Definition: Subspace

Let ($F,+,\cdot)$ be a field and $V$ be a vector space over $F$. A non-empty subset $U\subseteq V$ is called subspace of $V$, if 1. $0\in U,$ 1. $x + y\in U$ for all $x,y\in U,$ 1. $\alpha\cdot x\in U$ for all $\alpha \in F, x\in U.$

These properties are equivalent to those: * $(U, + )$ is a subgroup of $(V, + )$ and * $U$ is closed under the scalar multiplication by elements of $F$.

Examples: 1