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