Answered: Let x, y E Rn be given. (a) If ||x|| =…

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|.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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).

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