Group Theory and Linear Algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updated: 27 September 2014

Lecture 33: Revision: Proof machine

First important point: You cannot prove anything without having the definitions clearly stated.

Show that -(-5)=5 in the integers.

This problem is impossible without knowing, exactly, the definition of -x.

Types of proofs:

(1) To show: If A then B
Assume A
To show: B
(2) To show: There exists x such that C
Let x=_
To show: C
(3) To show: x such that C is unique.
Assume x satisfies C
Assume y satisfies C
To show: x=y
(4) Proofs by induction
To show: If n>0 then B
Base Case: n=1
Assume n=1.
To show: B

Base Case: n=2
Assume n=2.
To show: B

Base Case: n=3
Assume n=3.
To show: B

Do enough base cases so that the pattern is clear.
Induction step
Assume that if k>0 and k<n then B.
To show: B
This proof should be exactly the same as your last base case (say you did base cases to n=5) except with n replacing 5.
(5) Proofs by contradiction.
To show: If A and B and C then D
Assume A
Assume B
Assume C
Proof by contradiction.
Assume not D

Proceed until you derive a contradiction to something that was assumed.
Then not B
This is a contradiction to B
So if A and B and C then D.


A proof is the explanation of why something is true.

Proof machine is a way of formulating this explanation in an organised way which conforms to the conventions of logical sequencing.

A remark on the definition of isometry.

Definition 2.4.6 p. 56 of Groves-Hodgson:
Let f be a linear transformation on an inner product space V. f is an isometry if f*f=idV.

§3.8.4 p. 118 of Groves-Hodgson:
An isometry of 𝔼n is a function f:nn such that if x,yn then d(x,y)=d(f(x),f(y)).

Explain why these two uses of the term "isometry" are not inconsistent.

Assume f:nn is a linear transformation.

To show:
(a) If f satisfies f*f=id then f satisfies if x,yn then d(x,y)=d(fx,fy).
(b) If f satisfies if x,yn then d(x,y)=d(fx,fy) then f satisfies f*f=id.

Notes and References

These are a typed copy of Lecture 33 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on October 21, 2011.

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