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?
Best Answer
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.