Definition of span

Question

On an old midterm exam, my professor requested the students prove that

The span of $S$ (where $S$ is a subset of a vector space $V$) is equal to all vectors that can be expressed as linear combinations of the elements in $S$.

Does this make any sense? He's requesting we show that the span of $S$ equals what I believe to be the definition of span. Is there possibly some other definition of span that I should be aware of?

Answer 1

You can define $\text{span} (S)$ to be the smallest vector subspace containing $S$, or equivalently the intersection all vector subspaces containing $S$. Such a definition is very common in algebra.

Answer 2

Yes, there is another definition: let$$\mathcal{W}=\left\{W\subset V\,\middle|\,S\subset W\text{ and }W\text{ is a vector subspace of }V\right\}.$$Now, define $\operatorname{span}S=\bigcap_{W\in\mathcal W}W$.

Answer 3

Perhaps the definition of span that your professor is using is: The smallest vector space generated by the vectors in the spanning set.