###### Exercise1

Suppose that $$A$$ and its reduced row echelon form are

\begin{equation*} A = \left[\begin{array}{rrrrrr} 0 \amp 2 \amp 0 \amp -4 \amp 0 \amp 6 \\ 0 \amp -4 \amp -1 \amp 7 \amp 0 \amp -16 \\ 0 \amp 6 \amp 0 \amp -12 \amp 3 \amp 15 \\ 0 \amp 4 \amp -1 \amp -9 \amp 0 \amp 8 \\ \end{array}\right] \sim \left[\begin{array}{rrrrrr} 0 \amp 1 \amp 0 \amp -2 \amp 0 \amp 3 \\ 0 \amp 0 \amp 1 \amp 1 \amp 0 \amp 4 \\ 0 \amp 0 \amp 0 \amp 0 \amp 1 \amp -1 \\ 0 \amp 0 \amp 0 \amp 0 \amp 0 \amp 0 \\ \end{array}\right]\text{.} \end{equation*}
1. The null space $$\nul(A)$$ is a subspace of $$\real^p$$ for what $$p\text{?}$$ The column space $$\col(A)$$ is a subspace of $$\real^p$$ for what $$p\text{?}$$

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

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

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

in-context