The vectors v 1 = ( a 11 ⋮ a m 1 ) , … , v n = ( a 1 n ⋮ a m n ) ∈ K m {\displaystyle {}v_{1}={\begin{pmatrix}a_{11}\\\vdots \\a_{m1}\end{pmatrix}},\ldots ,v_{n}={\begin{pmatrix}a_{1n}\\\vdots \\a_{mn}\end{pmatrix}}\in K^{m}} are linearly dependent, if and only if the homogeneous linear system
has a nontrivial solution.