Does span(A) < span(B) imply A < B?
Given $A \subseteq B \subseteq V$ for some vector space $V$, we know that $\mathrm{span}(A) \subseteq \operatorname{span}(B) $. We can see that as span of $B$ can make any elements of $A$.
However does the inverse of the statement hold? So if $\operatorname{span}(A) \subseteq \operatorname{span}(B) $, can we say $A \subseteq B$?
$\endgroup$2 Answers
$\begingroup$No. For example, take $V = \mathbb{R}^2$, $A = \{(2,0)\}$ and $B = \{(1,0), (0,1)\}$. Since $B$ is a basis for $V$, $\operatorname{span}(B) = V$, so $\operatorname{span}(A) \subseteq \operatorname{span}(B)$. However, $A \not\subset B$.
$\endgroup$ 2 $\begingroup$No. In $\mathbb{R}^2$, take $A=\{(1,0)\}$, $B=\{(2,0),(0,1)\}$. Then $\mathrm{Span}\, A=\mathbb{R}\subseteq\mathbb{R}^2$ and $\mathrm{Span}\, B=\mathbb{R}^2$ (so $\mathrm{Span}\, A\subseteq\mathrm{Span}\, B$), but $A\not\subseteq B$, since $(1,0)\notin B$.
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