Let B= {cos² x, sin r cos x, sin? x}, let V = Span(B), and let L: V f(x) → f'(x) V (a) Compute the matrix representation of L with respect to B, denoted M = [ L ]g¬2 Note that M is the unique matrix in M3(R) that makes the following diagram сотmute L V R³. M (b) characteristic polynomial of M. It has exactly one real root, so M has exactly one real eigenvalue, A. The eigenspace, E, is one-dimensional. Compute the i. FIRST What is X? ii. SECOND What is the unique standard basis vector, va, of E,? iii. THIRD What is the unique function f(x) E V such that [ f(x) ], = vx? iv. FOURTH Double check that L[f(x)] = \f(x) by using a trig identity.

Let B= {cos² x, sin r cos x, sin? x}, let V = Span(B), and let L: V f(x) → f'(x) V (a) Compute the matrix representation of L with respect to B, denoted M = [ L ]g¬2 Note that M is the unique matrix in M3(R) that makes the following diagram сотmute L V R³. M (b) characteristic polynomial of M. It has exactly one real root, so M has exactly one real eigenvalue, A. The eigenspace, E, is one-dimensional. Compute the i. FIRST What is X? ii. SECOND What is the unique standard basis vector, va, of E,? iii. THIRD What is the unique function f(x) E V such that [ f(x) ], = vx? iv. FOURTH Double check that L[f(x)] = \f(x) by using a trig identity.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 27EQ

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