Difference between revisions of "Quotient vector space"

Revision as of 15:51, 21 April 2016

Note: see Quotient for other types of quotient

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

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

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

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

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.

References