Problem 9. Let V be the vector space {B € M2(R)|B BT}, and define the matrices P = 0 1 Q- 6 Define the linear transformations T :V → V and S :V →V by T(B) = PBP", S(B) = QBQ". Show that S inverts T in the sense that S(T(B)) = B. Problem 10. Let V and T be the vector space and linear transformation defined in Problem 9, and let be a basis of V. Find the matrix A of T with respect to C, and show that A-1 is the matrix of S with respect to C.

Problem 9. Let V be the vector space {B € M2(R)|B BT}, and define the matrices P = 0 1 Q- 6 Define the linear transformations T :V → V and S :V →V by T(B) = PBP", S(B) = QBQ". Show that S inverts T in the sense that S(T(B)) = B. Problem 10. Let V and T be the vector space and linear transformation defined in Problem 9, and let be a basis of V. Find the matrix A of T with respect to C, and show that A-1 is the matrix of S with respect to C.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.6: The Matrix Of A Linear Transformation

Problem 6AEXP

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