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In this part we ask you to write a formal proof for each of the statements given below. Giving an example and verifying the statement on that example does not constitute a proof. You need to argue generally. Indicate which properties you used (for example, if you are using det(QR) = (det Q)(det R), just say so). Use this definition of an orthogonal matrix: We say that Q E Mnxn(R) is an orthogonal matrix if QtrQ = In, where In is the n x n identity matrix. (a) that AB is an orthogonal matrix. Suppose that A and B are orthogonal n x n matrices. Show (b) det A = ±1. Suppose that A is an n x n orthogonal matrix. Show that (c) || AŬ|| = ||T||, for all vectors i E R". Suppose that A is an n x n orthogonal matrix. Show that

In this part we ask you to write a formal proof for each of the statements given below. Giving an example and verifying the statement on that example does not constitute a proof. You need to argue generally. Indicate which properties you used (for example, if you are using det(QR) = (det Q)(det R), just say so). Use this definition of an orthogonal matrix: We say that Q E Mnxn(R) is an orthogonal matrix if QtrQ = In, where In is the n x n identity matrix. (a) that AB is an orthogonal matrix. Suppose that A and B are orthogonal n x n matrices. Show (b) det A = ±1. Suppose that A is an n x n orthogonal matrix. Show that (c) || AŬ|| = ||T||, for all vectors i E R". Suppose that A is an n x n orthogonal matrix. Show that

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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In this part we ask you to write a formal proof for each of the statements given below. Giving an example and verifying the statement on that example does not constitute a proof. You need to argue generally. Indicate which properties you used (for example, if you are using det(QR) = (det Q)(det R), just say so). Use this definition of an orthogonal matrix: We say that Q E Mnxn(R) is an orthogonal matrix if QtrQ = In, where In is the n x n identity matrix. (a) that AB is an orthogonal matrix. Suppose that A and B are orthogonal n x n matrices. Show (b) det A = ±1. Suppose that A is an n x n orthogonal matrix. Show that (c) || AT|| = ||T||, for all vectors Ủ E R". Suppose that A is an n x n orthogonal matrix. Show that (d) in R": Show that the formula below is true for all vectors i and w ||ū+ w||? + ||7 – w||? = 2|| ||? + 2||1||. (This proof is easy: Start from the left hand side and write ||U+w||² = (7+ u) · (7+w) and similarly for the other term. Distribute, simplify, and you should be all set.)
Transcribed Image Text:In this part we ask you to write a formal proof for each of the statements given below. Giving an example and verifying the statement on that example does not constitute a proof. You need to argue generally. Indicate which properties you used (for example, if you are using det(QR) = (det Q)(det R), just say so). Use this definition of an orthogonal matrix: We say that Q E Mnxn(R) is an orthogonal matrix if QtrQ = In, where In is the n x n identity matrix. (a) that AB is an orthogonal matrix. Suppose that A and B are orthogonal n x n matrices. Show (b) det A = ±1. Suppose that A is an n x n orthogonal matrix. Show that (c) || AT|| = ||T||, for all vectors Ủ E R". Suppose that A is an n x n orthogonal matrix. Show that (d) in R": Show that the formula below is true for all vectors i and w ||ū+ w||? + ||7 – w||? = 2|| ||? + 2||1||. (This proof is easy: Start from the left hand side and write ||U+w||² = (7+ u) · (7+w) and similarly for the other term. Distribute, simplify, and you should be all set.)
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