If $A$ is an $m \times n$ matrix, prove that $Ax=0$ has infinite solutions $\iff$ $\text{rank}(A)<n$

The key (since $A \lambda x = \lambda Ax$) is to observe that $Ax = 0$ has infinitely many solutions if and only if $Ax = 0$ has a nonzero solution.

Observe that $Ax$ is a linear combination of the columns of $A$, so $Ax = 0$ has a nonzero solution if and only if the columns of $A$ are linearly dependent.

Can you relate this to the rank of the matrix?