3.4b Span

Span

The span of a set of vectors $\mathcal{P}=\left\{{\vec{v}}_1,\dots ,{\vec{v}}_n\right\}\subseteq {\mathcal{R}}^n$ is the set of all vectors that can be built using the vectors in $\mathcal{P}$ by scaling and adding them together (i.e., the linear combinations of vectors in $\mathcal{P}$). Formally,
$\mathrm{span}(\mathcal{P})=\left\{{\sum }_{i=1}^n{\alpha }_i{\vec{v}}_i:{\alpha }_i\in \mathbb{R}\right\}\subseteq {\mathbb{R}}^n,$
where ${\alpha }_i$ is the scaling factor (or weight) of each vector in $\mathcal{P}$.

If $\mathcal{P}$ contains $k$ linearly independent vectors, then the dimension $d$ of its span is $n\ge d\ge k$. Note that the maximum amount of linearly independent vectors in ${\mathbb{R}}^n$ is $n$.

Infinite Property

All nonzero vectors in an infinite vector space have an infinitely extending span.

Intuition

The span is not too hard of a concept.