Let T: R2→R2 be a linear transformation with standard matrix a, where a, and a, are the vectors shown in the figure. Using the figure, draw the image of under the 3 transformation T. Choose the correct graph below. O A. OB. OC. OD. X2 X2 X2 T(-1,3) T(-1,3) a2 a2 a1 a, a a1 T(-1,3) T(-1,3)
Let T: R2→R2 be a linear transformation with standard matrix a, where a, and a, are the vectors shown in the figure. Using the figure, draw the image of under the 3 transformation T. Choose the correct graph below. O A. OB. OC. OD. X2 X2 X2 T(-1,3) T(-1,3) a2 a2 a1 a, a a1 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
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![# Linear Transformation and Matrix Multiplication
## Understanding Transformations using a Standard Matrix
Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation with a standard matrix
\[ A = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 \end{bmatrix} \]
where \(\mathbf{a}_1\) and \(\mathbf{a}_2\) are the vectors shown in the figure.
### Step-by-Step Process to Draw the Image of a Vector under Transformation T
**Given:**
\[
\mathbf{v} = \begin{bmatrix} -1 \\ 3 \end{bmatrix}
\]
**Find:** The image of \(\mathbf{v}\) under transformation \(T\) using the matrix \(A\).
**Using the figure, the vectors are:**
\[ \mathbf{a}_1 \]
\[ \mathbf{a}_2 \]
**Transformation involves matrix multiplication:**
\[ T(\mathbf{v}) = A \mathbf{v} = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 \end{bmatrix} \begin{bmatrix} -1 \\ 3 \end{bmatrix} \]
### Explanation of Graphs
Given four choices below, determine which graph correctly represents the image of the vector under the transformation \(T\):
- **Graph A:**
- Coordinates \(\mathbf{a}_1\), \(\mathbf{a}_2\)
- Shows transformation \(T\left( \begin{bmatrix} -1 \\ 3 \end{bmatrix} \right)\)
- **Graph B:**
- Coordinates \(\mathbf{a}_1\), \(\mathbf{a}_2\)
- Shows transformation \(T\left( \begin{bmatrix} -1 \\ 3 \end{bmatrix} \right)\)
- **Graph C:**
- Coordinates \(\mathbf{a}_1\), \(\mathbf{a}_2\)
- Shows transformation \(T\left( \begin{bmatrix} -1 \\ 3 \end{bmatrix} \right)\)
- **Graph D:**
- Coordinates \(\mathbf{a}_1\), \(\mathbf{a}_](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0867650-b445-489d-98f3-e27e64467d14%2Ff8ab3afe-bddc-47f4-9f99-f67d3420f237%2F1i7u48_processed.png&w=3840&q=75)
Transcribed Image Text:# Linear Transformation and Matrix Multiplication
## Understanding Transformations using a Standard Matrix
Let \( T: \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation with a standard matrix
\[ A = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 \end{bmatrix} \]
where \(\mathbf{a}_1\) and \(\mathbf{a}_2\) are the vectors shown in the figure.
### Step-by-Step Process to Draw the Image of a Vector under Transformation T
**Given:**
\[
\mathbf{v} = \begin{bmatrix} -1 \\ 3 \end{bmatrix}
\]
**Find:** The image of \(\mathbf{v}\) under transformation \(T\) using the matrix \(A\).
**Using the figure, the vectors are:**
\[ \mathbf{a}_1 \]
\[ \mathbf{a}_2 \]
**Transformation involves matrix multiplication:**
\[ T(\mathbf{v}) = A \mathbf{v} = \begin{bmatrix} \mathbf{a}_1 & \mathbf{a}_2 \end{bmatrix} \begin{bmatrix} -1 \\ 3 \end{bmatrix} \]
### Explanation of Graphs
Given four choices below, determine which graph correctly represents the image of the vector under the transformation \(T\):
- **Graph A:**
- Coordinates \(\mathbf{a}_1\), \(\mathbf{a}_2\)
- Shows transformation \(T\left( \begin{bmatrix} -1 \\ 3 \end{bmatrix} \right)\)
- **Graph B:**
- Coordinates \(\mathbf{a}_1\), \(\mathbf{a}_2\)
- Shows transformation \(T\left( \begin{bmatrix} -1 \\ 3 \end{bmatrix} \right)\)
- **Graph C:**
- Coordinates \(\mathbf{a}_1\), \(\mathbf{a}_2\)
- Shows transformation \(T\left( \begin{bmatrix} -1 \\ 3 \end{bmatrix} \right)\)
- **Graph D:**
- Coordinates \(\mathbf{a}_1\), \(\mathbf{a}_
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