In an
n-dimensional
vector space, a group of
vectors is dependent if some non-trivial
linear combination of the vectors can be found to cancel them out.
That is, if we can find a set of n numbers
l1, l2, ... , ln-1, ln that are not all 0
such that
l1v1+l2v2+...+ln-1vn-1+lnvn = 0,
the vectors are said to be
linearly dependent (or simply "dependent"). If they are not dependent, they are said to be
linearly independent.
Given a point p:
- Two vectors are dependent if p, (p+v1) and (p+v2) lie in the same straight line.
-
Three vectors are dependent if p, (p+v1) (p+v2), and (p+v3) lie in the same plane.
- Four vectors are dependent if p, (p+v1) (p+v2), (p+v3), and (p+v4) lie in the same three-dimensional hyperplane.
So, in a three-dimensional vector space, four vectors are always dependent.