Nullspace that spans $\mathbb{R}^n$? My professor said that if for a $n \times n$ matrix $A$, $\text{null}(A) = \mathbb{R}^n$, then $A = 0_{n}$. Why is this true? I understand what its saying - if everything times this matrix is zero, then the matrix has to be zero.
The intuition is simple enough with numbers, but could someone explain why this is true with matrices? Thanks 

Show that if the matrix $A$ contains one non-zero entry, then you can find a vector $x$ such that $Ax \ne 0$.  The easiest vectors to consider are ones that are all zero except for one entry, which is one.
