Let T:R² → R² be the linear transformation: T(x,y) = (7x+3y, 5x+2y), determine the inverse transformation. O a. T¹(x,y) = (-2x+3y, 5x-7y) Ob.T-¹(x,y)=(-2x+5y, 3x-7y) OCT ¹(x,y)=(7x+5y, 3x+2y) Od. T ¹(x,y) = (-7x-3y, -5x-2y) O e. none of these Let 7:R² → R² be the linear transformation T(x,y) = (5x+3y, 3x+y). Let B={(2,3), (-1,2) } be a basis for R². Determine the matrix of the linear transformation with respect to the given basis. O a. 53 31 A = O b. A = OC. A = O d. 5 91 10 4-[19] 01 e. none of these
Let T:R² → R² be the linear transformation: T(x,y) = (7x+3y, 5x+2y), determine the inverse transformation. O a. T¹(x,y) = (-2x+3y, 5x-7y) Ob.T-¹(x,y)=(-2x+5y, 3x-7y) OCT ¹(x,y)=(7x+5y, 3x+2y) Od. T ¹(x,y) = (-7x-3y, -5x-2y) O e. none of these Let 7:R² → R² be the linear transformation T(x,y) = (5x+3y, 3x+y). Let B={(2,3), (-1,2) } be a basis for R². Determine the matrix of the linear transformation with respect to the given basis. O a. 53 31 A = O b. A = OC. A = O d. 5 91 10 4-[19] 01 e. none of these
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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Question
![**Problem 1: Inverse of a Linear Transformation**
Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be the linear transformation:
\[ T(x,y) = (7x + 3y, \, 5x + 2y), \]
determine the inverse transformation.
Options:
- a. \( T^{-1}(x,y) = (-2x + 3y, \, 5x - 7y) \)
- b. \( T^{-1}(x,y) = (-2x + 5y, \, 3x - 7y) \)
- c. \( T^{-1}(x,y) = (7x + 5y, \, 3x + 2y) \)
- d. \( T^{-1}(x,y) = (-7x - 3y, \, -5x - 2y) \)
- e. none of these
---
**Problem 2: Matrix of a Linear Transformation with Respect to a Given Basis**
Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be the linear transformation:
\[ T(x,y) = (5x + 3y, \, 3x + y). \]
Let \( \mathcal{B} = \{(2, -3), \, (-1, 2)\} \) be a basis for \( \mathbb{R}^2 \).
Determine the matrix of the linear transformation with respect to the given basis.
Options:
- a.
\[
A = \begin{bmatrix}
5 & 3 \\
3 & 1
\end{bmatrix}
\]
- b.
\[
A = \begin{bmatrix}
1 & 1 \\
3 & -1
\end{bmatrix}
\]
- c.
\[
A = \begin{bmatrix}
9 & 1 \\
5 & 1
\end{bmatrix}
\]
- d.
\[
A = \begin{bmatrix}
1 & 0 \\
0 & 1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b8a4ad9-b66e-4dbe-8b3f-39bf0adb3cae%2F1a442224-ca1e-463c-a577-302a34ff6cc9%2Fykbgibc_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 1: Inverse of a Linear Transformation**
Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be the linear transformation:
\[ T(x,y) = (7x + 3y, \, 5x + 2y), \]
determine the inverse transformation.
Options:
- a. \( T^{-1}(x,y) = (-2x + 3y, \, 5x - 7y) \)
- b. \( T^{-1}(x,y) = (-2x + 5y, \, 3x - 7y) \)
- c. \( T^{-1}(x,y) = (7x + 5y, \, 3x + 2y) \)
- d. \( T^{-1}(x,y) = (-7x - 3y, \, -5x - 2y) \)
- e. none of these
---
**Problem 2: Matrix of a Linear Transformation with Respect to a Given Basis**
Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be the linear transformation:
\[ T(x,y) = (5x + 3y, \, 3x + y). \]
Let \( \mathcal{B} = \{(2, -3), \, (-1, 2)\} \) be a basis for \( \mathbb{R}^2 \).
Determine the matrix of the linear transformation with respect to the given basis.
Options:
- a.
\[
A = \begin{bmatrix}
5 & 3 \\
3 & 1
\end{bmatrix}
\]
- b.
\[
A = \begin{bmatrix}
1 & 1 \\
3 & -1
\end{bmatrix}
\]
- c.
\[
A = \begin{bmatrix}
9 & 1 \\
5 & 1
\end{bmatrix}
\]
- d.
\[
A = \begin{bmatrix}
1 & 0 \\
0 & 1
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