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The given problem asks to solve the homogeneous linear system corresponding to the matrix:

\[
\begin{bmatrix}
1 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}
\]

The solution is requested in terms of parameters \( t \) and \( s \).

### Step-by-step Solution:
This is a homogeneous system, meaning the system is of the form \( A \mathbf{x} = 0 \), where \( A \) is the given coefficient matrix and \( \mathbf{x} = (x_1, x_2, x_3, x_4) \).

To solve this:

1. **Convert the system to equations**:
   - From row 1: \( x_1 + x_4 = 0 \) \( \Rightarrow x_1 = -x_4 \)
   - From row 2: \( x_2 + x_4 = 0 \) \( \Rightarrow x_2 = -x_4 \)
   - From row 3: \( x_3 = 0 \)
   - Row 4 does not give any new information.

2. **Parameterize the free variables**:
   - Let \( x_4 = t \), where \( t \) is a free parameter.

3. **Write the solution in terms of \( t \)**:
   - From row 1 and row 2: \( x_1 = -t \) and \( x_2 = -t \).
   - From row 3: \( x_3 = 0 \).
   - \( x_4 = t \).

Thus, the solution vector can be written as:

\[
(x_1, x_2, x_3, x_4) = (-t, -t, 0, t)
\]

This is the general solution for the system in terms of \( t \).

Would you like more details on any specific part of the solution?

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Here are five related questions to deepen your understanding:
1. How would the solution change if a non-homogeneous term were added to the system?
2. What happens if there are multiple free variables in the system?
3. How do we interpret a row of all zeros in a matrix like this one?
4. How does Gaussian elimination work for solving systems like this?
5. How can you check if a system has infinitely many solutions?

**Tip:** When solving homogeneous systems, always identify free variables to express the general solution in terms of parameters!

Solve the homogeneous linear system corresponding to the given coefficient matrix for x₁, x₂, x₃, and x₄ in terms of the parameters t and s.

Learn how to solve a homogeneous linear system of equations with four variables. The solution is expressed in terms of the parameters t and s, covering key concepts in linear algebra such as Gaussian elimination and free variables.

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