Quotient vector space
 Note: see Quotient for other types of quotient
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Contents
Definition
Given:
 A vector space [ilmath](V,\mathbb{F})[/ilmath] over a field [ilmath]\mathbb{F} [/ilmath] and
 A vector subspace [ilmath]W\subseteq V[/ilmath]
We define an:
 Equivalence relation on [ilmath]V[/ilmath] defined as:
 [ilmath]v\sim v'[/ilmath] if [ilmath]vv'\in W[/ilmath]
Here [ilmath][v][/ilmath] denotes the equivalence class of [ilmath]v[/ilmath] under [ilmath]\sim[/ilmath], that is:
 [ilmath][v]:=\{u\in V\vert v\sim u\}[/ilmath]
Then the following two diagrams commute
Diagram for addition on equivalence classes
[math]\begin{xy}\xymatrix{ V\times V \ar@{>}[rr]^{+} \ar@{>}[drr]^{\pi\circ+} \ar@{>}[d]_{\pi\times\pi} & & V \ar@{>}[d]^\pi \\ \frac{V}{\sim}\times\frac{V}{\sim} \ar@{.>}[rr]^{+}&& \frac{V}{\sim} }\end{xy}[/math] 
Note that here:

Diagram  Key 

[ilmath]+:\frac{V}{\sim}\times\frac{V}{\sim}\rightarrow\frac{V}{\sim}[/ilmath] is given by [ilmath]\pi\circ+\circ(\pi\times\pi)^{1}[/ilmath]. This means that [ilmath][u]+[v]=\pi(\pi^{1}([u])+\pi^{1}([v]))=\underbrace{[x\in\pi^{1}([u])+y\in\pi^{1}([v])]}_\text{Welldefinedness}=[u+v][/ilmath]^{[Note 1]} 
Note that:
 The dashed arrow labeled [ilmath]+[/ilmath] denotes the induced binary operation on [ilmath]\frac{V}{\sim} [/ilmath], in the context of factoring functions we often write the function induced by [ilmath]f[/ilmath] as [ilmath]\tilde{f} [/ilmath] however (as usual) the meaning of addition is given by the context, so it is not ambiguous to define addition of [ilmath]\frac{V}{\sim} [/ilmath] where addition on [ilmath]V[/ilmath] is already defined.
 The 'welldefinedness' need not be checked as it is used in the proof of factorising functions  it is mentioned here only to explain the abuse of notation
Diagram for scalar multiplication
[math]\begin{xy}\xymatrix{ \mathbb{F}\times V \ar@{>}[rr]^{*} \ar@{>}[d]^{i\times\pi} \ar@{>}[drr]^{\pi\circ*} & & V \ar@{>}[d]^\pi \\ \mathbb{F}\times\frac{V}{\sim} \ar@{.>}[rr]^{*} & & \frac{V}{\sim} }\end{xy}[/math] 
Note that here:

Diagram  Key 

[ilmath]*:\mathbb{F}\times\frac{V}{\sim}\rightarrow\frac{V}{\sim}[/ilmath] is given by [ilmath]\pi\circ*\circ(i\times\pi)^{1} [/ilmath]. That is [ilmath]\alpha[v]=\pi(\alpha\pi^{1}([v]))=\underbrace{[\alpha x\ \text{for }x\in\pi^{1}([v])]}_\text{welldefinedness}=[\alpha v][/ilmath]^{[Note 2]} 
Overview of proofs
Usually we simply say:
 Addition defined by:
 [ilmath][v]+[u]=[v+u][/ilmath] and check it is well defined (this is to check that whichever representatives we choose of [ilmath]a\in[u][/ilmath] and [ilmath]b\in[v][/ilmath] that [ilmath][a+b]=[u+v][/ilmath] still
 Scalar multiplication defined by:
 [ilmath]\alpha[v]=[\alpha v][/ilmath] and again, check this is well defined (that is for whichever [ilmath]a\in[v][/ilmath] we choose to represent [ilmath][v][/ilmath] that [ilmath][\alpha a]=[\alpha v][/ilmath]
This isn't wrong. However by using diagrams we can get a much "purer" proof which only involves checking the conditions of factoring functions  this shifts the notion of "well defined" to this operation and we simply apply a theorem.
Proof of claims
Claim 1: [ilmath]v\sim v'[/ilmath] is indeed an equivalence relation
TODO: Be bothered, this is really easy to do
Claim 2: The diagram for addition commutes
TODO: Be bothered, this is really easy to do
Claim 1: The diagram for multiplication commutes
TODO: Be bothered, this is really easy to do
Todo notes
 This method is "purer" and more advanced then is seen when this concept is first introduced. A "simple" version ought to be created
 A summary section of factorising functions ought to be transcluded into this page.
 Some examples
TODO: These things
Notes
 ↑ This is where welldefinedness comes into play, but the Factor (function) theorem already takes this into account. We abuse the notation when writing [ilmath]\pi^{1} [/ilmath] as this is of course a subset, it's okay though because whichever member of the subset we take, the equivalence class of the addition with another representative of the second term is the same
 ↑ Note that [ilmath]\pi^{1}([v])[/ilmath] is actually a set but as Factor (function) shows it doesn't matter what representative we take. This is an abuse of notation.