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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 and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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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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