Linearly independent/Simple properties/Fact/Proof/Exercise

Let be a field, let be a -vector space, and let , , be a family of vectors in . Prove the following facts.

  1. If the family is linearly independent, then for each subset , also the family  , is linearly independent.
  2. The empty family is linearly independent.
  3. If the family contains the null vector, then it is not linearly independent.
  4. If a vector appears several times in the family, then the family is not linearly independent.
  5. A vector is linearly independent if and only if .
  6. Two vectors and are linearly independent if and only if is not a scalar multiple of , and vice versa.