3. Recall that P denotes the vector space of polynomials of degree at most k with real coefficients, andthat we've discussed a nice basis for this space. Also, recall that the derivative function : Pk → Pk-1is a linear transformation. (In the following, you'll want to refer to our discussion from class onWednesday 2/23.)dr(a) Determine the numbers n, m such that P2 - R" and P1theorem 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 usingthe 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

The complete solutions are given below

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).

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Linear Transformations

Section6.CR: Review Exercises

Problem 65CR

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