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== Problem 1. Recall that O2 (R) = {A = M₂(R): A'A = 1} is the group of 2 × 2 orthogonal matrices, and let A = O2(R). 1.1. Show that det(A) = 1 or det(A) = −1. 1.2. Show that the transformation TA: R² → R² defined by TA(v) = Av is a linear isometry. This will complete the proof of Corollary 1.14 from the course notes. 1.3. Let v R² be the first column of A and let w R² be the second, so that A=(vw). Show that v and w satisfy ||v|| = ||w|| = 1 and v.w=0.

== Problem 1. Recall that O2 (R) = {A = M₂(R): A'A = 1} is the group of 2 × 2 orthogonal matrices, and let A = O2(R). 1.1. Show that det(A) = 1 or det(A) = −1. 1.2. Show that the transformation TA: R² → R² defined by TA(v) = Av is a linear isometry. This will complete the proof of Corollary 1.14 from the course notes. 1.3. Let v R² be the first column of A and let w R² be the second, so that A=(vw). Show that v and w satisfy ||v|| = ||w|| = 1 and v.w=0.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.9: Properties Of Determinants
Problem 34E
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== Problem 1. Recall that O2 (R) = {A = M₂(R): A'A = 1} is the group of 2 × 2 orthogonal matrices, and let A = O2(R). 1.1. Show that det(A) = 1 or det(A) = −1. 1.2. Show that the transformation TA: R² → R² defined by TA(v) = Av is a linear isometry. This will complete the proof of Corollary 1.14 from the course notes. 1.3. Let v R² be the first column of A and let w R² be the second, so that A=(vw). Show that v and w satisfy ||v|| = ||w|| = 1 and v.w=0.
Transcribed Image Text:== Problem 1. Recall that O2 (R) = {A = M₂(R): A'A = 1} is the group of 2 × 2 orthogonal matrices, and let A = O2(R). 1.1. Show that det(A) = 1 or det(A) = −1. 1.2. Show that the transformation TA: R² → R² defined by TA(v) = Av is a linear isometry. This will complete the proof of Corollary 1.14 from the course notes. 1.3. Let v R² be the first column of A and let w R² be the second, so that A=(vw). Show that v and w satisfy ||v|| = ||w|| = 1 and v.w=0.
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