<s>[INST] Basis for Matrix A 
Find a basis for all $2\times2$ matrices $A$ for which
  $A\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ = $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$.

Maybe I'm dumb-- but isn't $A$ just the $0$ matrix? In which case, the base is simply the $0$ matrix as well?
 [/INST] If you let $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ then your equation says,
$$ \begin{pmatrix} a + b & a + b \\ c+d & c+d \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$$
i.e $a = -b$ and $c = -d$ which means we can rewrite $A$ as,
$$\begin{pmatrix} a & -a \\ -d & d \end{pmatrix} = a \begin{pmatrix} 1 & -1 \\ 0 & 0 \end{pmatrix} + d \begin{pmatrix} 0 & 0 \\ -1 & 1 \end{pmatrix}$$
And so you are interested in the subspace $S = \textbf{span}\left\{\begin{pmatrix} 1 & -1 \\ 0 & 0 \end{pmatrix},\begin{pmatrix} 0 & 0 \\ -1 & 1 \end{pmatrix}\right\} $
</s>