Show that $$span (v_1, ..., v_k)$$ is a subspace of $$R^n$$ and is the smallest subspace containing $$v_1, ..., v_k$$.
I know if we assume $$v_1, ..., v_k$$ is an element of V where V is a subspace of $$R^n$$ and there is a w that is an element of $$span (v_1, ..., v_k)$$ then w is an element of $$a_1v_k+...+a_kv_k$$. Thus, w is an element of V because V is a subspace. Thus the $$span (v_1, ..., v_k)$$ is subspace in R^n. Now how do I show it is the smallest subspace. What does that mean?
I am confused with your proof that the span is a subspace of $R^n$. You show that any member of the span is a member of the subspace but that only shows it is a subset, not that the span is a vector space itself. To do that you need to show that it is closed under vector addition and scalar multiplication.
To show it is the smallest subspace that contains the given vectors, you need to show that any subspace of $R^n$ of lower degree cannot contain all of the given vectors.
$\textbf{Hint}$: The smallest subspace (of a vector space $V$) containing a subset $S$ of $V$ is the intersection of all subspaces of $V$ which contain $S$ since the (arbitrary) intersection of subspaces is again a subspace.
