Basis

Def: Basis

If $V$ is a vector space, a basis for $V$ is a linearly independent spanning set. dim(V) = |B| where B is a basis of $V$

Thm: Every Vector Space Has a Basis

Thm: Invariance Theorem

In a finite vector space V, all bases have the same cardinality.
uses the fundamental theorem
$| B_{1} | \leq | B_{2} |$ and $| B_{2} | \leq | B_{1} |$
It follows that any spanning set, or linearly independent set with that same cardinality are a basis.

Thm: Extension or Reduction to a Basis

If $V$ is a finite dimensional vector space fdvs then:

Any linearly independent set can be extended to a basis (axiom of choice)
Any spanning set can be reduced to a basis.
Uses span theorems, the fundamental theorem, and the fact that if $V = span {v_{1}, v_{2}, \dots, v_{n}}$ and $v_{1} \in span {v_{2}, \dots, v_{n}}$ then $V = span {v_{2}, \dots, v_{n}}$ . When we take the minimal spanning set it has to be L.I or we get a contradiction, therefore it is a basis.

Thm: More Theorems

If V is a fdvs and $U \subseteq$ V is a subspace:
1. dim( $U$ ) $\leq$ dim(V)
2. if dim( $U$ )=dim(V) then $U$ =V
3. any basis for $U$ extends to a basis of V