(a) Prove that an orthogonal 2 x 2 matrix must have the form b] or a b -a where is a unit vector. b (b) Using part (a), show that every orthogonal 2 x 2 matrix is of the form cos 0 - sin 0 cos 0 cos e or sin 0 -cos 0 sin 0 sin 0 where 0<0<27. (c) Show that every orthogonal 2 x 2 matrix corre- sponds to either a rotation or a reflection in R. (d) Show that an orthogonal 2 x 2 matrix Q cor- responds to a rotation in R' if det Q = 1 and a reflection in R? if det Q = -1.
(a) Prove that an orthogonal 2 x 2 matrix must have the form b] or a b -a where is a unit vector. b (b) Using part (a), show that every orthogonal 2 x 2 matrix is of the form cos 0 - sin 0 cos 0 cos e or sin 0 -cos 0 sin 0 sin 0 where 0<0<27. (c) Show that every orthogonal 2 x 2 matrix corre- sponds to either a rotation or a reflection in R. (d) Show that an orthogonal 2 x 2 matrix Q cor- responds to a rotation in R' if det Q = 1 and a reflection in R? if det Q = -1.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.6: Matrices
Problem 17E
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