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The given problem asks to describe all solutions of \( A\vec{x} = \vec{0} \), where matrix \( A \) and vector \( \vec{x} \) are defined.

### Matrix \( A \):

\[
A = \begin{bmatrix}
1 & 4 & 3 & 1 & -1 & -1 \\
0 & 0 & 1 & -1 & 0 & -3 \\
0 & 0 & 0 & 0 & 1 & 2 \\
0 & 0 & 0 & 0 & 0 & 0
\end{bmatrix}
\]

This is a linear system where we solve for \( \vec{x} \in \mathbb{R}^6 \) such that \( A \vec{x} = \vec{0} \). The goal is to find the null space of the matrix \( A \), i.e., the set of all vectors \( \vec{x} \) that satisfy the equation.

### Approach:

1. **Matrix structure and row reduction**:
   We observe that the matrix \( A \) already contains a lot of zeros, so it can be simplified. Specifically:
   - The first row gives the equation:
     \[
     x_1 + 4x_2 + 3x_3 + x_4 - x_5 - x_6 = 0
     \]
   - The second row gives the equation:
     \[
     x_3 - x_4 - 3x_6 = 0 \quad \Rightarrow \quad x_3 = x_4 + 3x_6
     \]
   - The third row gives the equation:
     \[
     x_5 + 2x_6 = 0 \quad \Rightarrow \quad x_5 = -2x_6
     \]

2. **Express free variables**:
   We will use the free variables \( x_2, x_4, x_6 \) to express the solution. 
   
   Substituting \( x_3 \) and \( x_5 \) from the above equations into the first equation:
   \[
   x_1 + 4x_2 + 3(x_4 + 3x_6) + x_4 - (-2x_6) - x_6 = 0
   \]
   Simplify:
   \[
   x_1 + 4x_2 + 3x_4 + 9x_6 + x_4 + 2x_6 - x_6 = 0 \quad \Rightarrow \quad x_1 + 4x_2 + 4x_4 + 10x_6 = 0
   \]
   Therefore:
   \[
   x_1 = -4x_2 - 4x_4 - 10x_6
   \]

### Solution in parametric form:

The solution to \( A\vec{x} = \vec{0} \) can now be expressed in terms of the free variables \( x_2, x_4, x_6 \):

\[
\vec{x} = x_2 \begin{bmatrix} -4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}
+ x_4 \begin{bmatrix} -4 \\ 0 \\ 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}
+ x_6 \begin{bmatrix} -10 \\ 0 \\ 3 \\ 0 \\ -2 \\ 1 \end{bmatrix}
\]

Thus, the general solution to the system is:

\[
\vec{x} = x_2 \begin{bmatrix} -4 \\ 1 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}
+ x_4 \begin{bmatrix} -4 \\ 0 \\ 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}
+ x_6 \begin{bmatrix} -10 \\ 0 \\ 3 \\ 0 \\ -2 \\ 1 \end{bmatrix}
\]

This is the parametric description of all solutions.

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Would you like any further details or clarifications on the solution?

Here are some related questions:
1. How is the rank of a matrix related to the number of free variables?
2. What is the significance of the null space in linear algebra?
3. How do we interpret the free variables in a system of linear equations?
4. Can the solution to \( A \vec{x} = \vec{0} \) ever be unique? Under what conditions?
5. How would the solution change if \( A \) had full rank?

**Tip:** Always double-check the consistency of dependent and independent variables when interpreting the null space solution.

Describe all solutions of A\vec{x} = \vec{0}, where A is the matrix given in the image.

This solution describes the null space of a 4x6 matrix A, providing the parametric form for the vector solution to A\vec{x} = \vec{0}. The problem explores linear algebra concepts like systems of linear equations, row reduction, and null space solutions.

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