Basis

A Set of Basis Vectors. A set of Linearly Indepedent Vectors whose Linear Combinations can represent any vector in that space.

Standard Basis Vectors in are i, j, k, but you can use other vectors to span the same amount of space if you want.

• Let
• is some basis for the subspace

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

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Basis Theorem

Any two bases for a Subspace have the same Dimension

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