In my linear algebra class, my teacher stated the following theorem:
Let $E$ be a vector space over a field $K$ and let $S_1=\{u_1,...,u_p\}$ and $S_2=\{v_1,...,v_n\}$ subsets of $E$ such that $S_1$ is linearly independent and $S_1\subseteq L(S_2)$, where $L(S_2)$ is the set of all linear combinations of $S_2$. Then:
- $p\leq n$
- We can add to $S_1$ $n-p$ vectors of $S_2$ such that the resulting set generates $L(S_2)$
I was trying to prove this myself but I wasn't able to. My teacher used Mathematical Induction to do so. Is there a way to do this more directly?