A set of vectors is said to be linearly independent if there exists no vector in the set that is a linear combination of any other vector in the set. If such a vector exists the set of vectors is said to be linearly dependent.

Formally a set of vectors is linearly independent if the vector equation

has only the trivial solution . The set is linearly dependent otherwise (ie. if there is a solution where any of is nonzero).

Quick Checks

Dimensionality

If the length of the set is greater than the dimensionality of the vectors within it, the set is automatically linearly dependent.

Zero Vectors

If any of the vectors in the set are , the whole set must be linearly dependent.

Determinant

If the length of the set is equal to the dimensionality of the vectors we can check the determinant of the vectors arranged in a matrix to determine linear independence. If the determinant is nonzero they are independent.