We have now found a test for determining whether a given set of vectors is linearly independent: A set of n vectors of length n is linearly independent if the ...
A set of vectors { v 1 , v 2 ,..., v k } is linearly independent if and only if, for every j , the vector v j is not in Span { v 1 , v 2 ,..., v j − 1 } .
In the theory of vector spaces, a set of vectors is said to be linearly dependent if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be linearly independent. These concepts are central to the definition of dimension.