6 4 A = 5 -5 4 -3 1 Find the images of u = = -7 41 -2 and 7 = a [B] с b under T.
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
![## Linear Transformation Application
### Problem Statement
Define the linear transformation \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) by \( T(\vec{x}) = A\vec{x} \).
Given the matrix \( A \):
\[ A = \begin{bmatrix}
6 & 4 & -7 \\
5 & -5 & -7 \\
4 & -3 & 1
\end{bmatrix} \]
### Task
Find the images of vectors \( \vec{u} \) and \( \vec{v} \) under the transformation \( T \).
#### Given Vectors
\[ \vec{u} = \begin{bmatrix}
-3 \\
-2 \\
-1
\end{bmatrix} \]
\[ \vec{v} = \begin{bmatrix}
a \\
b \\
c
\end{bmatrix} \]
### Required
Calculate \( T(\vec{u}) \) and \( T(\vec{v}) \).
### Solution Space
\[ T(\vec{u}) = \begin{bmatrix}
\boxed{} \\
\boxed{} \\
\boxed{}
\end{bmatrix} \]
\[ T(\vec{v}) = \begin{bmatrix}
\boxed{} \\
\boxed{} \\
\boxed{}
\end{bmatrix} \]
**Explanation:**
- To find \( T(\vec{u}) \), multiply the matrix \( A \) by the vector \( \vec{u} \).
- To find \( T(\vec{v}) \), multiply the matrix \( A \) by the vector \( \vec{v} \).
Insert the results into the respective boxes.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad281aeb-79af-4eda-b9b4-346ba477171b%2F8152b1a7-43ad-4be1-9d1e-41b7e76b1371%2Fl5vwbxc_processed.png&w=3840&q=75)
Transcribed Image Text:## Linear Transformation Application
### Problem Statement
Define the linear transformation \( T : \mathbb{R}^3 \to \mathbb{R}^3 \) by \( T(\vec{x}) = A\vec{x} \).
Given the matrix \( A \):
\[ A = \begin{bmatrix}
6 & 4 & -7 \\
5 & -5 & -7 \\
4 & -3 & 1
\end{bmatrix} \]
### Task
Find the images of vectors \( \vec{u} \) and \( \vec{v} \) under the transformation \( T \).
#### Given Vectors
\[ \vec{u} = \begin{bmatrix}
-3 \\
-2 \\
-1
\end{bmatrix} \]
\[ \vec{v} = \begin{bmatrix}
a \\
b \\
c
\end{bmatrix} \]
### Required
Calculate \( T(\vec{u}) \) and \( T(\vec{v}) \).
### Solution Space
\[ T(\vec{u}) = \begin{bmatrix}
\boxed{} \\
\boxed{} \\
\boxed{}
\end{bmatrix} \]
\[ T(\vec{v}) = \begin{bmatrix}
\boxed{} \\
\boxed{} \\
\boxed{}
\end{bmatrix} \]
**Explanation:**
- To find \( T(\vec{u}) \), multiply the matrix \( A \) by the vector \( \vec{u} \).
- To find \( T(\vec{v}) \), multiply the matrix \( A \) by the vector \( \vec{v} \).
Insert the results into the respective boxes.
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