Subspaces of Dimension 2 Real Vector Space

Theorem
Take the $\R$-vector space $\left({\R^2, +, \times}\right)_\R$.

Let $S$ be a subspace of $\left({\R^2, +, \times}\right)_\R$.

Then $S$ is one of:
 * $(1): \quad \left({\R^2, +, \times}\right)_\R$
 * $(2): \quad \left\{{0}\right\}$
 * $(3): \quad$ A line through the origin.