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]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:
|
| 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:
|
| 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
Notes
- ↑ 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
- ↑ 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.
