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 ProblemWe 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

Answered: With T defined by T(x) = Ax, find a…

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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Question

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

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