Suppose $U$ is an $m\times n$ orthogonal matrix. Show that $m \geq n$.
I'm having trouble with this proof --
I understand that the columns of $~U~$ can only be linearly independent in the cases where
$(i) ~~~m > n~$ and
$(ii)~~~ m = n~$,
but how do I go on to discuss whether or not this indicates that the column vectors themselves are orthogonal or not?
And why this is not the case when $~m < n~$?