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

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

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 24EQ
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Let V be an inner product space. For a fixed vector v in V, define T: V → R by T(v) = (v, v₁). Prove that 7 is a linear transformation. Let v Vo be a vector in inner product space V, and define T: V→ R by T(v) (V, V ---Select--- V T(v + w) = = = = V + W (v, v = C V V ---Select--- T(v) + T(w) Next, suppose c is a scalar. By the properties of the inner product, we have the following. T(cv) = CV 0 cT(v) Therefore, T: V→ R is a linear transformation. V₁ = ✓---Select--- Now suppose v and w are vectors in V. By the properties of the inner product, we have the following.
Transcribed Image Text:Let V be an inner product space. For a fixed vector v in V, define T: V → R by T(v) = (v, v₁). Prove that 7 is a linear transformation. Let v Vo be a vector in inner product space V, and define T: V→ R by T(v) (V, V ---Select--- V T(v + w) = = = = V + W (v, v = C V V ---Select--- T(v) + T(w) Next, suppose c is a scalar. By the properties of the inner product, we have the following. T(cv) = CV 0 cT(v) Therefore, T: V→ R is a linear transformation. V₁ = ✓---Select--- Now suppose v and w are vectors in V. By the properties of the inner product, we have the following.
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