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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb39617aa-1a85-4ff5-98c6-3f7fe0e80deb%2F8d9e3dfa-d10a-44a2-bc3d-39f2f803daa8%2Fbofderb_processed.png&w=3840&q=75)
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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