Exercise 1

Suppose that $A$ and its reduced row echelon form are $$A = \left[\begin{array}{rrrrrr} 0 & 2 & 0 & -4 & 0 & 6 \\ 0 & -4 & -1 & 7 & 0 & -16 \\ 0 & 6 & 0 & -12 & 3 & 15 \\ 0 & 4 & -1 & -9 & 0 & 8 \\ \end{array}\right] \sim \left[\begin{array}{rrrrrr} 0 & 1 & 0 & -2 & 0 & 3 \\ 0 & 0 & 1 & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array}\right]$$.

1. The null space $\nul(A)$ is a subspace of $\real^p$ for what $p$? The column space $\col(A)$ is a subspace of $\real^p$ for what $p$?

2. What are the dimensions $\dim~\nul(A)$ and $\dim~\col(A)$?

3. Find a basis for the column space $\col(A)$.

4. Find a basis for the null space $\nul(A)$.