If a vector space had bases of two different finite sizes $m < n$ say, then expressing one in terms of the other gives $m$ by $n$ and $n$ by $m$ matrices $A$ and $B$ such that $BA=I_n$. Now use Gaussian elimination (quote their first-year course) to "prove" (with an appropriate amount of hand-waving) that $Av=0$ has a nonzero solution, and thus derive a contradiction. Surely less than a lecture :-)