Exercise 1.7.6

Let $F$ be the family of all linearly independent subsets of $S_2$ that contain $S_1$. We may check there is some set containing each member of a chain for all chains of $F$ just as the proof in Theorem 1.13. So by the Maximal principle, there is a maximal element $\beta$ in $F$. By the maximality, we know $\beta$ can generate $S_2$ and hence can generate $V$. In addition to its independence, we know $\beta$ is a basis.