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Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ spanned [9 by the vector 3 Let S R³ R³ denote the reflection through the plane P: it takes a vector in R³ and 7 transforms it into its mirror image, the mirror being the plane P. Let T : R³ → R denote the counterclockwise rotation around L by an angle of 90°. Both S and T are linear transformations (you don't have to prove that). Find the matrix A such that (T○ S)(x) = Ax for all vectors & R³. Here is some information that you might find useful: [1] • The vector 2 is perpendicular to the plane. 21 • The vectors and -7 are perpendicular to the line. They are also perpendicular to each other. 30 Finally, here is a sugestion: don't hesitate to use a 3D graphing calculator (like Desmos) to help you picture the plane and the line.

Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ spanned [9 by the vector 3 Let S R³ R³ denote the reflection through the plane P: it takes a vector in R³ and 7 transforms it into its mirror image, the mirror being the plane P. Let T : R³ → R denote the counterclockwise rotation around L by an angle of 90°. Both S and T are linear transformations (you don't have to prove that). Find the matrix A such that (T○ S)(x) = Ax for all vectors & R³. Here is some information that you might find useful: [1] • The vector 2 is perpendicular to the plane. 21 • The vectors and -7 are perpendicular to the line. They are also perpendicular to each other. 30 Finally, here is a sugestion: don't hesitate to use a 3D graphing calculator (like Desmos) to help you picture the plane and the line.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
Problem 13AEXP
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Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ spanned [9 by the vector 3 Let S R³ R³ denote the reflection through the plane P: it takes a vector in R³ and 7 transforms it into its mirror image, the mirror being the plane P. Let T : R³ → R denote the counterclockwise rotation around L by an angle of 90°. Both S and T are linear transformations (you don't have to prove that). Find the matrix A such that (T○ S)(x) = Ax for all vectors & R³. Here is some information that you might find useful: [1] • The vector 2 is perpendicular to the plane. 21 • The vectors and -7 are perpendicular to the line. They are also perpendicular to each other. 30 Finally, here is a sugestion: don't hesitate to use a 3D graphing calculator (like Desmos) to help you picture the plane and the line.
Transcribed Image Text:Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ spanned [9 by the vector 3 Let S R³ R³ denote the reflection through the plane P: it takes a vector in R³ and 7 transforms it into its mirror image, the mirror being the plane P. Let T : R³ → R denote the counterclockwise rotation around L by an angle of 90°. Both S and T are linear transformations (you don't have to prove that). Find the matrix A such that (T○ S)(x) = Ax for all vectors & R³. Here is some information that you might find useful: [1] • The vector 2 is perpendicular to the plane. 21 • The vectors and -7 are perpendicular to the line. They are also perpendicular to each other. 30 Finally, here is a sugestion: don't hesitate to use a 3D graphing calculator (like Desmos) to help you picture the plane and the line.
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