? Using part (a), what is TS(p(x))? d) Given another basis C2 : E1 – E2, E3, E1 + E2 + E3 for W, find the matrix for S from (b) with respect to B, and C2.
Transcribed Image Text:For the following let V = P2(R), the real vector space of polynomials with degree at most 2 and W = R1, the real vector space of 3x1 matrices (column vectors). a) Explain why V is isomorphic to W. Using the standard bases B, : 1, æ, x² for V and C1 : E1, E2, E3 (where E is a column vector of O's except for a 1 in the ith row) for W, define an isomorphism T from V to W. What is T-1? For these bases find M(T) and M(T 1). How does T compare to the linear map from V to W given by [p(x)]B- for any p(x) E V? b) Consider another basis of V given by B2 : a – 1, x² – x – 1,1. Let T be the linear map from V to W given by T (a(x – 1) + b(x² – x – 1) + c) = [a(x – 1) + b(x² – x – 1) + c]B,. What is the matrix for Î with respect to B2 and C1? What is the matrix for (T) 1 with respect to C1 and B2? Explain how to use your answer in part (a) and these matrices to obtain [p(z)], from [p(x)]B, for any p(x) E V. c) Suppose that S :V → W is a linear map whose matrix with respect to the bases B1 and C1 is given by: [1 1 1] M(S) = |1 1 0 |20 1 For p(x) = a + bx + cx² e V, what is S(p(x))? Using part (a), what is TS(p(x))? d) Given another basis C2 : E1 – E2, E3, E1 + E2 + E3 for W, find the matrix for S from (b) with respect to B, and C2.
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