Prove if $v \in \text{span}(S)$, then $ \text{span}(S) = \text{span}(S ∪ \{v\})$

Question

Suppose $S$ is a subset of a vector space $V$. Show that if $v\in \text{span}(S)$, then $\text{span}(S) = \text{span}(S ∪ \{v\})$

Any help is appreciated. I don't even know where to start really, thanks for your time!

Answer 1

Since $S\subseteq S\cup\{v\}$ it's clear that $span(S)\subseteq span(S\cup\{v\})$. For other hand, $S\subseteq span(S)$ and $v\in span(S)$ implies that any linear combination between elements of $S$ and $v$ lies in $span(S)$, i.e., $span(S\cup\{v\})\subseteq span(S)$.