How to determine if a set of vectors can be written as a span?

Question

Let $A =\{ (a,b)∈\Bbb{R}^2 : a = 0$ or $b = 0\}$

Specifically, I'm wondering if this set can be written as a span, if so, how? and if not, are there rules or ways to tell if something can be written as a span?

Answer 1

If it is a span, it must be a vector space. One has $(1,0) \in A$ and $(0,1) \in A$, but $(1,0)+(0,1) = (1,1) \notin A$, so $A$ is not a vector space, thus not a span.