## Plan

## 1.6 Bases

- Defintion: A basis is a linearly independent and spanning set. The dimension is the cardinality of the basis.
- Does it exist? Is it unique? Is there any invariant?
- Examples:
- Standard basis is ever basis is nice.
*F*^{n}, polynomial

- Theorem: Linear independence gives you unique representation.
- Theorem: Span gives you span.
- Theorem: Basis gives you both.
- Examples:
- Do problem 3 in au17 final, both null and col

- In this course, in terms of basis, we are primarily interested in finite dimensional spaces.
- Theorem: If a space is finitely generated by
*S* then some subset of *S* is a basis.