Subspaces of Dimension 2 Real Vector Space/Proof 2

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.

Proof

Follows directly from Dimension of Proper Subspace is Less Than its Superspace.

$\blacksquare$