Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Linear Transformation with Matrix Multiplication
**Objective:** Define a linear transformation \( T \) from \( \mathbb{R}^3 \) to \( \mathbb{R}^2 \) by \( T(\vec{x}) = A\vec{x} \). Find the images of vectors \( \vec{u} \) and \( \vec{v} \) under the transformation \( T \).
#### Given Data:
- Matrix \( A \):
\[ A = \begin{bmatrix}
-1 & -4 & 3 \\
-4 & -3 & 3
\end{bmatrix} \]
- Vector \( \vec{u} \):
\[ \vec{u} = \begin{bmatrix}
3 \\
1 \\
-3
\end{bmatrix} \]
- Vector \( \vec{v} \):
\[ \vec{v} = \begin{bmatrix}
a \\
b \\
c
\end{bmatrix} \]
### Steps to Find the Image of Vectors:
1. **Linear Transformation Definition:**
The transformation \( T \) is defined as \( T(\vec{x}) = A\vec{x} \).
2. **Image of \( \vec{u} \):**
To find the image of \( \vec{u} \) under \( T \):
Calculate \( A\vec{u} \),
\[
A\vec{u} = \begin{bmatrix}
-1 & -4 & 3 \\
-4 & -3 & 3
\end{bmatrix}
\begin{bmatrix}
3 \\
1 \\
-3
\end{bmatrix}
\]
3. **Image of \( \vec{v} \):**
To find the image of \( \vec{v} \) under \( T \):
Calculate \( A\vec{v} \),
\[
A\vec{v} = \begin{bmatrix}
-1 & -4 & 3 \\
-4 & -3 & 3
\end{bmatrix}
\begin{bmatrix}
a \\
b \\
c
\end{bmatrix}
\]
### Additional Notes](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad281aeb-79af-4eda-b9b4-346ba477171b%2F0fb4bb78-e982-46b1-87fc-1f767e2ae043%2Ficc6a4f_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Transformation with Matrix Multiplication
**Objective:** Define a linear transformation \( T \) from \( \mathbb{R}^3 \) to \( \mathbb{R}^2 \) by \( T(\vec{x}) = A\vec{x} \). Find the images of vectors \( \vec{u} \) and \( \vec{v} \) under the transformation \( T \).
#### Given Data:
- Matrix \( A \):
\[ A = \begin{bmatrix}
-1 & -4 & 3 \\
-4 & -3 & 3
\end{bmatrix} \]
- Vector \( \vec{u} \):
\[ \vec{u} = \begin{bmatrix}
3 \\
1 \\
-3
\end{bmatrix} \]
- Vector \( \vec{v} \):
\[ \vec{v} = \begin{bmatrix}
a \\
b \\
c
\end{bmatrix} \]
### Steps to Find the Image of Vectors:
1. **Linear Transformation Definition:**
The transformation \( T \) is defined as \( T(\vec{x}) = A\vec{x} \).
2. **Image of \( \vec{u} \):**
To find the image of \( \vec{u} \) under \( T \):
Calculate \( A\vec{u} \),
\[
A\vec{u} = \begin{bmatrix}
-1 & -4 & 3 \\
-4 & -3 & 3
\end{bmatrix}
\begin{bmatrix}
3 \\
1 \\
-3
\end{bmatrix}
\]
3. **Image of \( \vec{v} \):**
To find the image of \( \vec{v} \) under \( T \):
Calculate \( A\vec{v} \),
\[
A\vec{v} = \begin{bmatrix}
-1 & -4 & 3 \\
-4 & -3 & 3
\end{bmatrix}
\begin{bmatrix}
a \\
b \\
c
\end{bmatrix}
\]
### Additional Notes
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