Subspace

definitionvector-spaces

Definition

A subset $U$ of $V$ is a subspace of $V$ if $U$ is a vector space with the same $0$, addition, and scalar multiplication as in $V$.

Verifying a Subspace

In order to check that a subset $U\subseteq V$ is a subspace of $V$, we can check the following.

1. $0_v\in U$.
2. $u,w\in U$ implies $u+w\in U$ (closure under addition).
3. $u\in U$ and $\lambda \in \mathbb{F}$ implies $\lambda u\in U$ (closure under scalar multiplication).

Examples