# Axler 1.A

## Key sequence

- In this chapter, we defined complex numbers, their definition, their closeness under addition and multiplication, and their properties
- These properties make them a field: namely, they have, associativity, commutativity, identities, inverses, and distribution.
- 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.
- These combined (with some algebra) shows that \(\mathbb{F}^n\) under addition is a commutative group.
- 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

## 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 )))