Last updated: 27 September 2014
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 $x=\underset{\_}{\phantom{\rule{1em}{0ex}}}$ 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 $n\in {\mathbb{Z}}_{>0}$ then B Base Case: $n=1$ Assume $n=1\text{.}$ To show: B $\phantom{\rule{1em}{0ex}}\vdots $ Base Case: $n=2$ Assume $n=2\text{.}$ To show: B $\phantom{\rule{1em}{0ex}}\vdots $ Base Case: $n=3$ Assume $n=3\text{.}$ To show: B $\phantom{\rule{1em}{0ex}}\vdots $ 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 $\phantom{\rule{1em}{0ex}}\vdots $ 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}}\vdots $ 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 GrovesHodgson:
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 GrovesHodgson:
An isometry of ${\mathbb{E}}^{n}$ is a function $f:{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$
such that if $x,y\in {\mathbb{R}}^{n}$ 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 $f:{\mathbb{R}}^{n}\to {\mathbb{R}}^{n}$
is a linear transformation.
To show: 

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.