Quotient vector space
- Note: see Quotient for other types of quotient
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[hide]Definition
Given:
- A vector space (V,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
\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:
|
Diagram | Key |
---|---|
+:\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
\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:
|
Diagram | Key |
---|---|
*:\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
Claim 1: v\sim v' is indeed an equivalence relation
Claim 2: The diagram for addition commutes
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.