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title: "Axler 1.A"
source: https://www.jemoka.com/posts/kbhaxler_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&rsquo;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&rsquo;s identity), and call it a group (groups (vector spaces (fields )))
