Linear Span

linear-algebra

Definition

Linear Span

A linear span $[M]$ of a non-empty subset $M\subseteq V$, where $V$ is a vector space over a scalar field $F$, is the set of all vectors that can be formed by linear combinations of a finite set of vectors from $M$.

$$[M]:=\{\mathbf{v}\in V\mid \exists {\mathbf{w}}_1,\dots ,{\mathbf{w}}_n\in M:\mathbf{v}=\sum _{i=1}^n{\lambda }_i\cdot {\mathbf{w}}_i\}$$

where ${\lambda }_i\in F$.

The linear span of an empty set is defined as:

$$[\varnothing ]=\{0\}$$

Smallest Subspace

Smallest Subspace

Let $M$ be a subset of a vector space $V$. Then a linear span $[M]$ is a subspace of a vector space $V$, with respect to set-theoretical inclusion, the smallest subspace that contains all vectors from $M$.