Definition
Let
be vectors in a vector space V. If the only linear combination of the vectors satisfying is the trivial linear combination (all the a’s is zero), then the vectors are linearly independent.
Linear Dependence Test
See: rank Two or more vectors are linearly dependent iff one of the vectors is a linear combination of the others.
Extending Linear Dependence
Let
- Suppose
. Then the vectors are linearly independent (see span) - Remove a vector from
and the remaining vectors are still linearly independent
Let
- For any vector
the vectors are linearly dependent - At least one vector can be removed from
without changing its span.