Let $\newcommand{\x}{\boldsymbol{x}} T(\x)=A\x$ be a linear transformation that maps $\Bbb{R}^n$ into $\Bbb{R}^m$, and assume that $m>n$. Prove that $T(\x)$ can not possibly be onto.
Proof: Suppose $m>n$, and that $T(\x)=A\x$ is a linear transformation that maps $\Bbb{R}^n\to\Bbb{R}^m$. Since $m>n$, we see that $A\x$ has more equations than variables, thus there will not be a pivot in every row, by definition of an onto transformation, hence $T(\x)$ cannot possibly be onto. ∎
Considering this is for an elementary linear algebra course, how was this proof?