Editing Linear independence (section)

== Space of linear dependencies ==

A '''linear dependency''' or [[linear relation]] among vectors {{math|'''v'''<sub>1</sub>, ..., '''v'''<sub>''n''</sub>}} is a [[tuple]] {{math|(''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub>)}} with {{mvar|n}} [[scalar (mathematics)|scalar]] components such that

:<math>a_1 \mathbf{v}_1 + \cdots + a_n \mathbf{v}_n= \mathbf{0}.</math>

If such a linear dependence exists with at least a nonzero component, then the {{mvar|n}} vectors are linearly dependent. Linear dependencies among {{math|'''v'''<sub>1</sub>, ..., '''v'''<sub>''n''</sub>}} form a vector space.

If the vectors are expressed by their coordinates, then the linear dependencies are the solutions of a homogeneous [[system of linear equations]], with the coordinates of the vectors as coefficients. A [[basis (linear algebra)|basis]] of the vector space of linear dependencies can therefore be computed by [[Gaussian elimination]].