does the zero vector lie in the span of every sequence of vectors?

Question

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)?

Answer 1

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).

Answer 2

Yes

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

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