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 $A$.
Best Answer
An element of the column space is a column vector of the form $Av$. Since $A(Av)=A^2v=0v=0$, we have $Av\in\operatorname{null}(A)$.