Lauren Diamante
MAE 501 class notes: November 29, 2010
Review of Homework #8 Problems
Discussion of question #3- Given a point (x,y) and an angle , find the image of the
point (x,y) under the rotation R (C, ) .
The class tackled this problem in three different ways:
 use a matrix.
 One way was to
cos sin  x 

 = x cos  ysin , xsin   ycos  by matrix multiplication.
sin  cos y 

How did people know that this matrix would produce a rotation? Many students
remembered
 it from a previous Linear Algebra course.

 Others decided to convert coordinates (x,y) into polar coordinates and was
able to define the rotation about an angle , by adding .
(x,y)~(r,) (r, +)
 Another student decided to decompose (x, y) as:
(x,y )= (x, 0) + (0, y)
Therefore taking the rotation of each part separately.
R (C, ) (x, 0) = (xcos, xsin)
R (C, ) (0, y) = (-ysin, ycos)


Then, the rotation of the point (x, y) is represented as adding the two rotations
together.
R (C, ) (x, y)= R (C, ) (x, 0) + R (C, ) (0, y)
This rotation is displayed through graphs below:



Lauren Diamante
By this step, the student is assuming that there exists Linearity within rotations.
Obviously, this is not true for all functions, but is stated true for rotations.
For example, it is not true for the sine function: sin(x+y)  sin(x) + sin(y)
To prove that rotations are linear mappings (i.e. that the graphs drawn above yield
the same result) you must use general points ( x1, y1) and ( x2 y2 ).

Everyone should try to write out the generalized proof that rotations are linear
mappings.


Discussion of question #1: Write out a clear, concise definition of an isometry of the
plane.
Some students did not write the direct definition from our classnotes, and
instead wrote different definitions of an isometry of a plane. Here are a couple of
examples which we discussed.
First Example:
Isometry: A linear map that preserves distance. For example, a rotation, a
reflection, etc.
Initially, one student questioned what does the statement “preserves
distance” include, and whether or not that includes preserving angle measurements.
The professor pointed out that although it does not explicitly state that preserving
distance includes angle measurements, it can be proved through the definition, thus
it is not necessary to state it in the definition.
Lauren Diamante
What everyone should focus on in this example of a definition should be the
term “linear map”. What does it mean to be a linear map? Should this be included in
the definition of an Isometry.
Let’s remember the definition from our class notes…
Isometry: An isometry of the plane is a distance preserving transformation of
the plane. It can be written as a mapping
Do you think that that the first definition is stronger or weaker than the definition
from our class notes? To answer, we need to consider two important questions:
1) Is every isometry a linear map?
2) Is every linear map an isometry?
We have decided earlier that rotations are a linear mapping. So let’s consider an
isometry, which is not a linear map. If we can find one isometry that is not a linear
map, then this definition is not valid.
Translations are not linear because the origin is moved! Thus, every isometry is not
a linear map!
Now, what about a linear map which is not an isometry. Let’s consider a matrix
linear mapping:
x  3x 
    This is a linear map, but it is not an isometry, thus not all linear maps
y  3y 
are isometries!

Although we have discovered two specific examples of linear maps, which are not
 isometries and isometries, which are not linear maps, there does exist isometries
which are linear maps and there does exist linear maps which are isometries.
Second example:
Isometry: Is a map from one metric space to another (that preserves distance).
[ A metric space has a distance function.]
The way this definition is written can be a bit confusing. First of all, we need
to understand what a metric space is and what a distance function is.
What is a metric space?
Many students are able describe metric spaces as the common planes we work
within, such as R, R2 , R 3 .

Lauren Diamante
What is a distance function?
Space
Formula
RR
R2  R2
R3  R3


Generalized Function
x1  x2 
2
2
x1  x2   y1  y2 
2
2
2
x1  x2   y1  y2   z
1  z2 
2

x1  x2
x1  x2  y1  y2
x1  x2  y1  y2  z1  z2


 According

to the website www.mathworld.wolfram.com,


A metric space is a set S, with a global distance function (the metric g) that, for every
two points(x, y) in S, gives the distance between them as a nonnegative real
number g(x, y). A metric space must also satisfy:
1) g(x, y)=0 iff x=y
2) g(x, y)= g(y, x)
3) The triangle inequality g(x,y) + g(y, z)  g(x, z)
In class, we defined properties of distance as:




d(x, y)= d(y, x) symmetry
d(x, y)= 0 iff x=y
d(x, y)  0
d(x, y) + d(y, z)  d(x, z)

All of these properties hold true for the generalized absolute value distance

function.

These properties are commonly used, and are important to teach our future high
school students, since these properties are useful to high school students especially
when solving word problems.
Discussion of question # 4:
The important thing for all students to realize about question #4 is that
everyone needs to make sure they are proving in a generalized way and not for two
specific lines. Many students may have meant to prove for two arbitrary
intersecting lines, but it was not clear in the proof.
Lauren Diamante
Interesting Statements from Homework:

“If rotating through an angle 60° repeatedly, you eventually get back to
where you started in 1 cycle”
This is true for 60°… but is it true for every angle rotation?
After some deliberation in class, we have decided that this is not always true.
For instance, looking at the rotation of 46°, we will not get back to where we
started.
z  cos  i sin 
z 2  (cos   i sin  )2



… z n  cos  i sin   = 1
n
This is not always true. After deliberation we have discovered that this does not
hold true for every angle.
For example, by plugging 1.05 for  , we can never reach 1.
This is because our  value must be a multiple of 2. Therefore we know, any
integer multiple of 2 will always be irrational and any integer multiple of 1.05 will
never be irrational, thus 
for 1.05 we will never reach 1.

Thoughts for students to consider:



Which set is bigger, the Rationals or the Irrationals?
Are both sets equally dense around the origin?
What is a clear, concise definition of density in a set?