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.

 

 

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