Consider the transformation 7(x, y) = (x - 2y, x + 2y). (a) Compute the image under T of each vertex in the below grid and make a careful plot of them, which should be fairly large as you will add to it later. To speed this up, divide the task up among all members of the group. (b) For each pair A and B of vertices of the grid joined by a line, add the line segment joining T(A) to T(B) to your plot. This gives a rough picture of what T is doing. Check your answer with the instructor. (c) What is the image of the x-axis under T? The y-axis? (d) Consider the line £ given by x+y=1. What is the image of £ under T? Is it a circle, an ellipse, a hyperbola, or something else? Hint: First, parameterize L by r: R → R² and then consider f(t) = T(r(t)). (e) Consider the circle C given by x² + y² = 1. What is the image of C under T? Ⓒ (f) Add T(L), T(C) and T( ) to your picture. Check your answer with the instructor. Note: The transformation T is a particularly simple sort called a linear transformation.
Consider the transformation 7(x, y) = (x - 2y, x + 2y). (a) Compute the image under T of each vertex in the below grid and make a careful plot of them, which should be fairly large as you will add to it later. To speed this up, divide the task up among all members of the group. (b) For each pair A and B of vertices of the grid joined by a line, add the line segment joining T(A) to T(B) to your plot. This gives a rough picture of what T is doing. Check your answer with the instructor. (c) What is the image of the x-axis under T? The y-axis? (d) Consider the line £ given by x+y=1. What is the image of £ under T? Is it a circle, an ellipse, a hyperbola, or something else? Hint: First, parameterize L by r: R → R² and then consider f(t) = T(r(t)). (e) Consider the circle C given by x² + y² = 1. What is the image of C under T? Ⓒ (f) Add T(L), T(C) and T( ) to your picture. Check your answer with the instructor. Note: The transformation T is a particularly simple sort called a linear transformation.
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:6. Consider the transformation T(x, y) = (x - 2y, x + 2y).
(a) Compute the image under T of each vertex in the below grid and make
a careful plot of them, which should be fairly large as you will add to
it later.
To speed this up, divide the task up among all members of the group.
(b) For each pair A and B of vertices of the grid joined by a line, add the
line segment joining T(A) to T(B) to your plot. This gives a rough
picture of what T is doing.
Check your answer with the instructor.
(c) What is the image of the x-axis under T? The y-axis?
(d) Consider the line L given by x+y=1. What is the image of L under
T? Is it a circle, an ellipse, a hyperbola, or something else?
Hint: First, parameterize L by r: R→ R² and then consider f(t) T(r(t)).
(e) Consider the circle C given by rx² + y² = 1. What is the image of C under T?
(f) Add T(L), T(C) and T() to your picture. Check your answer with the instructor.
Note: The transformation T is a particularly simple sort called a linear transformation.
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