(d) A linear transformation T: R" → R" is called an isometry if ||T(u)|| = ||u||, for all u E R. Show that R is an isometry. (e) (f) If n is any normal vector to W, find R(n). If v is any vector in W, find R(v).
(d) A linear transformation T: R" → R" is called an isometry if ||T(u)|| = ||u||, for all u E R. Show that R is an isometry. (e) (f) If n is any normal vector to W, find R(n). If v is any vector in W, find R(v).
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Linear Transformations
Section6.1: Introduction To Linear Transformations
Problem 76E: A translation in R2 is a function of the form T(x,y)=(xh,yk), where at least one of the constants h...
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especially question (d)and(e)and(f), thanks!
![4.
W
=
(a)
(b)
(c)
(d)
(e)
(f)
Consider the transformation R: R2 R2 given by the reflection about the line
{(x, y) | 2x + 3y = 0}.
Write the formula for R(u), u € R2, in terms of the projection of u onto a
normal vector of the line W.
Use the formula from Part (a) to show that the transformation R is linear by
verifying (LT1) and (LT2) (see the definition of a linear transformation).
Find the standard matrix A = [R] for R.
A linear transformation T: R → Rn is called an isometry if ||T(u)|| = ||u||,
for all u E Rn. Show that R is an isometry.
If n is any normal vector to W, find R(n).
If v is any vector in W, find R(v).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b3ba425-73b4-4287-b33a-2515a4443977%2F3a4c91db-6446-4785-8dda-362344c8b395%2F71fa7ot_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4.
W
=
(a)
(b)
(c)
(d)
(e)
(f)
Consider the transformation R: R2 R2 given by the reflection about the line
{(x, y) | 2x + 3y = 0}.
Write the formula for R(u), u € R2, in terms of the projection of u onto a
normal vector of the line W.
Use the formula from Part (a) to show that the transformation R is linear by
verifying (LT1) and (LT2) (see the definition of a linear transformation).
Find the standard matrix A = [R] for R.
A linear transformation T: R → Rn is called an isometry if ||T(u)|| = ||u||,
for all u E Rn. Show that R is an isometry.
If n is any normal vector to W, find R(n).
If v is any vector in W, find R(v).
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