Subspace of $ \mathbb{R}^2 $

Question

I wanted to know whether $$ V=\{ (a, b) :a, b, \in \mathbb{R}\} $$ is a subspace of $\mathbb{R}^2$.

Now, it is clear $V $ contains the origin of $\mathbb{R} ^2$ and is also closed under vector addition and scalar multiplication. But still my book says that it is not a subspace. Is there anything which I am missing?

Answer 1

As stated, your space is certainly a subspace of $\mathbb{R}^2$, since it is all of $\mathbb{R}^2$. Assuming that this is the problem in the book, their answer is wrong.