## 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~~because we know 1 already, and you can follow the behavor of scalar multiplication*some*\(0\) exists?~~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 \(\{0\}\)