I think that I have somewhat of an idea of what to do, but not fully.
So if
$$ N(A) =\text{span}\left(\left[
\begin{array}
11\\
0
\end{array}
\right]\right) $$
how would I go about finding a matrix that has this property.
Also, is it possible for the null space to be $\mathbb R^2$? If it is possible, what kind of matrix would that be?
Are you familiar with how to find the matrix representing a linear transformation in some basis? One way would be to manufacture a linear transformation which sends $(1,0)$ to $0$, and then find the associated matrix. In this case, $f(x,y) = (0,y)$ would work. The associated matrix would be $\begin{pmatrix}0&0\\0&1\end{pmatrix}$
The only matrix whose null space is all of $\mathbb{R}^2$ is the $0$ matrix.