Subspace

Definition

A subset $U$ of $V$ is called a subspace of $V$ if $U$ is also a vector_space (using the same addition and scalar muliplication as on $V$).

Conditions for a Subspace

A subset $U$ of $V$ is a subspace of $V$ if and only if $U$ satisfies the following three conditions:

1. Additive identity: $0 \in U$

2. Closed under addition: $u,w \in U$ implies $u + w \in U$

3. Closed under scalar multiplication: $a \in \mathbb{F}$ and $u \in U$ implies $au \in U$

Proof: