) Let B = {1,1+x, 2+x+x2) and D Pn is the set of polynomials of degree n). a) (1+x, 1-x) be bases of P, and P₁, respectively (recall Let H be the map from P₂ to P₁ given by the derivative operation: H(f) - Find the matrix representation of H with respect to the bases B and D. b) Let W be the subset of P₂ that is spanned by the vectors {1+x, 2+x+x²) and let A be the map from W to P₁ given by the derivative map. Prove or disprove: A is an isomorphism. c) Let U be the subset of P₂ that is spanned by the vectors {1,2+x+x2) and let B be the map from U to P₁ given by the derivative map. Prove or disprove: U is an isomorphism. d) Find a basis B' of W and a basis D' of P₁ such that the matrix representation of A is a diagonal matrix.

) Let B = {1,1+x, 2+x+x2) and D Pn is the set of polynomials of degree n). a) (1+x, 1-x) be bases of P, and P₁, respectively (recall Let H be the map from P₂ to P₁ given by the derivative operation: H(f) - Find the matrix representation of H with respect to the bases B and D. b) Let W be the subset of P₂ that is spanned by the vectors {1+x, 2+x+x²) and let A be the map from W to P₁ given by the derivative map. Prove or disprove: A is an isomorphism. c) Let U be the subset of P₂ that is spanned by the vectors {1,2+x+x2) and let B be the map from U to P₁ given by the derivative map. Prove or disprove: U is an isomorphism. d) Find a basis B' of W and a basis D' of P₁ such that the matrix representation of A is a diagonal matrix.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 27EQ

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