Sets larger than a finitely generated space's bases are linearly dependent

up:: Steinitz Exchange Lemma

Let $(V,+,\cdot )$ be a Finitely Generated Vector Space, with Spanning Set $S=\{s_1,\dots ,s_n\}$. Let $U=\{u_1,\dots ,u_m\}\subseteq V$.

From the Steinitz Exchange Lemma, by contrapositive, we have that $m>n$ implies that $U$ is Linearly Dependent (assuming $S$ is a spanning set as hypothesis).