13. Let B = {(1, 3), (-2,-2)} and B' = {(-12, 0), (-4,4)} be bases for R², and let 2 A = [3 be the matrix for T: R² R² relative to B. (a) Find the transition matrix P from B' to B. (b) Use the matrices P and A to find [v] and [7(v)], where [v] = [-1 2]T. (c) Find P-¹ and A' (the matrix for T relative to B'). (d) Find [7(v)] two ways. 14. Repeat Exercise 13 for B' = {(1, -1), (0, 1)}, and [v] (Use matrix A in Exercise 13.) B = {(1, 1), (2,3)}, = [1 −3]¹.
13. Let B = {(1, 3), (-2,-2)} and B' = {(-12, 0), (-4,4)} be bases for R², and let 2 A = [3 be the matrix for T: R² R² relative to B. (a) Find the transition matrix P from B' to B. (b) Use the matrices P and A to find [v] and [7(v)], where [v] = [-1 2]T. (c) Find P-¹ and A' (the matrix for T relative to B'). (d) Find [7(v)] two ways. 14. Repeat Exercise 13 for B' = {(1, -1), (0, 1)}, and [v] (Use matrix A in Exercise 13.) B = {(1, 1), (2,3)}, = [1 −3]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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10. Please solve only Question#14
![Finding a Matrix for a Linear Transformation In
Exercises 1-12, find the matrix A' for T relative to the
basis B'.
1. T: R² R², T(x, y) = (2x - y, y - x),
B' = {(1, 2), (0, 3)}
2. T: R² R², T(x, y) = (2x + y, x - 2y),
B' = {(1, 2), (0,4)}
3. T: R² R², T(x, y) = (x + y, 4y),
B' = {(-4, 1), (1, -1)}
4. T: R² R², T(x, y) = (x - 2y, 4x),
B' = {(2, 1), (-1, 1)}
5. T: R² R², T(x, y) = (-3x + y, 3x - y),
B' = {(1, 1), (1,5)}
6. T: R² R², T(x, y) = (5x + 4y, 4x + 5y),
B' = {(12, 13), (13, -12)}
7. T. R³
R³, T(x, y, z) = (x, y, z),
B' =
{(1, 1, 0), (1, 0, 1), (0, 1, 1)}
8. T: R³ R³, T(x, y, z) = (0, 0, 0),
B' = {(1, 1, 0), (1, 0, 1), (0, 1, 1)}
9. T. R³ R³, T(x, y, z) = (y + z, x + z, x + y),
B' = {(5, 0, 1), (-3, 2, 1), (4, -6,5)}
10. T: R³ R³, T(x, y, z) = (-x, x - y, y - z),
B' = {(0, 1, 2), (-2, 0, 3), (1, 3, 0)}
11. T: R³ R³,
T(x, y, z)=(x-y + 2z, 2x + y - z, x + 2y+z),
B' = {(1, 0, 1), (0, 2, 2), (1, 2, 0)}
12. T: R³ → R³,
T(x, y, z) = (x, x + 2y, x + y + 3z),
B' = {(1, 1, 0), (0, 0, 1), (0, 1, -1)}
13. Let B = {(1, 3), (-2,-2)} and B' = {(−12, 0), (-4,4)}
be bases for R², and let
A =
[3
be the matrix for T: R²R² relative to B.
(a) Find the transition matrix P from B' to B.
(b) Use the matrices P and A to find [v] and [7(v)],
where [v] = [1 2]¹.
(c) Find P-¹1 and A' (the matrix for T relative to B').
(d) Find [7(v)] two ways.
14. Repeat Exercise 13 for
B' = {(1, 1), (0, 1)}, and [v]
(Use matrix A in Exercise 13.)
B = {(1, 1), (2, 3)},
= [1-3].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabd5e741-0235-409f-80de-883f5f0e5da7%2Fc63fa7c4-2ccb-4ebc-b1dc-ecf4faa6db8e%2Fq2g3m0m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Finding a Matrix for a Linear Transformation In
Exercises 1-12, find the matrix A' for T relative to the
basis B'.
1. T: R² R², T(x, y) = (2x - y, y - x),
B' = {(1, 2), (0, 3)}
2. T: R² R², T(x, y) = (2x + y, x - 2y),
B' = {(1, 2), (0,4)}
3. T: R² R², T(x, y) = (x + y, 4y),
B' = {(-4, 1), (1, -1)}
4. T: R² R², T(x, y) = (x - 2y, 4x),
B' = {(2, 1), (-1, 1)}
5. T: R² R², T(x, y) = (-3x + y, 3x - y),
B' = {(1, 1), (1,5)}
6. T: R² R², T(x, y) = (5x + 4y, 4x + 5y),
B' = {(12, 13), (13, -12)}
7. T. R³
R³, T(x, y, z) = (x, y, z),
B' =
{(1, 1, 0), (1, 0, 1), (0, 1, 1)}
8. T: R³ R³, T(x, y, z) = (0, 0, 0),
B' = {(1, 1, 0), (1, 0, 1), (0, 1, 1)}
9. T. R³ R³, T(x, y, z) = (y + z, x + z, x + y),
B' = {(5, 0, 1), (-3, 2, 1), (4, -6,5)}
10. T: R³ R³, T(x, y, z) = (-x, x - y, y - z),
B' = {(0, 1, 2), (-2, 0, 3), (1, 3, 0)}
11. T: R³ R³,
T(x, y, z)=(x-y + 2z, 2x + y - z, x + 2y+z),
B' = {(1, 0, 1), (0, 2, 2), (1, 2, 0)}
12. T: R³ → R³,
T(x, y, z) = (x, x + 2y, x + y + 3z),
B' = {(1, 1, 0), (0, 0, 1), (0, 1, -1)}
13. Let B = {(1, 3), (-2,-2)} and B' = {(−12, 0), (-4,4)}
be bases for R², and let
A =
[3
be the matrix for T: R²R² relative to B.
(a) Find the transition matrix P from B' to B.
(b) Use the matrices P and A to find [v] and [7(v)],
where [v] = [1 2]¹.
(c) Find P-¹1 and A' (the matrix for T relative to B').
(d) Find [7(v)] two ways.
14. Repeat Exercise 13 for
B' = {(1, 1), (0, 1)}, and [v]
(Use matrix A in Exercise 13.)
B = {(1, 1), (2, 3)},
= [1-3].
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