Plan

• 1.6

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.
  • Fⁿ, 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.