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Let x, y E Rn be given. (a) If ||x|| = ||y|| = 1 and x y = 1, show that x = y. (b) If ||x|| ≤ 1 and ||y|| ≤ 1, prove that √1 - ||x||2²√1 - ||y||² ≤ 1 − |x · y|.

Let x, y E Rn be given. (a) If ||x|| = ||y|| = 1 and x y = 1, show that x = y. (b) If ||x|| ≤ 1 and ||y|| ≤ 1, prove that √1 - ||x||2²√1 - ||y||² ≤ 1 − |x · y|.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.1: Inner Product Spaces
Problem 15BEXP
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Let x, y E Rn be given. (a) If ||x|| = ||y|| = 1 and x y = 1, show that x = y. (b) If ||x|| ≤ 1 and ||y|| ≤ 1, prove that √1 - ||x||2²√1 - ||y||² ≤ 1 − |x · y|.
Transcribed Image Text:Let x, y E Rn be given. (a) If ||x|| = ||y|| = 1 and x y = 1, show that x = y. (b) If ||x|| ≤ 1 and ||y|| ≤ 1, prove that √1 - ||x||2²√1 - ||y||² ≤ 1 − |x · y|.
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Let A E Mn(R) be given. (a) If det (A) = 1, prove that adj (adj(A)) = A. (b) If A is nonsingular, prove that adj(A) is nonsingular and adj(A)-¹ = adj(A-¹).
Transcribed Image Text:Let A E Mn(R) be given. (a) If det (A) = 1, prove that adj (adj(A)) = A. (b) If A is nonsingular, prove that adj(A) is nonsingular and adj(A)-¹ = adj(A-¹).
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