Vector space/Characterizations of basis/Maximal/Minimal/Fact

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Let be a field, and let be a -vector space. Let be a family of vectors. Then the following statements are equivalent.

  1. The family is a basis of .
  2. The family is a minimal generating system, that is, as soon as we remove one vector , the remaining family is not a generating system any more.
  3. For every vector , there is exactly one representation
  4. The family is maximal linearly independent, that is, as soon as some vector is added, the family is not linearly independent any more.