# Linear Algebra II

## September 8, 2014

### Lecture 4

Filed under: 2014 Fall — Y.K. Lau @ 10:18 PM

Last week we learnt the definitions of vector spaces and subspaces. Today we want to introduce a new and important concept — quotient vector space.

To start with we need some fundamental concepts in set theory, that is, equivalence relation, equivalence class and quotient set. Given a set ${X}$, we may specify a rule, called relation, to link up pairs of elements in ${X}$. As there is no requirement on how to specify the rule, we shall not get a nice object to play with. Hence, we consider a special kind of relations that satisfy the following 3 properties:

1. (Reflexive)     ${\forall}$ ${x\in X}$, ${x\sim x}$.
2. (Symmetric)   ${\forall}$ ${x,y\in X}$, ${x\sim y}$ implies ${y\sim x}$.
3. (Transitive)     ${\forall}$ ${x,y,z\in X}$, ${x\sim y}$ & ${y\sim z}$ implies ${x\sim z}$.

A relation satisfying the above 3 properties is called an equivalence relation.

To help understanding, below are some examples and non-examples: let ${X}$ be the set of all men in Hong Kong,

1. For any ${x,y\in X}$, define ${x\sim y}$ iff ${x}$ is the father of ${y}$. This relation satisfies NONE of the 3 properties.
2. For any ${x,y\in X}$, define ${x\sim y}$ iff ${x}$ and ${y}$ have the relationship of father and son. Then ${\sim}$ satisfies the symmetric property but not the others.
3. For any ${x,y\in X}$ define ${x\sim y}$ iff ${x}$ and ${y}$ have the same ancestor in the Tang dynasty. Now ${\sim}$ is an equivalence relation.

Next we define equivalence class and quotient set. Given a set ${X}$ with an equivalence relation ${\sim}$, we define for any ${x\in X}$,

$\displaystyle [x]=\{y\in X: \ y\sim x\}.$

This set ${[x]}$ is called the equivalence class of ${x}$. Then we have the following important result:

Given a set ${X}$ with the equivalence relation ${\sim}$, ${X}$ is a disjoint union of equivalence classes. (Often people say that ${X}$ is partitioned into equivalence classes.)

Finally, we define

$\displaystyle X/\!\!\sim\,\ =\{[x]: \ x\in X\}$

which is the quotient set of ${X}$ by ${\sim}$.

Recall that in lecture, we illustrated the concepts with the example ${{\mathbb N}}$ and the relation: ${x\sim y}$ iff ${x}$ and ${y}$ have the same parity (${x,y\in{\mathbb N}}$).

With these fundamental concepts, we return to our concerned object — quotient space.

Suppose ${V}$ is a vector space and ${W}$ is a subspace of ${V}$. For any ${x,y\in V}$, define the relation

${x\sim y}$ iff ${x-y\in W}$.

Claim 1. The relation ${\sim}$ is an equivalence relation.

Denote the quotient set ${V/\!\!\sim}$ by ${V/W}$. Now we impose two operations on ${V/W}$, defined as follows:

1. Addition:   ${\forall}$ ${[x],[y]\in V/W}$,

${[x]+[y]=\{a+b: \ a\in [x], \, b\in [y]\}}$.

2. Scalar multiplication:   ${\forall}$ ${\alpha\in{\mathbb R}}$,   ${\forall}$ ${[x]\in V/W}$,

${\alpha\cdot [x]=\{\alpha a: \ a\in [x]\}}$.

Claim 2. ${(V/W, +, \cdot\,)}$ is a vector space, called the quotient (vector) space of ${V}$ by ${W}$.

We haven’t completed the proof of Claim 2 yet and will do next lecture. To end this post, let us have an exercise.

Exercise: Consider ${{\mathbb R}^2}$ and ${W={\mathbb R}\begin{pmatrix} 1\\ 1\end{pmatrix}}$   (see Example 2 in p.297 of the textbook). Let

$\displaystyle \underline{a} = \begin{pmatrix} 1\\ 0\end{pmatrix}$,    ${\displaystyle\underline{b} = \begin{pmatrix} 3\\ 3\end{pmatrix}}$,   ${\displaystyle\underline{c} = \begin{pmatrix} 0\\ 1\end{pmatrix}}$.

1. Describe ${[a]}$ and ${[b]}$.
2. Draw out ${[a]}$, ${[b]}$ and ${[c]}$ if ${{\mathbb R}^2}$ is represented as the ${xy}$-plane.