Question: Let $S =$ { $(a,b) \in \mathbb{R}^{2}$ | $(a,b)$ has the form $(1, x)$ }.
Redefine addition and scalar multiplication operations as follows:
(1, y) + (1, y') = (1, y + y')
$k(1, y) = (1, ky)$
Can I now say that because $\mathbb{R}^{2}$ is a vector space and $S \subset \mathbb{R}^{2}$, that if additive & scalar multiplicative closures are shown to hold, then $S$ is a vector space?
Or do I first have to show that $\mathbb{R}^{2}$ is still a vector space with the newly defined operations?