11 [1] [3] [4] (²22) into 12 and maps v = into Let T: be a linear transformation that maps u = Use the fact that T is linear to find the image of 3u + v.
11 [1] [3] [4] (²22) into 12 and maps v = into Let T: be a linear transformation that maps u = Use the fact that T is linear to find the image of 3u + v.
Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter12: Angle Relationships And Transformations
Section12.6: Rotations And Symmetry
Problem 1E
Related questions
Question
![### Linear Transformation Problem
**Problem Statement:**
Solve the problem.
Let \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a linear transformation that maps \( \mathbf{u} =
\begin{bmatrix}
-6 \\
4
\end{bmatrix}
\) into
\begin{bmatrix}
-22 \\
12
\end{bmatrix}
\) and maps \( \mathbf{v} =
\begin{bmatrix}
2 \\
-5
\end{bmatrix}
\) into
\begin{bmatrix}
11 \\
-4
\end{bmatrix}
\).
Use the fact that \( T \) is linear to find the image of \( 3\mathbf{u} + \mathbf{v} \).
**Options:**
1.
\[
\begin{bmatrix}
-16 \\
7
\end{bmatrix}
\]
2.
\[
\begin{bmatrix}
-11 \\
8
\end{bmatrix}
\]
3.
\[
\begin{bmatrix}
-33 \\
24
\end{bmatrix}
\]
4.
\[
\begin{bmatrix}
-55 \\
32
\end{bmatrix}
\]
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Please find the image of \( 3\mathbf{u} + \mathbf{v} \) using the given linear transformation information and choose the correct option from the list provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F157e66f5-e794-4af2-988e-1885b818a80a%2F33ddb97b-6509-4d9f-8d92-6f293032a82f%2F4sgou6g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Transformation Problem
**Problem Statement:**
Solve the problem.
Let \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a linear transformation that maps \( \mathbf{u} =
\begin{bmatrix}
-6 \\
4
\end{bmatrix}
\) into
\begin{bmatrix}
-22 \\
12
\end{bmatrix}
\) and maps \( \mathbf{v} =
\begin{bmatrix}
2 \\
-5
\end{bmatrix}
\) into
\begin{bmatrix}
11 \\
-4
\end{bmatrix}
\).
Use the fact that \( T \) is linear to find the image of \( 3\mathbf{u} + \mathbf{v} \).
**Options:**
1.
\[
\begin{bmatrix}
-16 \\
7
\end{bmatrix}
\]
2.
\[
\begin{bmatrix}
-11 \\
8
\end{bmatrix}
\]
3.
\[
\begin{bmatrix}
-33 \\
24
\end{bmatrix}
\]
4.
\[
\begin{bmatrix}
-55 \\
32
\end{bmatrix}
\]
**Navigation:**
- **Previous** button to navigate to the previous question.
- **Next** button to navigate to the next question.
- **Submit Quiz** button to submit the answers.
**Status Bar:**
- Indicates "No new data to save. Last checked at 2:48am."
Please find the image of \( 3\mathbf{u} + \mathbf{v} \) using the given linear transformation information and choose the correct option from the list provided.
![### Linear Algebra: Finding the Preimage of a Vector Under a Transformation
Given:
\[
A = \begin{bmatrix} 1 & -3 & 2 \\ -3 & 4 & -1 \\ 2 & -5 & 3 \end{bmatrix}
\]
\[
b = \begin{bmatrix} 2 \\ 4 \\ -4 \end{bmatrix}
\]
We define a transformation \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) by \( T(x) = Ax \).
**Problem Statement:**
If possible, find a vector \( x \) whose image under \( T \) is \( b \). Otherwise, state that \( b \) is not in the range of the transformation \( T \).
**Options:**
- \[
\begin{bmatrix} 4 \\ 0 \\ -4 \end{bmatrix}
\]
- \[
\begin{bmatrix} 4 \\ 2 \\ 2 \end{bmatrix}
\]
- \[
\begin{bmatrix} 4 \\ 4 \\ 0 \end{bmatrix}
\]
- \[
b \text{ is not in the range of the transformation } T.
\]
**Explanation:**
1. Perform the matrix-vector multiplication \( Ax \) for each candidate vector \( x \) to determine if it equals \( b \).
2. If \( Ax = b \) for any candidate vector \( x \), then that vector \( x \) is the solution.
3. If no candidate vector \( x \) satisfies \( Ax = b \), then \( b \) is not in the range of \( T \).
**Quiz Saved Information:**
The quiz was saved at 2:49am.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F157e66f5-e794-4af2-988e-1885b818a80a%2F33ddb97b-6509-4d9f-8d92-6f293032a82f%2F6gxdzg8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra: Finding the Preimage of a Vector Under a Transformation
Given:
\[
A = \begin{bmatrix} 1 & -3 & 2 \\ -3 & 4 & -1 \\ 2 & -5 & 3 \end{bmatrix}
\]
\[
b = \begin{bmatrix} 2 \\ 4 \\ -4 \end{bmatrix}
\]
We define a transformation \( T: \mathbb{R}^3 \rightarrow \mathbb{R}^3 \) by \( T(x) = Ax \).
**Problem Statement:**
If possible, find a vector \( x \) whose image under \( T \) is \( b \). Otherwise, state that \( b \) is not in the range of the transformation \( T \).
**Options:**
- \[
\begin{bmatrix} 4 \\ 0 \\ -4 \end{bmatrix}
\]
- \[
\begin{bmatrix} 4 \\ 2 \\ 2 \end{bmatrix}
\]
- \[
\begin{bmatrix} 4 \\ 4 \\ 0 \end{bmatrix}
\]
- \[
b \text{ is not in the range of the transformation } T.
\]
**Explanation:**
1. Perform the matrix-vector multiplication \( Ax \) for each candidate vector \( x \) to determine if it equals \( b \).
2. If \( Ax = b \) for any candidate vector \( x \), then that vector \( x \) is the solution.
3. If no candidate vector \( x \) satisfies \( Ax = b \), then \( b \) is not in the range of \( T \).
**Quiz Saved Information:**
The quiz was saved at 2:49am.
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