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)

  • $\begingroup$ Do you mean the linear span? $\endgroup$ – saulspatz Feb 1 '18 at 2:03
  • $\begingroup$ By $U$ do you mean $\cup$? The latter can be escaped as $\cup$. Also, $\mathrm{span}$ escapes as $\mathrm{span},$ and is usually preferred to $span$ ($span$) $\endgroup$ – stochasticboy321 Feb 1 '18 at 2:04
  • $\begingroup$ @saulspatz union of the linear spans of S1 and S2 vs linear span of the union of S! and S2! $\endgroup$ – Jonathan Low Feb 1 '18 at 2:05
  • $\begingroup$ @stochasticboy321 oh yes I'll go edit that $\endgroup$ – Jonathan Low Feb 1 '18 at 2:05
  • 3
    $\begingroup$ Using your example, $\text{span}(S_1)\cup \text{span}(S_2)$ is $\{(a,0)~:~a\in\Bbb R\}\cup \{(0,b)~:~b\in\Bbb R\}=\{(a,b)\in\Bbb R^2~:~a=0~\text{or}~b=0\}$, i.e. the axes and only the axes. Meanwhile $\text{span}(S_1\cup S_2)=\Bbb R^2$. $\endgroup$ – JMoravitz Feb 1 '18 at 2:05

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 $\mathbb{R}^2$.


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