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 be linearly independent vectors.

  • Suppose . Then the vectors are linearly independent (see span)
  • Remove a vector from and the remaining vectors are still linearly independent

Let be linearly dependent vectors.

  • For any vector the vectors are linearly dependent
  • At least one vector can be removed from without changing its span.