Vector Subspace Test
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Theorem
Let $V$ be a vector space over a division ring $K$.
Let $U \subseteq V$ be a non-empty subset of $V$ such that:
- $\forall u, v \in U: \forall \lambda \in K: u + \lambda v \in U$
Then $U$ is a subspace of $V$.
Corollary
Let $U \subseteq V$ be a non-empty subset of $V$ such that:
- $(1): \qquad \forall u \in U, \lambda \in K: \lambda u \in U$
- $(2): \qquad \forall u, v \in U: u + v \in U$
Then $U$ is a subspace of $V$.
Proof
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