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Show that M indeed is a reflection. To do this, you need to find a (nonzero) column vector v such that Mv = v, another (nonzero) column vector w such that Mw=-w, and you need v and w to be perpendicular to each other. (This shows that M gives reflection across the line spanned by v, or in other words, Rv is the line of symmetry. Your job is to find v and w.

Show that M indeed is a reflection. To do this, you need to find a (nonzero) column vector v such that Mv = v, another (nonzero) column vector w such that Mw=-w, and you need v and w to be perpendicular to each other. (This shows that M gives reflection across the line spanned by v, or in other words, Rv is the line of symmetry. Your job is to find v and w.

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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I need help with the second bullet point in d
cos and sin 0 simple and memorable form for the inverse.) sin 0 une nonzero), (6 (All but the first of these is a "special kind of matrix" and there is a 090). (d) Consider the matrix M - Cos 0 sin 0 0 (13) (3) ( cos sin 0 It is a fact that this gives counter- cos ( (c) Consider the matrix M = clockwise rotation around the origin by angle 0. On the basis of this fact, thinking geomet- rically, what should the inverse matrix be? Explain how this agrees with your calculation in (b) (using elementary trigonometry). - sin sin 0 - cos 0 utrices: cos 0 sin 0 (e) Going back to (b): - • (i 3¹); is a rotation. What is the angle? Sino 1 1 2 3 4 - sin 0 Cos 0 It is a fact that this is reflection Sing-Co across a a line. . What should the inverse of a reflection be? Explain how this agrees with your answer in (b). Show that M indeed is a reflection. To do this, you need to find a (nonzero) column vector v such that My = v, another (nonzero) column vector w such that Mw = -w, and you need v and w to be perpendicular to each other. (This shows that M gives reflection across the line spanned by v, or in other words, Rv is the line of symmetry. Your job is to find v and w.
Transcribed Image Text:cos and sin 0 simple and memorable form for the inverse.) sin 0 une nonzero), (6 (All but the first of these is a "special kind of matrix" and there is a 090). (d) Consider the matrix M - Cos 0 sin 0 0 (13) (3) ( cos sin 0 It is a fact that this gives counter- cos ( (c) Consider the matrix M = clockwise rotation around the origin by angle 0. On the basis of this fact, thinking geomet- rically, what should the inverse matrix be? Explain how this agrees with your calculation in (b) (using elementary trigonometry). - sin sin 0 - cos 0 utrices: cos 0 sin 0 (e) Going back to (b): - • (i 3¹); is a rotation. What is the angle? Sino 1 1 2 3 4 - sin 0 Cos 0 It is a fact that this is reflection Sing-Co across a a line. . What should the inverse of a reflection be? Explain how this agrees with your answer in (b). Show that M indeed is a reflection. To do this, you need to find a (nonzero) column vector v such that My = v, another (nonzero) column vector w such that Mw = -w, and you need v and w to be perpendicular to each other. (This shows that M gives reflection across the line spanned by v, or in other words, Rv is the line of symmetry. Your job is to find v and w.
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