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 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 )))