Editing Linear combination (section)

== Linear independence ==
{{main|Linear independence}}
Suppose that, for some sets of vectors '''v'''<sub>1</sub>,...,'''v'''<sub>''n''</sub>,
a single vector can be written in two different ways as a linear combination of them:
:<math>\mathbf v = \sum_i a_i \mathbf v_i = \sum_i b_i \mathbf v_i\text{ where } a_i \neq b_i.</math>
This is equivalent, by subtracting these (<math>c_i := a_i - b_i</math>), to saying a non-trivial combination is zero:<ref>{{Harvard citation text|Axler|2015}} pp. 32-33, §§ 2.17, 2.19</ref><ref>{{Harvard citation text|Katznelson|Katznelson|2008}} p. 14, § 1.3.2</ref>
:<math>\mathbf 0 = \sum_i c_i \mathbf v_i.</math>

If that is possible, then '''v'''<sub>1</sub>,...,'''v'''<sub>''n''</sub> are called ''[[linearly dependent]]''; otherwise, they are ''linearly independent''.
Similarly, we can speak of linear dependence or independence of an arbitrary set ''S'' of vectors.

If ''S'' is linearly independent and the span of ''S'' equals ''V'', then ''S'' is a [[basis (linear algebra)|basis]] for ''V''.