For each of the following linear transformations of the plane IR", determine where the transformation sends e, and eż, writing your answers as linear combinations of e, and ez. Enter exact values, not decimal approximations. (A) A horizontal shear that transforms e₂ into ₂-3e₁ (leaving e₁ unchanged). e₁ H e₁+ C₂H e₁+ (B) Rotation of the plane counter-clockwise about the origin by radians. e₁+ CH e₂ → e₁+ (C) Reflection of the plane through the line y = -2. e₁+ e₁+ (D) The composite linear transformation T that first performs (A), then performs (B), and finally performs (C). e₁ → e₁+ e → e₂ What is the standard matrix for T'? e₁+ e₂ e₂ e₂
For each of the following linear transformations of the plane IR", determine where the transformation sends e, and eż, writing your answers as linear combinations of e, and ez. Enter exact values, not decimal approximations. (A) A horizontal shear that transforms e₂ into ₂-3e₁ (leaving e₁ unchanged). e₁ H e₁+ C₂H e₁+ (B) Rotation of the plane counter-clockwise about the origin by radians. e₁+ CH e₂ → e₁+ (C) Reflection of the plane through the line y = -2. e₁+ e₁+ (D) The composite linear transformation T that first performs (A), then performs (B), and finally performs (C). e₁ → e₁+ e → e₂ What is the standard matrix for T'? e₁+ e₂ e₂ e₂
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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Transcribed Image Text:For each of the following linear transformations of the plane IR², determine where the transformation sends e, and e2, writing your answers as linear combinations of e, and e₂. Enter exact
values, not decimal approximations.
(A) A horizontal shear that transforms e₂ into ₂ - 3e₁ (leaving ₁ unchanged).
e₁+
→
e₂ →
e₁ H
e₂ →
(B) Rotation of the plane counter-clockwise about the origin by radians.
e₁+
e₂ H
e₁+
(C) Reflection of the plane through the line y = -2.
е₁ →
e₁+
e₂ →
e₁+
What is the standard matrix for T?
e₂
e₁+
€₂
e₁+
€₂
$
e₁+
(D) The composite linear transformation T that first performs (A), then performs (B), and finally performs (C).
e₂ →
e₂
$
e₂
€₂
e₂
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