(a) Recall that two nxn matrices A and B are similar if there is an invertible n x n matrix S so that A=SBs-1. Show if two matrices are similar, then they have the same determinant. (b) Let V be a finite dimensional vector space and let T be a linear operator on V. Define the determinant of T to be the determinant of [T], where B is an ordered basis of V. Show that the determinant of T is well-defined in the sense that it does not depend on the choice of basis ß (that is, choosing a different basis for V would result in the same answer).

(a) Recall that two nxn matrices A and B are similar if there is an invertible n x n matrix S so that A=SBs-1. Show if two matrices are similar, then they have the same determinant. (b) Let V be a finite dimensional vector space and let T be a linear operator on V. Define the determinant of T to be the determinant of [T], where B is an ordered basis of V. Show that the determinant of T is well-defined in the sense that it does not depend on the choice of basis ß (that is, choosing a different basis for V would result in the same answer).

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.2: Determinants

Problem 7AEXP

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