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To solve this, we need to determine the solution set to the given system of linear equations, represented by the matrix equation:

\[
\begin{pmatrix}
1 & 0 & -1 \\
0 & 1 & 2 \\
0 & 0 & 0
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix} =
\begin{pmatrix}
0 \\
0 \\
0
\end{pmatrix}
\]

This is a homogeneous system, meaning we are solving for the null space (or kernel) of the matrix \(A\), which consists of all vectors \( \mathbf{x} \) such that \( A\mathbf{x} = 0 \).

### Step 1: Set up the system of equations
From the matrix multiplication, we get the following system of equations:

1. \( x_1 - x_3 = 0 \)
2. \( x_2 + 2x_3 = 0 \)
3. \( 0 = 0 \) (this is redundant)

### Step 2: Solve the system
From the first equation:
\[
x_1 = x_3
\]

From the second equation:
\[
x_2 = -2x_3
\]

So the general solution for this system is:
\[
\begin{pmatrix}
x_1 \\
x_2 \\
x_3
\end{pmatrix}
=
\begin{pmatrix}
x_3 \\
-2x_3 \\
x_3
\end{pmatrix}
=
x_3
\begin{pmatrix}
1 \\
-2 \\
1
\end{pmatrix}
\]
where \( x_3 \) is a free parameter. Let \( t = x_3 \). Then the solution can be written as:
\[
\mathbf{x} = t \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}, \quad t \in \mathbb{R}
\]

### Step 3: Check the given options
The correct form for the solution is \( t \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} \), where \( t \in \mathbb{R} \). By inspecting the options provided:

- The correct option is **F**, which corresponds to the solution \( \mathbf{x} = t \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix} \).

### Final Answer: F

Let me know if you would like further clarification on any part of the solution.

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Here are 5 related questions to deepen your understanding:

1. How do you determine the null space of a matrix in general?
2. What is the significance of a homogeneous system of equations?
3. What is the geometric interpretation of the solution space in this example?
4. How can you tell if a solution set is unique or has infinitely many solutions?
5. How does the rank of a matrix relate to the number of free variables in its solution?

**Tip:** The null space of a matrix corresponds to the set of vectors that, when multiplied by the matrix, yield the zero vector. For square matrices, this is often linked to the concept of linear dependence among the columns of the matrix.

What is the correct solution set to the system of linear equations represented by the given matrix?

Find the correct solution set for the given homogeneous system of linear equations. Learn how to compute the null space of a matrix and understand the fundamental concepts of linear algebra.

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