relationship between rows and columns of an orthogonal matrix

Question

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 $~m < n~$?

Answer 1

Orthogonal matrices are by definition square matrices?

Edit: Recall that the $rank(U) \leq min(m, n)$. Then note that since it must have linearly independent columns since each column is by definition orthogonal to one another, it must be at least $n$. Therfore, $n \leq rank(U) \leq min(n, m) \leq m$.