Quotient vector space

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Note: see Quotient for other types of quotient
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We define an:

  • Equivalence relation on [ilmath]V[/ilmath] defined as:
    • [ilmath]v\sim v'[/ilmath] if [ilmath]v-v'\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:
  • [ilmath]\pi:V\rightarrow\frac{V}{\sim} [/ilmath] given by [ilmath]\pi:v\mapsto [v][/ilmath]
  • The diagonal arrow labeled as [ilmath]\pi\circ+[/ilmath] exists by composition of the:
    1. [ilmath]+:V\times V\rightarrow V[/ilmath] arrow with
    2. [ilmath]\pi:V\rightarrow\frac{V}{\sim} [/ilmath] arrow.
  • [ilmath]\pi\times\pi:V\times V\rightarrow\frac{V}{\sim}\times\frac{V}{\sim} [/ilmath] is simply the function:
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{Well-defined-ness}=[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 '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

[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:
  • [ilmath]*:\mathbb{F}\times V\rightarrow V[/ilmath] denotes scalar multiplication, that is:
    • [ilmath]*:(\alpha,v)\mapsto \alpha v[/ilmath]
  • Again the diagonal arrow, [ilmath]\pi\circ*[/ilmath] is the composition of the top and right arrows
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{well-defined-ness}=[\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

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


  1. 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 [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
  2. 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.