Subspace

A non-empty subset $S$ of $V$ is a subspace of $V$ over $F$ iff $S \subseteq V$ and $S$ is a vector space over $F$.

There are 2 trivial subspaces of $V$ over $F$:

• $\{\underline{0}\}$
• $V$

Suppose $S$ is a non-empty subset of $V$.

Criterion 1

$S$ is a subspace of $V\;\text{over}\;F$ iff $S$ is (1) closed under vector subtraction and (2) scalar multiplication of vectors.

Closed under vector subtraction forces $S$ to be closed under vector addition. And have a zero vector.

Criterion 2

$S$ is a subspace of $V$ iff for all $x,y \in S$ and $a,b \in F$ satisfies: $ax + by \in S$.