Given Column Space and Null Space construct a matrix. Construct a matrix whose column space contains (1 1 1) and (0 1 1) and whose null space contains (1 0 1) and (0 1 0), or explain why none can exist.
So I am not sure if I did this right, but after applying the null space to A I got that 
$A=\begin{bmatrix}
a& 0 &-a\\ 
b& 0 &-b\\[0ex]
c& 0 &-c
\end{bmatrix}.$
But now I am not sure how to apply the column space to find a matrix. I couldn't find any examples of how to do this so I am just kind of guessing. 

Before beginning, I just consider the case where the matrix we want to construct is a $3\times 3$ matrix (since you tried to construct such a matrix). I claim that we cannot construct such a matrix.
Indeed, since the $2$ vectors which belong to the column space are linearly independent, we have that $$2\le \text{ rank } A.\tag 1$$
Also, since the $2$ vectors which belong to the nullspace of $A$ are linearly independent, we have that:
$$2\le \text{ null } A\tag 2.$$
Adding $(1),(2)$ yields:
$$4\le  \text{ rank } (A) + \text{ null }(A).\tag 3$$
By rank-nullity theorem we have: $$\text{rank } A + \text{ null } A = 3.$$
Thus, $(3)$ cannot hold!
