(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).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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