With T defined by T(x) = Ax, find a vector x whose image under T is b, and determine whethe x is unique. 1 0 -31 A = B³3 -3 1 31 3 2 -2 6,b -11
With T defined by T(x) = Ax, find a vector x whose image under T is b, and determine whethe x is unique. 1 0 -31 A = B³3 -3 1 31 3 2 -2 6,b -11
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
![### Linear Algebra Problem: Solving for a Vector
**Problem Statement:**
With \( T \) defined by \( T(x) = Ax \), find a vector \( x \) whose image under \( T \) is \( b \), and determine whether \( x \) is unique.
The given matrix \( A \) and vector \( b \) are as follows:
\[ A = \begin{bmatrix}
1 & 0 & -3 \\
-3 & 1 & 6 \\
2 & -2 & -1 \\
\end{bmatrix}, \quad b = \begin{bmatrix}
-2 \\
3 \\
-1 \\
\end{bmatrix}, \]
### Explanation of the Problem
We are given a linear transformation \( T \) represented by the matrix \( A \). The goal is to find a vector \( x \) such that when multiplied by \( A \), it results in the vector \( b \).
Formally, we need to solve the matrix equation:
\[ Ax = b \]
### Steps to Solve the Problem:
1. **Set up the matrix equation:**
\[ \begin{bmatrix}
1 & 0 & -3 \\
-3 & 1 & 6 \\
2 & -2 & -1 \\
\end{bmatrix} \begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
\end{bmatrix} = \begin{bmatrix}
-2 \\
3 \\
-1 \\
\end{bmatrix} \]
2. **Reduce to Row Echelon Form:**
- Applying Gaussian elimination or other row reduction techniques to the augmented matrix \([A|b]\) to find the solution vector \( x \).
3. **Determine Uniqueness:**
- Check if the reduced row echelon form has a pivot in every column of \( A \). If so, the solution \( x \) is unique; otherwise, there are infinitely many solutions or no solution.
### Diagram/Graph Explanation:
There are no diagrams or graphs directly involved in this specific problem. However, the matrix \( A \) and vector \( b \) together can be considered as a form of a coefficient matrix and a constant vector in a system of linear equations.
### Conclusion:
By transforming \( A \) into its reduced row echelon form and analyzing the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9baea4c0-9d9e-491a-8271-bc2a2bbd8b45%2Fbd455cdc-54ed-4ad8-927c-35d0573a1dd6%2Flcb62k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Problem: Solving for a Vector
**Problem Statement:**
With \( T \) defined by \( T(x) = Ax \), find a vector \( x \) whose image under \( T \) is \( b \), and determine whether \( x \) is unique.
The given matrix \( A \) and vector \( b \) are as follows:
\[ A = \begin{bmatrix}
1 & 0 & -3 \\
-3 & 1 & 6 \\
2 & -2 & -1 \\
\end{bmatrix}, \quad b = \begin{bmatrix}
-2 \\
3 \\
-1 \\
\end{bmatrix}, \]
### Explanation of the Problem
We are given a linear transformation \( T \) represented by the matrix \( A \). The goal is to find a vector \( x \) such that when multiplied by \( A \), it results in the vector \( b \).
Formally, we need to solve the matrix equation:
\[ Ax = b \]
### Steps to Solve the Problem:
1. **Set up the matrix equation:**
\[ \begin{bmatrix}
1 & 0 & -3 \\
-3 & 1 & 6 \\
2 & -2 & -1 \\
\end{bmatrix} \begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
\end{bmatrix} = \begin{bmatrix}
-2 \\
3 \\
-1 \\
\end{bmatrix} \]
2. **Reduce to Row Echelon Form:**
- Applying Gaussian elimination or other row reduction techniques to the augmented matrix \([A|b]\) to find the solution vector \( x \).
3. **Determine Uniqueness:**
- Check if the reduced row echelon form has a pivot in every column of \( A \). If so, the solution \( x \) is unique; otherwise, there are infinitely many solutions or no solution.
### Diagram/Graph Explanation:
There are no diagrams or graphs directly involved in this specific problem. However, the matrix \( A \) and vector \( b \) together can be considered as a form of a coefficient matrix and a constant vector in a system of linear equations.
### Conclusion:
By transforming \( A \) into its reduced row echelon form and analyzing the
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