Why does $A^2=0$ imply that the column space is a subset of the null space So we have a matrix n by n matrix $A$ such that $A^2 =0$.  This means that $A^2 = [Aa_1 \space Aa_2 \space \dots Aa_n] = 0$, so $Aa_1 = \dots = Aa_n = 0$.
But why does this imply that col(A) $\subset$ null(A)? The column space is the space spanned by linear combinations of the columns of $A$. I don't see how $Aa_1 \dots Aa_n$ are all possible linear combinations of the columns of $A$. 

You noted that $Aa_j=0$ for every column $a_j$ of $A$. That precisely means that each $a_j$ is in the nullspace of $A$. But the nullpsace being a subspace (hence closed under linear combinations), this implies that any linear combination of the $a_j$ is also in the nullspace, in other words the column space is contained in the nullspace.
