1. General comprehension questions:
1) What types of transformations do you know? Are all the transformations linear?
2) Try to remember theorems dealing with linear transformations.
3) How can you visualize linear transformations?
A list of words you might need for the answer:
To set up/establish a correspondence; a one-to-one correspondence; parallel displacement; an extended plane; inverse; translation; rotation; inversion; a homothetic transformation; decomposition
2. Learn to read the following formulas:
1) z→z+a
z goes over to z plus a;
2) z→Tz
z goes over to T of z;
3) w=
w is equal to the fraction with the numerator az plus b and the denominator cz plus d; or w is equal to az plus b over cz plus d;
4) ad-bc≠0
ad minus bc is not equal to zero;
5) z= - d/c
z is equal to minus d over c;
6) w=∞
w is equal to infinity;
7) T-1
T to the power of minus one;
8)
A two by two matrix with the elements λα,λb,λc,λd;
9) (T1T2)T3=T1(T2T3)
The product of T1and T2 multiplied by T3 is equal to T1 multiplied by the product of T2and T3;
10) |k|=0
The modulus of k is equal to zero;
11) k > 0
k is greater than zero;
12)
az plus b over cz pluc d is equal to bc minus ad over c squared multiplied by, parenthesis, z plus d over c, close parenthesis, plus a over c;
13) Tz=
T of z is equal to the fraction with the numerator z minus z sub two and the denominator z minus z sub four divided by the fraction with the numerator z sub three minus z sub two and the denominator a sub three minus z sub four;
14) (sz1,sz2,sz3,sz4)=TS-1(sz1)=Tz1=(z1,z2,z3,z4)
The cross ratio sz1,sz2,sz3,sz4 is equal to T multiplied by S to the power of minus one of sz1 is equal to T of z1, is equal to the cross ratio z1,z2,z3,z4;
15)
|w-
|=|
|
The modulus of w minus the fraction with the numerator a barred d minus c barred b and the denominator a barred c minus c barred a is equal to the modulus of the fraction with the numerator ad minus bc and the denominator a barred c minus c barred a.
Read Text 1 aloud with the formulas:
Text 1.
A linear
transformation between two vector
spaces
and
is
a map
such
that the following hold:
1.
for
any vectors
and
in
,
and
2.
for
any scalar
.
A linear
transformation may or may not be injective
or surjective.
When
and
have
the same dimension,
it is possible for
to
be invertible, meaning there exists a
such
that
.
It is always the case that
.
Also, a linear transformation always maps lines
to lines
(or to zero).
The main
example of a linear transformation is given by matrix
multiplication.
Given an
matrix
,
define
,
where
is
written as a column
vector
(with
coordinates).
For example, consider
|
(1) |
then
is
a linear transformation from
to
,
defined by
|
(2) |
When
and
are
finite
dimensional, a general linear transformation can be written as a
matrix multiplication only after specifying a vector
space basis
for
and
.
When
and
have
an inner
product,
and their vector
space bases,
and
,
are orthonormal,
it is easy to write the corresponding matrix
.
In particular,
.
Note that when using the standard basis for
and
,
the
-th
column corresponds to the image of the
th
standard basis vector.
When
and
are
infinite
dimensional, then it is possible for a linear transformation to not
be continuous.
For example, let
be the space of polynomials in one variable, and
be the derivative.
Then
,
which is not continuous
because
while
does
not converge.
Linear two-dimensional transformations have a simple classification. Consider the two-dimensional linear transformation
|
|
|
(3) |
|
|
|
(4) |
Now rescale
by defining
and
.
Then the above equations become
|
(5) |
where
and
,
,
,
and
are
defined in terms of the old constants. Solving
for
gives
|
(6) |
so the
transformation is one-to-one.
To find the fixed
points
of the transformation, set
to obtain
|
(7) |
This gives two fixed points, which may be distinct or coincident. The fixed points are classified as follows.
variables |
type |
|
hyperbolic fixed point |
|
elliptic fixed point |
|
parabolic fixed point |
*Source: Rowland, Todd and Weisstein, Eric W. "Linear Transformation." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LinearTransformation.html