Vector subspace

Let $V$ be a vector space over $\mathbb{F}$.

If:

$$U=\operatorname{span}\{\vec u_1,\dots,\vec u_n\} =\left\{ \sum_{i=1}^n \lambda_i \vec u_i \;\middle|\; \lambda_i \in \mathbb{F} \right\}$$

for $\vec u_1,\cdots, \vec u_n \in V$.

Then Set $U$ is a vector subspace of $V$.

Set $\{u_1,\dots,u_n\}$ is then known as a generating set of $U$

Intersection and union

For vector subspaces $F,G$ $\in V$

$$\begin{gathered} \vec u \in F \cap G \iff \vec u \in F \land \vec u \in G \text{ intersection}\\ \vec u \in F \cup G \iff \vec u \in F \lor \vec u \in G \text{ union} \end{gathered}$$

Sum

For vector subspaces $F,G$ $\in V$

$$ F+G = \{\vec u + \vec v |\vec u \in F, \vec v \in G \} $$

Property

$$ \dim(F+G)=\dim(F)+\dim(G)-\dim(F \cap G) $$

Composition

$$ \operatorname{span}\{u_1,\dots,u_n\} = \operatorname{span}\{u_1\} + \cdots + \operatorname{span}\{u_n\} $$

for $U=\{u_1,\cdots, u_n\} \in V$.

Supplementary subspaces

For vector subspaces $F,G$ $\in V$, if

• $F+G=E$, and
• $F \cap G = \{ \vec 0_V \}$

Then $F,G$ are supplementary vector subspaces of $V$, noted as $E=F \oplus G$