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The problem asks to find all vectors \( \mathbf{x} \in \mathbb{R}^4 \) that are mapped to the zero vector by the transformation \( \mathbf{x} \to A\mathbf{x} \), where the matrix \( A \) is given as:

\[
A = \begin{bmatrix}
1 & 2 & -7 & 5 \\
0 & 1 & -4 & 0 \\
1 & 0 & 1 & 6 \\
2 & -1 & 6 & 8
\end{bmatrix}
\]

This problem is equivalent to solving the homogeneous system of linear equations \( A\mathbf{x} = \mathbf{0} \), or equivalently, finding the null space of the matrix \( A \).

### Step-by-Step Approach:
1. **Set up the system**: The equation \( A\mathbf{x} = \mathbf{0} \) becomes:
   \[
   \begin{aligned}
   1x_1 + 2x_2 - 7x_3 + 5x_4 &= 0, \\
   0x_1 + 1x_2 - 4x_3 + 0x_4 &= 0, \\
   1x_1 + 0x_2 + 1x_3 + 6x_4 &= 0, \\
   2x_1 - 1x_2 + 6x_3 + 8x_4 &= 0.
   \end{aligned}
   \]
2. **Form the augmented matrix**:
   \[
   \left[ A | \mathbf{0} \right] = \begin{bmatrix}
   1 & 2 & -7 & 5 & 0 \\
   0 & 1 & -4 & 0 & 0 \\
   1 & 0 & 1 & 6 & 0 \\
   2 & -1 & 6 & 8 & 0
   \end{bmatrix}
   \]

3. **Perform Gaussian elimination**: We reduce this augmented matrix to row echelon form (or reduced row echelon form if necessary) to find the solution for \( \mathbf{x} \).

Let me solve this system using Gaussian elimination to find the solutions.The matrix \( A \) has a rank of 3. Since \( A \) is a \( 4 \times 4 \) matrix, and the rank is 3, the nullity of the matrix is \( 4 - 3 = 1 \). This means the solution space is one-dimensional, implying that there is one free variable in the solution for \( A\mathbf{x} = 0 \).

Now, I will express the general solution by reducing the system to identify the dependent and independent variables.It seems like I can’t do more advanced data analysis right now. Please try again later.

You can, however, proceed with Gaussian elimination manually or use software tools like MATLAB, Python, or a calculator for the row-reduction process.

Would you like more help on how to continue solving this problem by hand, or do you have any other questions? Here are 5 related questions you could explore:

1. What is the difference between the rank and nullity of a matrix?
2. How do you interpret the solution to a homogeneous system of equations?
3. What does it mean for a matrix to have full rank?
4. How is the null space related to linear transformations?
5. Can you describe how free variables arise in systems of linear equations?

**Tip:** When solving homogeneous systems, always check the rank of the matrix first—it helps determine the number of free variables in the solution.

Find all vectors x in R^4 that are mapped to the zero vector by the transformation x → Ax, where matrix A is given as a 4x4 matrix.

Learn how to solve the system Ax = 0 by finding the null space of a 4x4 matrix. The problem involves solving a homogeneous system using Gaussian elimination, and understanding the rank-nullity theorem. Suitable for college-level linear algebra students.

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