# Definition of span

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?

• And of course the true insight is that one proves the equivalence of all these properties and thereby is motivated to define the notion of span (as any and all of these) in the first place ... – Hagen von Eitzen Oct 12 '18 at 16:13
• Many definitions in mathematics have alternate formulations. In many cases, one definition is "bottom-up" and the other is "top-down". Top-down definitions are usually easy to construct and easy to show that they have certain properties, but it's usually hard to name the elements explicitly in them. Bottom-up definitions have the exact opposite drawbacks and advantages. – Robert Wolfe Oct 12 '18 at 18:24

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.
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 $$\operatorname{span}S=\bigcap_{W\in\mathcal W}W$$.