Let $V$ be an arbitrary $n$ dimensional vector space over $\mathbb{F}$, and let $\beta_1$ and $\beta_2$ be bases for $V$. Take elements $s_1, s_2, \cdots, s_m$ of $V$ with $m < n$, and let $S = \{[s_1]_{\beta_1}, [s_2]_{\beta_1}, \cdots, [s_m]_{\beta_1}\}$ be the set of coordinate vectors of $s_i$ relative to $\beta_1$.
Now take $[\sigma]_{\beta_1} \in span(S)$. There exists some $v \in V$ such that the n-tuple of scalars representing $[\sigma]_{\beta_1}$ is the same as the n-tuple of scalars representing $[v]_{\beta_2}$. Does this imply that $[v]_{\beta_1} \in span(S)$? Note that $span(S)$ is the set of all linear combinations of the coordinate vectors $[s_i]_{\beta_1}$.
Is there any general reading I can find studying the properties of vectors of the same space relative to different bases and their interactions?