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 , we may specify a rule, called relation, to link up pairs of elements in . 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:

- (Reflexive) , .
- (Symmetric) , implies .
- (Transitive) , & implies .

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

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

- For any , define iff is the father of . This relation satisfies NONE of the 3 properties.
- For any , define iff and have the relationship of father and son. Then satisfies the symmetric property but
__not__the others. - For any define iff and have the same ancestor in the Tang dynasty. Now is an equivalence relation.

Next we define equivalence class and quotient set. Given a set with an equivalence relation , we define for any ,

This set is called the equivalence class of . Then we have the following important result:

Given a set with the equivalence relation , is a disjoint union of equivalence classes. (Often people say that is partitioned into equivalence classes.)

Finally, we define

which is the quotient set of by .

Recall that in lecture, we illustrated the concepts with the example and the relation: iff and have the same parity ().

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

Suppose is a vector space and is a subspace of . For any , define the relation

iff .

**Claim 1.** The relation is an equivalence relation.

Denote the quotient set by . Now we impose two operations on , defined as follows:

- Addition: ,
.

- Scalar multiplication: , ,
.

**Claim 2.** is a vector space, called the quotient (vector) space of by .

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 and (see Example 2 in p.297 of the textbook). Let

, , .

- Describe and .
- Draw out , and if is represented as the -plane.

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