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is the zero vector in the span of every sequence of vectors? For example, would (0,0,0) lie in the span of (1,0,1),(2,3,0)?

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  • $\begingroup$ The 0 vector is in every vector space, and a verification of "is this abstract space a vector space?" should include checking the 0 vector. $\endgroup$
    – Doug M
    Commented Apr 17, 2018 at 16:44

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Yes. Depending on your definition of span, it is either the smallest subspace containing a set of vectors (and hence $0$ belongs to it because $0$ is a member of any subspace) or it is the set of all linear combinations in which case the empty sum convention kicks in. Even $\mathrm{span}\,\varnothing = \{0\}$ contains $0$ (and nothing else).

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  • $\begingroup$ thank you! that's a very helpful answer $\endgroup$
    – user553480
    Commented Apr 17, 2018 at 16:49
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Yes

$0\times (1,0,1)+0\times (2,3,0)=(0,0,0) $

so $(0,0,0) \in span\{(2,3,0);(1,0,1)\}$

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