(c) Which of the following statements are FALSE? (1) Let A E Mnxn(F). If (A1, v1) and (A2, v2) are eigenpairs of A, then (A1, v1 + v2) is an eigenpair of A. (2) Let A € Mnxn(F). If A is diagonalizable, then there exist an invertible matrix P and a unique diagonal matrix D such that P-AP = D. (3) Let T = {v1, V2, ..., V,} € F" and let w € F". If T is linearly independent, then TU{w} is linearly dependent. Answer for (c): (d) Which of the following statements are FALSE? 2+3i 4 - 5i (1) If C = then Cx = 0 has only the trivial solution. 2- 3і —3— 21, (2) Every subset of a vector space V that contains the zero vector in V is a subspace of V. (3) If V is a non-trivial subspace of F" and B is a basis for F", then it is possible to reduce B to obtain a basis for V. Answer for (d):

(c) Which of the following statements are FALSE? (1) Let A E Mnxn(F). If (A1, v1) and (A2, v2) are eigenpairs of A, then (A1, v1 + v2) is an eigenpair of A. (2) Let A € Mnxn(F). If A is diagonalizable, then there exist an invertible matrix P and a unique diagonal matrix D such that P-AP = D. (3) Let T = {v1, V2, ..., V,} € F" and let w € F". If T is linearly independent, then TU{w} is linearly dependent. Answer for (c): (d) Which of the following statements are FALSE? 2+3i 4 - 5i (1) If C = then Cx = 0 has only the trivial solution. 2- 3і —3— 21, (2) Every subset of a vector space V that contains the zero vector in V is a subspace of V. (3) If V is a non-trivial subspace of F" and B is a basis for F", then it is possible to reduce B to obtain a basis for V. Answer for (d):

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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