This preview shows page 1. Sign up to view the full content.
10/4/2003, LINEAR TRANSFORMATIONS
Math 21b, O. Knill
HOMEWORK. For Wednesday: Section 2.1: 6,14,28,42,44,34*,(2430)*
TRANSFORMATIONS. A
transformation
T
from a set
X
to a set
Y
is a rule, which assigns to every element
in
X
an element
y
=
T
(
x
) in
Y
.
One calls
X
the
domain
and
Y
the
codomain
.
A transformation is also
called a
map
.
LINEAR TRANSFORMATION. A map
T
from
R
n
to
R
m
is called a
linear transformation
if there is a
m
×
n
matrix
A
such that
T
(
~x
) =
A~x
.
EXAMPLES.
•
To the linear transformation
T
(
x, y
) = (3
x
+4
y, x
+5
y
) belongs the matrix
±
3
4
1
5
²
. This transformation
maps the plane onto itself.
•
T
(
x
) =

3
x
is a linear transformation from the real line onto itself. The matrix is
A
= [

3].
•
To
T
(
~x
) =
~
y
·
~x
from
R
3
to
R
belongs the matrix
A
=
~
y
=
³
y
1
y
2
y
3
´
. This 1
×
3 matrix is also called
a
row vector
. If the codomain is the real axes, one calls the map also a
function
. function de±ned on
space.
•
T
(
x
) =
x~
y
from
R
to
R
3
.
A
=
~
y
=
y
1
y
2
y
3
is a 3
×
1 matrix which is also called a
column vector
. The
map de±nes a line in space.
•
T
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.
 Spring '03
 JUDSON
 Linear Algebra, Algebra, Transformations

Click to edit the document details