Dimension

The dimension of a Subspace is the Cardinality of its Basis

This is because a basis requires a Set of Linearly Indepedent Vectors, and does not meet this requirement. Therefore, its only basis is . The cardinality of the empty set is 0, therefore so is

Examples

• dim
• dim(Nul A) is the number of
  • number of free vars of A
• dim(Col A) is the number of
  • number of pivots of A
• H =
  • n variables, solve for ito everything else. → one pivot everything else free vars. Therefore free vars