1. Assume that T : V → W is a linear transformation of vector spaces. (a) Assume that T is injective, and that (, :)w is some inner product on W. Prove that (:, )v defined by (x, y)v = (T(x),T(y))w for all x, y E V defines an inner product on V. (b) Show that one of the properties of an inner product is not true for (, )v if T is not injective.

1. Assume that T : V → W is a linear transformation of vector spaces. (a) Assume that T is injective, and that (, :)w is some inner product on W. Prove that (:, )v defined by (x, y)v = (T(x),T(y))w for all x, y E V defines an inner product on V. (b) Show that one of the properties of an inner product is not true for (, )v if T is not injective.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.4: Linear Transformations

Problem 34EQ

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