Is it possible to swap vectors into a basis to get a new basis? Let $V$ be a vector space in $\mathbb{R}^3$. Assume we have a basis, $B = (b_1, b_2, b_3)$, that spans $V$. Now choose some $v \in V$ such that $v \ne 0$. Is is always possible to swap $v$ with a column in $B$ to obtain a new basis for $V$?
My initial thought is that it is not always possible since if $v$ is some linear combination of any 2 vectors in $B$ then if we were to swap $w$ into $B$, $B$ would no longer span $V$. Am I missing a nuance here?

If $(v_1,\cdots, v_n)$ is a basis for any vector space $V$, and $w\in V$ is an arbitrary vector, then swapping $w$ for $v_i$ will result in a basis iff $w$ is not in the span of $\{v_j\mid j\ne i\}$. The proof is a good exercise.  
