). Consider the linear transformation T(x, y) = (x+y, 2x − y, 3x + 5y) and the basis 3 = {(1, 1), (0, −1)} and y = {(1, 1, 1), (1, 0, 1), (0, 0, 1)} of the domain and Image, respectively. Find: a) The change of basis matrix B from the basis y to the standard basis E in the image (that is, the change of basis matrix that satisfies that B[w] = [v]E). b) The change of basis C from the basis to the standard basis s of R2 (that is, the change of basis matrix that satisfies that C[v] = [v]s) c) The matrix A = = [T]E d) The matrix that represents the linear transformations with respect to the basis 3 (in the domain) and y in the image: [T] = [[T(1, 1)], [T(0, –1)]y]. e) Verify the formula A = = B[T] C-¹.
). Consider the linear transformation T(x, y) = (x+y, 2x − y, 3x + 5y) and the basis 3 = {(1, 1), (0, −1)} and y = {(1, 1, 1), (1, 0, 1), (0, 0, 1)} of the domain and Image, respectively. Find: a) The change of basis matrix B from the basis y to the standard basis E in the image (that is, the change of basis matrix that satisfies that B[w] = [v]E). b) The change of basis C from the basis to the standard basis s of R2 (that is, the change of basis matrix that satisfies that C[v] = [v]s) c) The matrix A = = [T]E d) The matrix that represents the linear transformations with respect to the basis 3 (in the domain) and y in the image: [T] = [[T(1, 1)], [T(0, –1)]y]. e) Verify the formula A = = B[T] C-¹.
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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Just answer C,D,E
![10. Consider the linear transformation T(x, y) = (x + y, 2x − y, 3x + 5y) and the basis 3 = {(1, 1), (0, -1)}
and y = {(1, 1, 1), (1, 0, 1), (0, 0, 1)} of the domain and Image, respectively. Find:
a) The change of basis matrix B from the basis y to the standard basis E in the image ( that is, the
change of basis matrix that satisfies that B[w] = [v]E).
b) The change of basis C from the basis 3 to the standard basis s of R² ( that is, the change of basis
matrix that satisfies that C[v] = [v]s)
[T]E
d) The matrix that represents the linear transformations with respect to the basis ß (in the domain)
and y in the image:
[T] = [[T(1, 1)], [T(0, -1)]y].
c) The matrix A
=
e) Verify the formula A = B[T]}C−¹.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F655e5d67-ab19-404c-b883-3aa3c693f6a1%2F082135e1-9561-46b2-adc6-e8e68f45d49a%2Ffg5739k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:10. Consider the linear transformation T(x, y) = (x + y, 2x − y, 3x + 5y) and the basis 3 = {(1, 1), (0, -1)}
and y = {(1, 1, 1), (1, 0, 1), (0, 0, 1)} of the domain and Image, respectively. Find:
a) The change of basis matrix B from the basis y to the standard basis E in the image ( that is, the
change of basis matrix that satisfies that B[w] = [v]E).
b) The change of basis C from the basis 3 to the standard basis s of R² ( that is, the change of basis
matrix that satisfies that C[v] = [v]s)
[T]E
d) The matrix that represents the linear transformations with respect to the basis ß (in the domain)
and y in the image:
[T] = [[T(1, 1)], [T(0, -1)]y].
c) The matrix A
=
e) Verify the formula A = B[T]}C−¹.
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Step 1: Define change of basis matrix
VIEWStep 2: Determine the matrix A
VIEWStep 3: Determine A using the definition of the change of basis matrix
VIEWStep 4: Determine the required matrix
VIEWStep 5: Determine the required matrix
VIEWStep 6: Determine B and C^-1
VIEWStep 7: Verify the given formula
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