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To find a basis for the null space of the matrix \( A \), we need to solve the equation:

\[
A \mathbf{x} = \mathbf{0}
\]

where \( A \) is the given matrix and \( \mathbf{x} \) is the vector of unknowns. Let's go step by step:

The matrix \( A \) is:

\[
A = \begin{bmatrix}
-4 & 4 & -1 & 2 \\
5 & -5 & 5 & -4 \\
19 & -19 & -14 & -14 \\
-7 & 7 & -8 & 2 \\
-2 & 2 & 12 & 4
\end{bmatrix}
\]

We'll find the null space by reducing the matrix to row echelon form (REF) or reduced row echelon form (RREF), then finding the solutions to the corresponding homogeneous system.

### Step 1: Perform row reduction (RREF) on matrix \( A \).
Once the matrix is in RREF, we can identify the free variables and express the solution to \( A \mathbf{x} = \mathbf{0} \).

### Step 2: Solve for the free variables.
This will give us the parametric vector form of the solution, which will then allow us to find a basis for the null space.

I'll perform these steps now.The basis for the null space of the matrix \( A \) is given by the vector:

\[
\begin{bmatrix}
-0.7071 \\
-0.7071 \\
0 \\
0
\end{bmatrix}
\]

This vector can be scaled by any constant, so the null space is spanned by:

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

Thus, the basis for the null space consists of the vector:

\[
\left\{
\begin{bmatrix}
1 \\
1 \\
0 \\
0
\end{bmatrix}
\right\}
\]

Would you like more details on any of the steps or further clarification?

Here are 5 related questions to expand on this topic:
1. What is the geometric interpretation of the null space in terms of the transformation defined by matrix \( A \)?
2. How do free and pivot variables relate to the null space basis?
3. Can a matrix have a null space dimension of zero? Under what conditions?
4. How does the rank of the matrix relate to the dimension of the null space?
5. What changes if you calculate the null space for a non-square matrix?

**Tip:** The dimension of the null space is also called the *nullity* of the matrix, and it is related to the rank-nullity theorem, which connects the rank and nullity to the total number of columns.

Find a basis for the null space of the matrix A = \begin{bmatrix}-4 & 4 & -1 & 2 \\ 5 & -5 & 5 & -4 \\ 19 & -19 & -14 & -14 \\ -7 & 7 & -8 & 2 \\ -2 & 2 & 12 & 4 \end{bmatrix}.

Learn how to find a basis for the null space of a 5x4 matrix A with step-by-step solutions. This problem covers key linear algebra concepts, including the null space, row reduction, and the Rank-Nullity Theorem. Suitable for undergraduate students.

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