Let V be an inner product space. For a fixed vector v, in V, define T: V - R by T(v) = (v, v.). Prove that Tis a linear transformation. Let v, be a vector in inner product space V, and define T: V → R by T(v) = (v, v). Now suppose v and w are vectors in V. By the properties of the inner product, we have the following. T(v + w) = (---Select--- v --Select--- v) = (v, ---Select-- 1) + --Select-- v.-Select-- v) = T(v) + T(w) Next, suppose c is a scalar. By the properties of the inner product, we have the following. T(cv) = -Select--- v --Select-- v) = c ---Select--- ---Select--- v = cT(v) Therefore, T: V -R is a linear transformation.

Let V be an inner product space. For a fixed vector v, in V, define T: V - R by T(v) = (v, v.). Prove that Tis a linear transformation. Let v, be a vector in inner product space V, and define T: V → R by T(v) = (v, v). Now suppose v and w are vectors in V. By the properties of the inner product, we have the following. T(v + w) = (---Select--- v --Select--- v) = (v, ---Select-- 1) + --Select-- v.-Select-- v) = T(v) + T(w) Next, suppose c is a scalar. By the properties of the inner product, we have the following. T(cv) = -Select--- v --Select-- v) = c ---Select--- ---Select--- v = cT(v) Therefore, T: V -R is a linear transformation.

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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