Find a basis for the null space of the matrix.
\[
A=\left[\begin{array}{rrrrr}
1 & 0 & -3 & 0 & -3 \\
0 & 1 & 4 & 0 & 5 \\
0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 0
\end{array}\right]
\]

Step 1 ：We are given the matrix A = \(\left[\begin{array}{rrrrr} 1 & 0 & -3 & 0 & -3 \\ 0 & 1 & 4 & 0 & 5 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\).

Step 2 ：The null space of a matrix A is the set of all vectors x such that Ax = 0. We need to find a basis for the null space, which involves solving the homogeneous system of linear equations Ax = 0.

Step 3 ：The solutions to this system will form a basis for the null space.

Step 4 ：After solving the system, we find that the basis for the null space of the matrix A is given by the vectors \(\left[\begin{array}{c} 3 \\ -4 \\ 1 \\ 0 \\ 0 \end{array}\right]\) and \(\left[\begin{array}{c} 3 \\ -5 \\ 0 \\ -1 \\ 1 \end{array}\right]\).

Step 5 ：\(\boxed{\text{The basis for the null space of the matrix A is given by the vectors } \left[\begin{array}{c} 3 \\ -4 \\ 1 \\ 0 \\ 0 \end{array}\right] \text{ and } \left[\begin{array}{c} 3 \\ -5 \\ 0 \\ -1 \\ 1 \end{array}\right]}\)