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--- title: "Axler 1.B" source: https://www.jemoka.com/posts/kbhaxler_1_b/ --- Key Sequence \(\mathbb{F}^{n}\) not being a field kinda sucks, so we made an object called a “vector space” which essentially does everything a field does except without necessitating a multiplicative inverse Formally, a vector space is closed over addition and have a scalar multiplication. Its addition is commutative, both addition and scalar multiplication is associative, and distributivity holds. There is an additive identity, additive inverse, and multiplicative identity. We defined something called \(\mathbb{F}^{S}\), which is the set of functions from a set \(S\) to \(\mathbb{F}\). Turns out, \(\mathbb{F}^{S}\) is a Vector Space Over \(\mathbb{F}\) and we can secretly treat \(\mathbb{F}^{n}\) and \(\mathbb{F}^{\infty}\) as special cases of \(\mathbb{F}^{s}\). We established that identity and inverse are unique additively in vector spaces. Lastly, we proved some expressions we already know: \(0v=0\), \(-1v=-v\). New Definitions addition and scalar multiplication vector space and vectors vector space “over” fields \(V\) denotes a vector space over \(\mathbb{F}\) \(-v\) is defined as the additive inverse of \(v \in V\) Results and Their Proofs \(\mathbb{F}^{\infty}\) is a Vector Space over \(\mathbb{F}\) \(\mathbb{F}^{S}\) is a Vector Space Over \(\mathbb{F}\) All vector spaces \(\mathbb{F}^{n}\) and \(\mathbb{F}^{\infty}\) are just special cases \(\mathbb{F}^{S}\): you can think about those as a mapping from coordinates \((1,2,3, \dots )\) to their actual values in the “vector” additive identity is unique in a vector space additive inverse is unique in a vector space \(0v=0\), both ways (for zero scalars and vectors) \(-1v=-v\) Questions for Jana The way Axler presented the idea of “over” is a tad weird; is it really only scalar multiplication which hinders vector spaces without \(\mathbb{F}\)? In other words, do the sets that form vector spaces, apart from the \(\lambda\) used for scalar multiplication, need anything to do with the \(\mathbb{F}\) they are “over”? The name of the field and what its over do not have to be the same—“vector space \(\mathbb{C}^2\) over \(\{0,1\}\)” is a perfectly valid statement If lists have finite length \(n\), then what are the elements of \(\mathbb{F}^{\infty}\) called? “we could think about \(\mathbb{F}^{\infty}\), but we aren’t gonna.” Why is \(1v=v\) an axiom, whereas we say that some \(0\) exists? because we know 1 already, and you can follow the behavor of scalar multiplication what’s that thing called again in proofs where you just steal the property of a constituent element?: inherits Interesting Factoids The simplest vector space is \(\{0\}\)