Subspace

Definition: Subspace

Let $(V,F,+,\cdot )$ be a vector space.

A vector space $(U,F,+,\cdot )$ where $U\subseteq V$ and $\mathbf{u}+{\mathbf{u}}^{\prime }\in U$ and $c\mathbf{u}\in U$ for all $\mathbf{u},{\mathbf{u}}^{\prime }\in U$ and all $c\in F$.

Theorem: Subspace Criterion

A vector space $(U,K,+,\cdot )$ is a subspace of $(V,K,+,\cdot )$ if and only if all of the following hold:

1. The zero vector of $V$ is also in $U$;

2. $\mathbf{u}+{\mathbf{u}}^{\prime }\in U\forall \mathbf{u},{\mathbf{u}}^{\prime }\in U$;

3. $c\mathbf{u}\in U\forall c\in F,\mathbf{u}\in U$

PROOF

TODO