## Group Theory and Linear Algebra

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 $-\left(-5\right)=5$ 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 $x=\underset{_}{\phantom{\rule{1em}{0ex}}}$ 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 $n\in {ℤ}_{>0}$ then B Base Case: $n=1$ Assume $n=1\text{.}$ To show: B $\phantom{\rule{1em}{0ex}}⋮$ Base Case: $n=2$ Assume $n=2\text{.}$ To show: B $\phantom{\rule{1em}{0ex}}⋮$ Base Case: $n=3$ Assume $n=3\text{.}$ To show: B $\phantom{\rule{1em}{0ex}}⋮$ Do enough base cases so that the pattern is clear. Induction step Assume that if $k\in {ℤ}_{>0}$ and $k then B. To show: B $\phantom{\rule{1em}{0ex}}⋮$ 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 $\phantom{\rule{1em}{0ex}}⋮$ 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 $d(x,y)=d(f(x),f(y)).$

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