Subspaces of R

[!Definition 4.4.1]

A subset U of ${\mathbb{R}}^n$ is a subspace of ${\mathbb{R}}^n$ if the following properties are all satisfied:

1. ${\vec{0}}_{{\mathbb{R}}^n}\in U$ U contains the zero vector of ${\mathbb{R}}^n$
2. If $\vec{x},\vec{y}\in U$, then $\vec{x}+\vec{y}\in U$ U is closed under vector addition
3. If $\vec{x}\in U$ and $c\in \mathbb{R}$, then $c\vec{x}\in U$ U is closed under scalar multiplication