Span

Definition

Let

• $V$ : Vector space
• $S=\{{\mathbf{w}}_1,{\mathbf{w}}_2,\dots ,{\mathbf{w}}_r\}$ : Set of vectors in $V$
• $W$ : Subspace of $V$

If $W$ consists of all possible linear combinations of the vectors in $S$

Then

• We say $W$ is the subspace of $V$ generated or spanned by $S$
• We say the vectors ${\mathbf{w}}_1,{\mathbf{w}}_2,\dots ,{\mathbf{w}}_r$ span $W$
• We say the set $S$ span $W$
• We denote $W=\mathrm{span}\{{\mathbf{w}}_1,{\mathbf{w}}_2,\dots ,{\mathbf{w}}_r\}=\text{span(S)}$

Interpretation

Look at the definition

By definition, if $W=\mathrm{span}(S)$, then ”$W$ consists of all possible linear combinations of the vectors in $S$“.

This means, a linear combination like ${\mathbf{w}}_1+3{\mathbf{w}}_2-4{\mathbf{w}}_3$ is in $W$ (Assuming $r=3$).