what's the difference between $span(S_1 \cup S_2)$ and $span(S_1) \cup span(S_2)$? supposing that $S_1$ and $S_2$ are finite subsets of $\Bbb R^n$
assuming the example of $S_1$ = {(1,0)} and $S_2$ = {(0,1)}
so to me the union of both sets gives me {(1,0),(0,1)}, wouldn't the span of that be the same as the union of span($S_1$) and span($S_2$)? since the former would be {(a,b)} where a,b are any real number and the latter be similar to that.
apologies for any fallacy in reasoning (I'm thinking my understanding of unions is flawed)
Assuming you mean the linear span of $\mathbb{R}^n$ as a vector space, the two expressions are not equivalent. As stated in a comment, $\mathrm{span}\{(1,0)\}$ is the $x$-axis and $\mathrm{span}\{(0,1)\}$ is the $y$-axis, so $\mathrm{span}\{(0,1)\}\cup\mathrm{span}\{(1,0)\}$ is simply the union of the two axes.
However, as you said yourself, $\mathrm{span}\{(1,0),(0,1)\}$ would indeed span the entirety of $\mathbb{R}^2$.