Vector Space Axioms

Question

I am having trouble understanding vector space axioms seen here (https://i.stack.imgur.com/X6VBD.jpg). As an example say we define our potential vector space to be the set of all pairs of real numbers of the form $(x, 0)$ with the standard operations on $\mathbb R^2$. What would $\mathbf{u}$ and $\mathbf{v}$ be in this example? How would I go about testing if this potential vector space is indeed a vector space? Any help would be appreciated! Thanks!

Answer 1

Your set is the vectors of the form $(x, 0)$. Just take two vectors $(a, 0), (b, 0)$ and a real number, say $c$, and test if the axioms hold. For example, for the first one, $(a, 0) + (b, 0) = (a+b, o)$. This is in the space, since it is on the form $(x, 0)$ and $a+b \in \mathbb{R}$, so the first axiom holds.