Span
High school
Consider a list of vectors $v_1, \dots, v_k$. The span of the vectors is the set of all vectors that can be built from them by stretching and adding them.
Undergraduate
Consider vectors $v_1, \dots, v_k$ in a vector space $V$. The span of the vectors is the set of their linear combinations:
\[\operatorname{span}(v_1, \dots, v_k) = \setdef{c_1 v_1 + \dots + c_k v_k \suchthat c_1, \dots, c_k \text{ scalars}}.\]The vectors span $V$ if
\[\operatorname{span}(v_1, \dots, v_k) = V.\]Examples
Examples
- The vectors $(1, 0)$ and $(0, 1)$ in $\reals^2$ High school
Properties
- The span of every list of vectors is a vector subspace, which is the smallest subspace that contains them.