## MAE 301 Fall 2002 Review for Midterm II

Understand the basis for the principle of mathematical
induction. Be able to derive the formulas for the sum of the
first `n` numbers and for the sum of the first
`n` squares. Understand why the "proof"
that any set of coins chosen from a
box will have the same denomination is not a correct induction
proof. Be able to use induction to prove elementary properties
of the Fibonacci sequence and to prove the
Binomial Theorem.
Be able to explain the connection between "long division"
and the ``Division Algorithm'' (the long division algorithm is
a schematic version of the *proof* of the Division Algorithm,
which is, strictly speaking, misnamed).

Understand why the Euclidean Algorithm takes as input
two positive integers `a,b` and gives their greatest
common divisor `(a,b)` as output. Be able to use the
Euclidean Algorithm to calculate greatest common divisors.
Be able to use the Euclidean Algorithm (by itself or in the
row-reduction form) to express `d = (a,b)`
in the form
`d = xa + yb`, with `x` and `y` integers.

Understand the statement and be able to prove this Theorem:
For any integer `b > 1` any positive integer has a
unique representation in base `b`. Be able to implement
the proof explicitly in switching back and forth between
decimal (base 10) and binary (base 2) representations of
given numbers.

Understand that the standard multiplication and long-division
algorithms work for numbers expressed in any base. Be able to
implement them in base 2 as well as in base 10.

Understand the definition of ``congruence transformation.''
We also call such a transformation an ``isometry.''

Be able to prove that a translation is an isometry. Be
able to prove that the composition of two translations is
a translation.

Be able to prove that a rotation is an isometry. Understand
how rotations about the origin in `R`^{2} are
represented by `2 x 2` matrices. Understand how writing
the matrix product for the composition of two rotations about
the origin leads to the addition formulas for `sin`
and `cos`. Be able to express a rotation about a point
`P` not the origin in terms of translations and a
rotation about the origin. Be able to implement this calculation
to give the `x,y`-coordinates of the image of a point
`(a,b)` after rotation by 45^{o} about the
point `P = (-1,2)` for example.

Be able to prove that a reflection is an isometry. Be able to
prove that the composition of two reflections is a rotation
(when the lines of reflection meet) or a translation (when the
lines are parallel). Be able to implement this calculation
to give the `x,y`-coordinates of the image of a point
`(a,b)` after reflection about the line `x-y=3`
and then reflection about the line `y=2x` for example.

November 30 2002