Let T: R2→R2 be a linear transformation with standard matrix X2 A- [a, :] where a, and a, are the vectors shown in the figure. Using the figure, draw the image of -1 under the transformation T. Choose the correct graph below. O A. OB. Oc. OD. X2 X2 X2 X2 T(-1,3) T(-1,3) a2 a2 a2 a, a T(-1,3) T(-1,3)
Let T: R2→R2 be a linear transformation with standard matrix X2 A- [a, :] where a, and a, are the vectors shown in the figure. Using the figure, draw the image of -1 under the transformation T. Choose the correct graph below. O A. OB. Oc. OD. X2 X2 X2 X2 T(-1,3) T(-1,3) a2 a2 a2 a, a T(-1,3) T(-1,3)
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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![### Linear Transformation Representation
#### Concept
Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation with standard matrix
\[ A = \left[ \begin{array}{cc}
a_1 & a_2
\end{array} \right], \]
where \(\mathbf{a_1}\) and \(\mathbf{a_2}\) are the vectors shown in the figure.
Using the figure, draw the image of
\[ \left[ \begin{array}{c}
-1 \\
3
\end{array} \right] \]
under the transformation \(T\).
#### Initial Figure
The given figure shows two vectors \( \mathbf{a_1} \) and \( \mathbf{a_2} \) in a 2-dimensional coordinate space with axes labeled \(x_1\) and \(x_2\).
#### Task
Choose the correct graph below.
#### Options
**A.**
**B.**
**C.**
**D.**
In each option, the vectors \( \mathbf{a_1} \) and \( \mathbf{a_2} \) are shown, and the point \( \mathbf{T(-1,3)} \) is indicated at different positions relative to these vectors.
#### Graph Explanation:
The graphs show the potential correct positions for the image of the vector
\[ \left[ \begin{array}{c}
-1 \\
3
\end{array} \right] \]
under the transformation \(T\). Each option (A, B, C, D) presents a unique location for the transformed vector \( T(-1,3) \) in the coordinate space defined by the standard basis vectors \( \mathbf{a_1} \) and \( \mathbf{a_2} \).
To find the correct position, matrix transformation basics and vector addition principles need to be applied. Select the graph where the image of the vector
\[ \left[ \begin{array}{c}
-1 \\
3
\end{array} \right] \]
is accurately represented.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0867650-b445-489d-98f3-e27e64467d14%2F044694ba-42dd-4c59-8789-6757cfd55bfe%2Fn9xkdif_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Transformation Representation
#### Concept
Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation with standard matrix
\[ A = \left[ \begin{array}{cc}
a_1 & a_2
\end{array} \right], \]
where \(\mathbf{a_1}\) and \(\mathbf{a_2}\) are the vectors shown in the figure.
Using the figure, draw the image of
\[ \left[ \begin{array}{c}
-1 \\
3
\end{array} \right] \]
under the transformation \(T\).
#### Initial Figure
The given figure shows two vectors \( \mathbf{a_1} \) and \( \mathbf{a_2} \) in a 2-dimensional coordinate space with axes labeled \(x_1\) and \(x_2\).
#### Task
Choose the correct graph below.
#### Options
**A.**
**B.**
**C.**
**D.**
In each option, the vectors \( \mathbf{a_1} \) and \( \mathbf{a_2} \) are shown, and the point \( \mathbf{T(-1,3)} \) is indicated at different positions relative to these vectors.
#### Graph Explanation:
The graphs show the potential correct positions for the image of the vector
\[ \left[ \begin{array}{c}
-1 \\
3
\end{array} \right] \]
under the transformation \(T\). Each option (A, B, C, D) presents a unique location for the transformed vector \( T(-1,3) \) in the coordinate space defined by the standard basis vectors \( \mathbf{a_1} \) and \( \mathbf{a_2} \).
To find the correct position, matrix transformation basics and vector addition principles need to be applied. Select the graph where the image of the vector
\[ \left[ \begin{array}{c}
-1 \\
3
\end{array} \right] \]
is accurately represented.
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