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Problem 2. Suppose that QE O2 (R). From Problem 1, we know that the determi- nant of Q is 1 or -1. 2.1. Show that if det(Q) = 1, then there is 0 << 2 such that Q = Ro, where cos Re = sin sin 0 cos 2.2. Show that if det(Q) = -1, then there is 0 << 2π such that Q = So, where Se = cos # sin sin 180). cos 0 2.3. Let v€ R². Determine the geometric relationship between the vector v and the vectors Rov and Sov. Hint. The first two parts of the problem can be solved simultaneously. As a first step, use Problem 1.3 to see that both columns of Q lay on the unit circle in R². What is another way to write any vector on the unit circle? For the third part of the problem, it might be helpful to write v in the form v = λ [cos sin for some non-negative real number A. Your answer should describe the relationship between v and Rev (in addition to v and Sav) using words like rotation and reflection. Make sure to justify your answer. Here is Problem 1 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|| = ||||= 1 and v.w=0.

Problem 2. Suppose that QE O2 (R). From Problem 1, we know that the determi- nant of Q is 1 or -1. 2.1. Show that if det(Q) = 1, then there is 0 << 2 such that Q = Ro, where cos Re = sin sin 0 cos 2.2. Show that if det(Q) = -1, then there is 0 << 2π such that Q = So, where Se = cos # sin sin 180). cos 0 2.3. Let v€ R². Determine the geometric relationship between the vector v and the vectors Rov and Sov. Hint. The first two parts of the problem can be solved simultaneously. As a first step, use Problem 1.3 to see that both columns of Q lay on the unit circle in R². What is another way to write any vector on the unit circle? For the third part of the problem, it might be helpful to write v in the form v = λ [cos sin for some non-negative real number A. Your answer should describe the relationship between v and Rev (in addition to v and Sav) using words like rotation and reflection. Make sure to justify your answer. Here is Problem 1 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|| = ||||= 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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Could you solve problem 2, please?

I have added problem 1 at the end, so please use it to solve the questions.

Thank you

Problem 2. Suppose that QE O2 (R). From Problem 1, we know that the determi- nant of Q is 1 or -1. 2.1. Show that if det(Q) = 1, then there is 0 << 2 such that Q = Ro, where cos Re = sin sin 0 cos 2.2. Show that if det(Q) = -1, then there is 0 << 2π such that Q = So, where Se = cos # sin sin 180). cos 0 2.3. Let v€ R². Determine the geometric relationship between the vector v and the vectors Rov and Sov. Hint. The first two parts of the problem can be solved simultaneously. As a first step, use Problem 1.3 to see that both columns of Q lay on the unit circle in R². What is another way to write any vector on the unit circle? For the third part of the problem, it might be helpful to write v in the form v = λ [cos sin for some non-negative real number A. Your answer should describe the relationship between v and Rev (in addition to v and Sav) using words like rotation and reflection. Make sure to justify your answer. Here is Problem 1 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|| = ||||= 1 and v.w=0.
Transcribed Image Text:Problem 2. Suppose that QE O2 (R). From Problem 1, we know that the determi- nant of Q is 1 or -1. 2.1. Show that if det(Q) = 1, then there is 0 << 2 such that Q = Ro, where cos Re = sin sin 0 cos 2.2. Show that if det(Q) = -1, then there is 0 << 2π such that Q = So, where Se = cos # sin sin 180). cos 0 2.3. Let v€ R². Determine the geometric relationship between the vector v and the vectors Rov and Sov. Hint. The first two parts of the problem can be solved simultaneously. As a first step, use Problem 1.3 to see that both columns of Q lay on the unit circle in R². What is another way to write any vector on the unit circle? For the third part of the problem, it might be helpful to write v in the form v = λ [cos sin for some non-negative real number A. Your answer should describe the relationship between v and Rev (in addition to v and Sav) using words like rotation and reflection. Make sure to justify your answer. Here is Problem 1 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|| = ||||= 1 and v.w=0.
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