Vectors , , ..., are linearly dependent Iff there
exist Scalars , , ..., , not all zero, such that
where Einstein Summation is used and , ..., . If no such Scalars exist,
then the vectors are said to be linearly independent. In order to satisfy the Criterion for linear dependence,
In order for this Matrix equation to have a nontrivial solution, the Determinant must be 0, so the
Vectors are linearly dependent if
and linearly independent otherwise.
Let and be -D Vectors. Then the following three conditions are equivalent
- 1. and are linearly dependent.
- 3. The Matrix
has rank less than two.
Gray, A. Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 186-187, 1993.
© 1996-9 Eric W. Weisstein