I'm not sure if it's is a duplicate question, but sure that I didn't find my answer in similar questions.
I found that $F+G= \operatorname{span}(F\cup G)$. However, while solving a question, my prof. used another equality which is, $F+G= \operatorname{span}(\operatorname{span}(F)\cup \operatorname{span}(G))$. Are the two same? If not, which is the right one? I'm really confused about this. Can anyone help me?
$F\subseteq \text{span}(F)$ and similarly for $G$. So, if $F+G = \text{span}(F\cup G)$, then the second equality is also true. Proving the reverse equality is fairly simple. Take any element of $\text{span}(\text{span}(F)\cup \text{span}(G))$ and show that it can be written as a linear combination only of elements in $F$ and/or $G$.