Quotient vector space

Note: see Quotient for other types of quotient

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Definition

Given:

A vector space $(V,\mathbb{F})$ over a field $\mathbb{F}$ and
A vector subspace $W\subseteq V$

We define an:

Equivalence relation on V defined as:
- $v\sim v'$ if $v-v'\in W$

Here $[v]$ denotes the equivalence class of $v$ under $\sim$ , that is:

$[v]:=\{u\in V\vert v\sim u\}$

Then the following two diagrams commute

Diagram for addition on equivalence classes

Diagram	Key
$\begin{xy}\xymatrix{ V\times V \ar@{->}[rr]^{+} \ar@{->}[drr]^{\pi\circ+} \ar@{->}[d]_{\pi\times\pi} & & V \ar@{->}[d]^\pi \\ \frac{V}{\sim}\times\frac{V}{\sim} \ar@{.>}[rr]^{+}&& \frac{V}{\sim} }\end{xy}$	Note that here: $\pi:V\rightarrow\frac{V}{\sim}$ given by $\pi:v\mapsto [v]$ The diagonal arrow labeled as $\pi\circ+$ exists by composition of the: $+:V\times V\rightarrow V$ arrow with $\pi:V\rightarrow\frac{V}{\sim}$ arrow. $\pi\times\pi:V\times V\rightarrow\frac{V}{\sim}\times\frac{V}{\sim}$ is simply the function: $\pi\times\pi:(u,v)\mapsto([u],[v])$
$+:\frac{V}{\sim}\times\frac{V}{\sim}\rightarrow\frac{V}{\sim}$ is given by $\pi\circ+\circ(\pi\times\pi)^{-1}$ . This means that $[u]+[v]=\pi(\pi^{-1}([u])+\pi^{-1}([v]))=\underbrace{[x\in\pi^{-1}([u])+y\in\pi^{-1}([v])]}_\text{Well-defined-ness}=[u+v]$ ^{[Note 1]}

Note that:

The dashed arrow labeled $+$ denotes the induced binary operation on $\frac{V}{\sim}$ , in the context of factoring functions we often write the function induced by $f$ as $\tilde{f}$ however (as usual) the meaning of addition is given by the context, so it is not ambiguous to define addition of $\frac{V}{\sim}$ where addition on $V$ is already defined.
The 'well-defined-ness' 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

Diagram	Key
$\begin{xy}\xymatrix{ \mathbb{F}\times V \ar@{->}[rr]^{} \ar@{->}[d]^{i\times\pi} \ar@{->}[drr]^{\pi\circ} & & V \ar@{->}[d]^\pi \\ \mathbb{F}\times\frac{V}{\sim} \ar@{.>}[rr]^{*} & & \frac{V}{\sim} }\end{xy}$	Note that here: $:\mathbb{F}\times V\rightarrow V$ denotes scalar multiplication, that is: $:(\alpha,v)\mapsto \alpha v$ Again the diagonal arrow, $\pi\circ*$ is the composition of the top and right arrows
$:\mathbb{F}\times\frac{V}{\sim}\rightarrow\frac{V}{\sim}$ is given by $\pi\circ\circ(i\times\pi)^{-1}$ . That is $\alpha[v]=\pi(\alpha\pi^{-1}([v]))=\underbrace{[\alpha x\ \text{for }x\in\pi^{-1}([v])]}_\text{well-defined-ness}=[\alpha v]$ ^{[Note 2]}

Overview of proofs

Usually we simply say:

Addition defined by:
$[v]+[u]=[v+u]$ and check it is well defined (this is to check that whichever representatives we choose of $a\in[u]$ and $b\in[v]$ that $[a+b]=[u+v]$ still
Scalar multiplication defined by:
$\alpha[v]=[\alpha v]$ and again, check this is well defined (that is for whichever $a\in[v]$ we choose to represent $[v]$ that $[\alpha a]=[\alpha v]$

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

[Expand]

Claim 1: $v\sim v'$ is indeed an equivalence relation

[Expand]

Claim 2: The diagram for addition commutes

[Expand]

Claim 1: The diagram for multiplication commutes

To-do 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

Jump up ↑ This is where well-defined-ness comes into play, but the Factor (function) theorem already takes this into account. We abuse the notation when writing $\pi^{-1}$ 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
Jump up ↑ Note that $\pi^{-1}([v])$ is actually a set but as Factor (function) shows it doesn't matter what representative we take. This is an abuse of notation.

References

[1] Jump up ↑ This is where well-defined-ness comes into play, but the Factor (function) theorem already takes this into account. We abuse the notation when writing $\pi^{-1}$ 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

[2] Jump up ↑ Note that $\pi^{-1}([v])$ is actually a set but as Factor (function) shows it doesn't matter what representative we take. This is an abuse of notation.

[Note 1]

[Note 2]

Diagram	Key
$\begin{xy}\xymatrix{ \mathbb{F}\times V \ar@{->}[rr]^{} \ar@{->}[d]^{i\times\pi} \ar@{->}[drr]^{\pi\circ} & & V \ar@{->}[d]^\pi \\ \mathbb{F}\times\frac{V}{\sim} \ar@{.>}[rr]^{*} & & \frac{V}{\sim} }\end{xy}$	Note that here: $:\mathbb{F}\times V\rightarrow V$ denotes scalar multiplication, that is: $:(\alpha,v)\mapsto \alpha v$ Again the diagonal arrow, $\pi\circ*$ is the composition of the top and right arrows
$:\mathbb{F}\times\frac{V}{\sim}\rightarrow\frac{V}{\sim}$ is given by $\pi\circ\circ(i\times\pi)^{-1}$ . That is $\alpha[v]=\pi(\alpha\pi^{-1}([v]))=\underbrace{[\alpha x\ \text{for }x\in\pi^{-1}([v])]}_\text{well-defined-ness}=[\alpha v]$ ^{[Note 2]}

Quotient vector space

Contents

Definition

Diagram for addition on equivalence classes

Diagram for scalar multiplication

Overview of proofs

Proof of claims

To-do notes

Notes

References

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