Left inverse and nullspace I am doing some independent studying, and came across one note from a professor, which states that if $A$ is an $m \times n$ matrix with zero nullspace, $m \geq n$. I simply have no idea on how to prove it. Hopefully someone can help me out. Thank you. 
 A: Finding the nullspace of $A$ is equivalent to finding all solutions to the matrix equation $A\mathbf{x}=\mathbf{0}$. 
Solving the matrix equation $A\mathbf{x}=\mathbf{0}$ is equivalent to solving a homogeneous system of linear equations. You have as many equations as there are rows in $A$, and as many unknowns as there are columns in $A$.
So if $A$ is an $m\times n$ matrix, then you have a system with $m$ equations in $n$ unknowns.
Rather than proving the statement, let's prove the contrapositive. Remember that the contrapositive of "If $P$, then $Q$" is "If (not $Q$), then (not $P$)". In this case, rather than proving:

If $A$ has zero nullspace, then $m\geq n$.

we will look at

If $m\lt n$, then the nullspace of $A$ is not just the zero vector.

So, suppose you have a homogeneous system of linear equation with $m$ equations in $n$ unknowns, and there are more unknowns than equations. Do you know anything about solutions to such systems? 
A: Hint: this follows from the rank-nullity theorem, using the fact that the rank of a matrix with $m$ rows is no greater than $m$.
