Editing Linear combination (section)

== The linear span ==
{{main article|Linear span}}

Take an arbitrary field ''K'', an arbitrary vector space ''V'', and let '''v'''<sub>1</sub>,...,'''v'''<sub>''n''</sub> be vectors (in ''V'').
It is interesting to consider the set of ''all'' linear combinations of these vectors.
This set is called the ''[[linear span]]'' (or just ''span'') of the vectors, say ''S'' = {'''v'''<sub>1</sub>, ..., '''v'''<sub>''n''</sub>}. We write the span of ''S'' as span(''S'')<ref>{{Harvard citation text|Axler|2015}} pp. 29-30, §§ 2.5, 2.8</ref><ref>{{Harvard citation text|Katznelson|Katznelson|2008}} p. 9, § 1.2.3</ref> or sp(''S''):
:<math> \operatorname{span}( \mathbf v_1 ,\ldots, \mathbf v_n) := \{ a_1 \mathbf v_1 + \cdots + a_n \mathbf v_n : a_1 ,\ldots, a_n \in K \}. </math>