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

• complex number
  • addition and multiplication of complex numbers
  • subtraction and division of complex numbers
• field: \(\mathbb{F}\) is \(\mathbb{R}\) or \(\mathbb{C}\)
  • power
• list
• \(\mathbb{F}^n\): F^n
  • coordinate
  • addition in \(\mathbb{F}^n\)
  • additive inverse of \(\mathbb{F}^n\)
  • \(0\): zero
  • scalar multiplication in \(\mathbb{F}^n\)

Results and Their Proofs

• properties of complex arithmetic
  • commutativity
  • associativity
  • identities
  • additive inverse
  • multiplicative inverse
  • distributive property
• properties of \(\mathbb{F}^n\)
  • addition in \(\mathbb{F}^n\) is associative
  • addition in \(\mathbb{F}^n\) is commutative
  • addition in \(\mathbb{F}^n\) has an identity (zero)
  • addition in \(\mathbb{F}^n\) has an inverse
  • scalar multiplication in \(\mathbb{F}^n\) is associative
  • scalar multiplication in \(\mathbb{F}^n\) has an identity (one)
  • scalar multiplication in \(\mathbb{F}^n\) is distributive

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 )))