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Here are the solutions for the given matrix \( A \):

### (a) Basis for the null space of \( A \):
The null space of the matrix has two basis vectors:
\[
\begin{pmatrix}
3 \\
4 \\
1 \\
0
\end{pmatrix}
\quad \text{and} \quad
\begin{pmatrix}
-1 \\
-1 \\
0 \\
1
\end{pmatrix}
\]

### (b) Rank and Nullity of \( A \):
- **Rank of \( A \)**: 2
- **Nullity of \( A \)**: 2 (calculated as the number of columns minus the rank)

Would you like more details or further explanation on any part of the solution?

Here are 5 follow-up questions that could expand your understanding:
1. How do we interpret the null space geometrically?
2. What is the significance of the rank-nullity theorem?
3. How would the solution change if the matrix was larger or smaller?
4. What is the relationship between the null space and the column space?
5. How can the rank of a matrix be used to determine the consistency of a system of linear equations?

**Tip:** The null space of a matrix represents the set of all solutions to the homogeneous equation \( A\mathbf{x} = 0 \). It tells us the freedom available in the solutions.

Consider the matrix A = [[2, -2, 2, 0], [2, 3, -18, 5], [2, -1, -2, 1]] (a) Determine a basis for the null space of A. (b) Determine the rank and nullity of A.

This problem involves determining the basis for the null space and calculating the rank and nullity of a given 3x4 matrix A. Using concepts from linear algebra, such as the null space, rank-nullity theorem, and matrix theory, the solution covers essential steps in solving for these properties.

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