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Exercise 2.5.1 Suppose T : R" + R" is a linear transformation. Prove that T is an isometry if and only if T(v) - T(w) = v.w. Recall that an isometry is a bijection that preserves distance Note: When proving that if T(v) - T(w) = v-w then T is an isometry, make sure you verify that T is a bijection.

Exercise 2.5.1 Suppose T : R" + R" is a linear transformation. Prove that T is an isometry if and only if T(v) - T(w) = v.w. Recall that an isometry is a bijection that preserves distance Note: When proving that if T(v) - T(w) = v-w then T is an isometry, make sure you verify that T is a bijection.

Linear Algebra: A Modern Introduction
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 18EQ
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Exercise 2.5.1 Suppose T : R" → R" is a linear transformation. Prove that T is an isometry if and only if T(v) T(w) = v. w. Recall that an isometry is a bijection that preserves distance Note: When proving that if T(v) T(w) = v w then T is an isometry, make sure you verify that T is a bijection.
Transcribed Image Text:Exercise 2.5.1 Suppose T : R" → R" is a linear transformation. Prove that T is an isometry if and only if T(v) T(w) = v. w. Recall that an isometry is a bijection that preserves distance Note: When proving that if T(v) T(w) = v w then T is an isometry, make sure you verify that T is a bijection.
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