# When can we say a set of vectors “spans” another set?

I know what the span of a set of vectors means, but sometimes we say a set "spans" another. These questions are about the specifics regarding the latter usage.

Question 1: Can we say that a set of vectors spans another set of vectors if that second set of vectors isn't a vector space?

Question 2:

Can a set of vectors $S$ be said to span a vector space $V$ if $Span(S)$ is larger than $V$? For example, does $\left\{\begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix}, \begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix}, \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}\right\}$ span the set of all vectors in the form $\begin{bmatrix} 0\\ a\\ b \end{bmatrix}$ for some $a, b\in\mathbb{R}$?

• The answer to question $2$ is yes: The span of $S$ has to contain $V$, and the word "larger" cannot be used in this context (what does it mean for a set to be larger than another, if both are infinite, for example?). As for question 1: not true.A set of vectors $S$ spans $V$ if $V \subset Span(S)$. There's nothing more to be said: $V$ could a vector space, infinite dimensional, large set, but it can still be spanned by some other set $S$. – Teresa Lisbon Feb 7 '17 at 3:29
• Have a look at this page: en.wikipedia.org/wiki/Linear_span – Cuhrazatee Feb 7 '17 at 4:32