Group Theory and Linear Algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $-x\text{.}$

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 $x=\underset{\_}{}$ 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 $n\in {\mathbb{Z}}_{>0}$ then B Base Case: $n=1$ Assume $n=1\text{.}$ To show: B $⋮$ Base Case: $n=2$ Assume $n=2\text{.}$ To show: B $⋮$ Base Case: $n=3$ Assume $n=3\text{.}$ To show: B $⋮$ Do enough base cases so that the pattern is clear. Induction step Assume that if $k\in {\mathbb{Z}}_{>0}$ 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 $n=5\text{)}$ 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 Then $⋮$ 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.

Proof

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 $V\text{.}$ $f$ is an isometry if $f^{*}f={\text{id}}_V\text{.}$

§3.8.4 p. 118 of Groves-Hodgson:
An isometry of ${𝔼}^n$ is a function $f:{ℝ}^n\to {ℝ}^n$ such that if $x,y\in {ℝ}^n$ then $$d(x,y)=d(f\left(x\right),f\left(y\right))\text{.}$$

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

Assume $f:{ℝ}^n\to {ℝ}^n$ is a linear transformation.

To show:

(a) If $f$ satisfies $f^{*}f=\text{id}$ then $f$ satisfies if $x,y\in {ℝ}^n$ then $d(x,y)=d(fx,fy)\text{.}$ (b) If $f$ satisfies if $x,y\in {ℝ}^n$ then $d(x,y)=d(fx,fy)$ then $f$ satisfies $f^{*}f=\text{id}\text{.}$

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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