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3. Recall that Pk denotes the vector space of polynomials of degree at most k with real coefficients, and that we've discussed a nice basis for this space. Also, recall that the derivative function : Pk → Pk-1 is a linear transformation. (In the following, you'll want to refer to our discussion from class on Wednesday 2/23.) (a) Determine the numbers n, m such that P2 ~ R" and P1 ~ R". (Explain your answer using the theorem from class and the basis we've discussed before for P,.) (b) For the values of n, m in (a), determine the linear transformation T: Rm R" obtained using the commutative diagram we get from the isomorphisms in (a) and the linear transformation (c) Determine the standard matrix corresponding to the transformation T from part (b).

3. Recall that Pk denotes the vector space of polynomials of degree at most k with real coefficients, and that we've discussed a nice basis for this space. Also, recall that the derivative function : Pk → Pk-1 is a linear transformation. (In the following, you'll want to refer to our discussion from class on Wednesday 2/23.) (a) Determine the numbers n, m such that P2 ~ R" and P1 ~ R". (Explain your answer using the theorem from class and the basis we've discussed before for P,.) (b) For the values of n, m in (a), determine the linear transformation T: Rm R" obtained using the commutative diagram we get from the isomorphisms in (a) and the linear transformation (c) Determine the standard matrix corresponding to the transformation T from part (b).

Advanced Engineering Mathematics
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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3. Recall that P denotes the vector space of polynomials of degree at most k with real coefficients, and that we've discussed a nice basis for this space. Also, recall that the derivative function : Pk → Pk-1 is a linear transformation. (In the following, you'll want to refer to our discussion from class on Wednesday 2/23.) dr (a) Determine the numbers n, m such that P2 - R" and P1 theorem from class and the basis we've discussed before for Pn.) - R". (Explain your answer using the (b) For the values of n, m in (a), determine the linear transformation T: Rm -→ R" obtained using the commutative diagram we get from the isomorphisms in (a) and the linear transformation : P2 → P1. (c) Determine the standard matrix corresponding to the transformation T from part (b). d
Transcribed Image Text:3. Recall that P denotes the vector space of polynomials of degree at most k with real coefficients, and that we've discussed a nice basis for this space. Also, recall that the derivative function : Pk → Pk-1 is a linear transformation. (In the following, you'll want to refer to our discussion from class on Wednesday 2/23.) dr (a) Determine the numbers n, m such that P2 - R" and P1 theorem from class and the basis we've discussed before for Pn.) - R". (Explain your answer using the (b) For the values of n, m in (a), determine the linear transformation T: Rm -→ R" obtained using the commutative diagram we get from the isomorphisms in (a) and the linear transformation : P2 → P1. (c) Determine the standard matrix corresponding to the transformation T from part (b). d
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