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Spanning sets

Definition

A subset $S$ of a vector space $V$ over a field $\mathbb{F}$ spans $V$ if we can write any vector as a linear combination of members of the set; i.e. for any vector $v \in V$ we can choose $v_1, ..., v_k \in V$ and $\alpha_1, ..., \alpha_k \in \mathbb{F}$ such that $v = \sum_{i=1}^k \alpha_i v_i$.

We call $S$ a spanning set and say $S$ spans $V$ and write $\langle S \rangle = V$. (We could also say "generating set"/"generates" like in group theory, but in linear algebra that wording is less common.)

CAUTION! Remember that in linear algebra, sums (and therefore linear combinations) are always finite in length!

Criterion for the finite case

A finite subset $S = \{v_1, ..., v_k\}$, it is spanning if for all $v \in V$ there exist $\alpha_1, ..., \alpha_k \in \mathbb{F}$ such that $v = \sum_{i=1}^k \alpha_i v_i$.

Properties

Any spanning set $S$ contains a basis for $V$. (proof - see the basis page).

Types

If a spanning set is linearly independent, we say it is a basis for $V$.

Related theorems

all these pages adapted with probably insufficient credit from my university's lecture notes