4. Consider the linear transformation Qm: R² → R² that reflects a vector over the line y = mx. [H]. Show that the matrix for the linear transformation T(T) = proj() is (a) Let u = given by (b) Now use the diagram below (and the fact that the diagonals of a parallelogram bisect each other) to show that Qm() = 2proj(x) - T. Y 1 A = uu¹ ||ū||² Qm (x) proj() (c) Use parts (a) and (b) to show that the matrix for Qm is [1 - m² - 2m 1+m² 1+m² 2m 1+m² (d) Show that (QmQm)(x) = for all in R². m² - 1 1+ m² X
4. Consider the linear transformation Qm: R² → R² that reflects a vector over the line y = mx. [H]. Show that the matrix for the linear transformation T(T) = proj() is (a) Let u = given by (b) Now use the diagram below (and the fact that the diagonals of a parallelogram bisect each other) to show that Qm() = 2proj(x) - T. Y 1 A = uu¹ ||ū||² Qm (x) proj() (c) Use parts (a) and (b) to show that the matrix for Qm is [1 - m² - 2m 1+m² 1+m² 2m 1+m² (d) Show that (QmQm)(x) = for all in R². m² - 1 1+ m² X
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question

Transcribed Image Text:4. Consider the linear transformation Qm : R² → R² that reflects a vector over the line y = mx.
1
H
(a) Let u
=
given by
Show that the matrix for the linear transformation T(7)
A =
=
1
||ū||²
uu¹
(b) Now use the diagram below (and the fact that the diagonals of a parallelogram bisect
each other) to show that Qm(x) 2projū(x) — ☎.
Y
44
Qm(T)
proj()
x
(c) Use parts (a) and (b) to show that the matrix for Qm is
1 - m² 2m
1+m²
1+m²
(d) Show that (Qm ° Qm)(x) = π for all ☞ in R².
1+m²
2m m²
1+ m²
=
X
proj() is
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