https://file.aiquickdraw.com/mathbot/file/172653548505992675toz.jpg

The image you uploaded contains a matrix \( A \) and a description of a set \( S \), which represents the column space of \( A \). The vector \( v \) belongs to this set if there exists a solution to the matrix equation:

\[
A \mathbf{x} = \mathbf{v}.
\]

The matrix equation is expanded as an augmented matrix:

\[
\begin{pmatrix}
2 & -1 & 4 & | & x \\
3 & -4 & 1 & | & y \\
-9 & 7 & -13 & | & z
\end{pmatrix}.
\]

The question asks to find the conditions under which this system has a solution and to check whether it has:
1. A unique solution.
2. Infinitely many solutions.
3. No solution.

It seems the row reduction of the augmented matrix leads to a condition where the solution set is dependent on satisfying a linear condition of the form \( ax + by + cz = 0 \). 

Let’s clarify and solve this problem.

### Steps:
1. **Perform row reduction on the augmented matrix** to analyze the number of solutions.
2. **Determine the linear condition** under which the system has solutions.
3. **Interpret the solution space** (unique, infinite, or no solution).

Would you like me to walk through the row reduction step-by-step?

For the matrix A = \begin{pmatrix} 2 & -1 & 4 \\ 3 & -4 & 1 \\ -9 & 7 & -13 \end{pmatrix}, determine if the system represented by the augmented matrix has a unique solution, infinitely many solutions, or no solution. The vector \mathbf{v} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} belongs to the set S = \{b \in \mathbb{R}^3 : b = Ax \text{ for some } x \in \mathbb{R}^3\}. Perform row reduction to find the linear condition that determines the solution.

Explore how to determine the solution to a system of linear equations represented by the matrix A. This involves performing row reduction on the augmented matrix to understand whether the system has a unique solution, infinitely many solutions, or no solution. Suitable for advanced high school students (Grades 10-12).

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