Houjun Liu

Axler 1.A

Key sequence

  1. In this chapter, we defined complex numbers, their definition, their closeness under addition and multiplication, and their properties
  2. These properties make them a field: namely, they have, associativity, commutativity, identities, inverses, and distribution.
  3. notably, they are different from a group by having 1) two operations 2) additionally, commutativity and distributivity. We then defined \(\mathbb{F}^n\), defined addition, additive inverse, and zero.
  4. These combined (with some algebra) shows that \(\mathbb{F}^n\) under addition is a commutative group.
  5. Lastly, we show that there is this magical thing called scalar multiplication in \(\mathbb{F}^n\) and that its associative, distributive, and has an identity. Technically scalar multiplication in \(\mathbb{F}^n\) commutes too but extremely wonkily so we don’t really think about it.

New Definitions

Results and Their Proofs

Question for Jana

  • No demonstration in exercises or book that scalar multiplication is commutative, why?

Interesting Factoids

  • You can take a field, look at an operation, and take that (minus the other op’s identity), and call it a group
  • (groups (vector spaces (fields )))