Subspaces of Dimension 2 Real Vector Space/Proof 2
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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.
$\blacksquare$