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1. Determine whether the following statements are true or false. If your answer is “true", then prove your answer; if your answer is “false", then give a counterexample that shows that the statement is false. (a) If A is an orthogonal matrix, then det(A) = ±1. (Hint: What is A-1?) (b) If det(A) = ±1, then A is an orthogonal matrix. Suppose T: R" → R" is a linear transformation. If, for any pair of orthogonal (c) vectors v and w in R", the vectors T(v) and T(w) are also orthogonal vectors in R", then T is an orthogonal transformation.

1. Determine whether the following statements are true or false. If your answer is “true", then prove your answer; if your answer is “false", then give a counterexample that shows that the statement is false. (a) If A is an orthogonal matrix, then det(A) = ±1. (Hint: What is A-1?) (b) If det(A) = ±1, then A is an orthogonal matrix. Suppose T: R" → R" is a linear transformation. If, for any pair of orthogonal (c) vectors v and w in R", the vectors T(v) and T(w) are also orthogonal vectors in R", then T is an orthogonal transformation.

Advanced Engineering Mathematics
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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1. Determine whether the following statements are true or false. If your answer is "true", then prove your answer; if your answer is “false", then give a counterexample that shows that the statement is false. (a) If A is an orthogonal matrix, then det(A) = ±1. (Hint: What is A-1?) (b) If det(A) = ±1, then A is an orthogonal matrix. Suppose T: R" → R" is a linear transformation. If, for any pair of orthogonal (c) vectors v and w in R", the vectors T(v) and T(w) are also orthogonal vectors in R", then T is an orthogonal transformation.
Transcribed Image Text:1. Determine whether the following statements are true or false. If your answer is "true", then prove your answer; if your answer is “false", then give a counterexample that shows that the statement is false. (a) If A is an orthogonal matrix, then det(A) = ±1. (Hint: What is A-1?) (b) If det(A) = ±1, then A is an orthogonal matrix. Suppose T: R" → R" is a linear transformation. If, for any pair of orthogonal (c) vectors v and w in R", the vectors T(v) and T(w) are also orthogonal vectors in R", then T is an orthogonal transformation.
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