Q:

union of spans versus span of union of sets

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)

A:

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 $\mathbb{R}^2$.

