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 $A$. 

Here's an intuitive explanation that perhaps you can make precise.
A matrix is a linear function on a vector space, and $A^2$ represents composing that function with itself. Now the column space of this matrix is essentially the image of the function, as it is the span of vectors you can get out. Now we apply $A$ once and get some linear subspace of the ambient vector space. Suppose we feed this into $A$ itself, and we get $0$. This means that everything that came out of $A$ the first time belongs to the null-space of $A$, as $A$ has taken them to $0$ on the second application.
