A. Ch-ch-ch-ch-changes Suppose all vectors = (x, y) in the unit square, i.e. 0 ≤ x ≤ 1, and 0 ≤ y ≤ 1 are transformed to Au where A is a 2 x 2 matrix. 1 2 37 1. What is the shape of the transformed region when A= Hint: What happens to the edges of the region? 2. In general, what is the shape of the transformed region? 3. For which matrices A is that region a square? 4. For which A is that region a line segment? 5. Optional Challenge: Show that the area of the transformed region is det(A). B. All about that basis III Let T: R³ R³ be the map given by T(x, y, z) = (x+y+z, 3x-y-z, 2x - 4y).
A. Ch-ch-ch-ch-changes Suppose all vectors = (x, y) in the unit square, i.e. 0 ≤ x ≤ 1, and 0 ≤ y ≤ 1 are transformed to Au where A is a 2 x 2 matrix. 1 2 37 1. What is the shape of the transformed region when A= Hint: What happens to the edges of the region? 2. In general, what is the shape of the transformed region? 3. For which matrices A is that region a square? 4. For which A is that region a line segment? 5. Optional Challenge: Show that the area of the transformed region is det(A). B. All about that basis III Let T: R³ R³ be the map given by T(x, y, z) = (x+y+z, 3x-y-z, 2x - 4y).
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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![A. Ch-ch-ch-ch-changes
Suppose all vectors = (x, y) in the unit square, i.e. 0 ≤ x ≤ 1, and 0 ≤ y ≤ 1 are
transformed to Au where A is a 2 x 2 matrix.
1 2
37
1. What is the shape of the transformed region when A=
Hint: What happens to the edges of the region?
2. In general, what is the shape of the transformed region?
3. For which matrices A is that region a square?
4. For which A is that region a line segment?
5. Optional Challenge: Show that the area of the transformed region is det(A).
B. All about that basis III
Let T: R³ R³ be the map given by T(x, y, z) = (x+y+z, 3x-y-z, 2x - 4y).
1. Show that T is a linear transformation.
2. Find the matrix A of T with respect to the standard unit basis E of R³.
3. Let S = {V1, V2, V3} = {(2,0, -2), (0, 2, 4), (1, 1, 0)}.
(a) Show that S is a basis for R³.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F50f06152-6ab7-40cf-844d-29583a7c45f7%2F10a7455d-f740-4387-becd-ba758a7e7b4a%2Fc3du3t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A. Ch-ch-ch-ch-changes
Suppose all vectors = (x, y) in the unit square, i.e. 0 ≤ x ≤ 1, and 0 ≤ y ≤ 1 are
transformed to Au where A is a 2 x 2 matrix.
1 2
37
1. What is the shape of the transformed region when A=
Hint: What happens to the edges of the region?
2. In general, what is the shape of the transformed region?
3. For which matrices A is that region a square?
4. For which A is that region a line segment?
5. Optional Challenge: Show that the area of the transformed region is det(A).
B. All about that basis III
Let T: R³ R³ be the map given by T(x, y, z) = (x+y+z, 3x-y-z, 2x - 4y).
1. Show that T is a linear transformation.
2. Find the matrix A of T with respect to the standard unit basis E of R³.
3. Let S = {V1, V2, V3} = {(2,0, -2), (0, 2, 4), (1, 1, 0)}.
(a) Show that S is a basis for R³.
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