Part I: Reflection about a line through the origin in R². In the context of linear transformations, the only lines we are permitted to reflect across in R? must go through the origin. (Why?) Simple examples such as reflections across the lines x = 0, y = 0, and y = x were presented in Section 6.2. In this part of the project, we aim to determine in general the matrix representation for the linear transformation T that reflects points of R? across an arbitrary line y = mx passing through the origin, where m is an arbitrary, fixed real number. (a) Determine a nonzero vector vị in R? such that T(v1) = v1. (b) Determine a nonzero vector v2 in R? such that T(v2) = -v2. (c) Explain why B = {v1, v2} is a basis for R2.
Part I: Reflection about a line through the origin in R². In the context of linear transformations, the only lines we are permitted to reflect across in R? must go through the origin. (Why?) Simple examples such as reflections across the lines x = 0, y = 0, and y = x were presented in Section 6.2. In this part of the project, we aim to determine in general the matrix representation for the linear transformation T that reflects points of R? across an arbitrary line y = mx passing through the origin, where m is an arbitrary, fixed real number. (a) Determine a nonzero vector vị in R? such that T(v1) = v1. (b) Determine a nonzero vector v2 in R? such that T(v2) = -v2. (c) Explain why B = {v1, v2} is a basis for R2.
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter6: Matrices And Determinants
Section: Chapter Questions
Problem 5P
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![Part I: Reflection about a line through the origin in R?. In the context of linear
transformations, the only lines we are permitted to reflect across in R? must go through
the origin. (Why?) Simple examples such as reflections across the lines x = 0, y = 0,
and y = x were presented in Section 6.2. In this part of the project, we aim to determine
in general the matrix representation for the linear transformation T that reflects points of
R? across an arbitrary line y = mx passing through the origin, where m is an arbitrary,
fixed real number.
(a) Determine a nonzero vector vi in R² such that T (v1) = V1.
(b) Determine a nonzero vector v2 in R? such that T(v2) = -v2.
%3D
(c) Explain why B = {v1, V2} is a basis for R?.
(d) Compute the matrix [T].
(e) Let I : R? → R? denote the identity linear transformation defined by I(x) = x
for all x in R?, and let C be any basis for R?. Use matrix representations for I to
find a formula for [7TIE in terms of [T].
(f) Let C = {(1,0), (0, 1)} denote the standard ordered basis on R2. Use part (e) to
determine [T]E.
(g) Use part (f) to derive a formula for T (x, y) for an arbitrary point (x, y) in R?.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F292fe4ee-a631-470e-961f-ce797a810e96%2F3aad2c0c-4d03-4528-a9b0-77e6499f3080%2Fi2zcrao_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Part I: Reflection about a line through the origin in R?. In the context of linear
transformations, the only lines we are permitted to reflect across in R? must go through
the origin. (Why?) Simple examples such as reflections across the lines x = 0, y = 0,
and y = x were presented in Section 6.2. In this part of the project, we aim to determine
in general the matrix representation for the linear transformation T that reflects points of
R? across an arbitrary line y = mx passing through the origin, where m is an arbitrary,
fixed real number.
(a) Determine a nonzero vector vi in R² such that T (v1) = V1.
(b) Determine a nonzero vector v2 in R? such that T(v2) = -v2.
%3D
(c) Explain why B = {v1, V2} is a basis for R?.
(d) Compute the matrix [T].
(e) Let I : R? → R? denote the identity linear transformation defined by I(x) = x
for all x in R?, and let C be any basis for R?. Use matrix representations for I to
find a formula for [7TIE in terms of [T].
(f) Let C = {(1,0), (0, 1)} denote the standard ordered basis on R2. Use part (e) to
determine [T]E.
(g) Use part (f) to derive a formula for T (x, y) for an arbitrary point (x, y) in R?.
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