Chapter 2: Matrices and Determinants
March 16, 2009
1
Linear transformations
Lecture 7
Definition 1.1.
Let
X
and
Y
be nonempty sets. A
function
from
X
to
Y
is a rule
f
:
X
→
Y
such that each element
x
in
X
is assigned a unique element
y
in
Y
, written as
y
=
f
(
x
);
the element
y
is called the
image
of
x
under
f
, and the element
x
is called the
preimage
of
f
(
x
).
Functions are also called
maps
, or
mappings
, or
transformations
.
Definition 1.2.
A function
T
:
R
n
→
R
m
is called a
linear transformation
if for any vectors
u
,
v
in
R
n
and scalar
c
,
(a)
T
(
u
+
v
) =
T
(
u
) +
T
(
v
),
(b)
T
(
c
u
) =
cT
(
u
).
Example 1.1.
(a) The function
T
:
R
2
→
R
2
, defined by
T
(
x
1
, x
2
) = (
x
1
+ 2
x
2
, x
2
), is a linear
transformation, see Figure 1
(1,0)
x
x
y
y
(0,0)
(0,1)
(1,1)
(0,0)
(1,0)
(2,1)
(3,1)
Figure 1: The geometric shape under a linear transformation.
(b) The function
T
:
R
2
→
R
3
, defined by
T
(
x
1
, x
2
) = (
x
1
+ 2
x
2
,
3
x
1
+ 4
x
2
,
2
x
1
+ 3
x
2
), is a linear
transformation.
(c) The function
T
:
R
3
→
R
2
, defined by
T
(
x
1
, x
2
, x
3
) = (
x
1
+ 2
x
2
+ 3
x
3
,
3
x
1
+ 2
x
2
+
x
3
), is a
linear transformation.
Example 1.2.
The transformation
T
:
R
n
→
R
m
by
T
(
x
) =
A
x
, where
A
is an
m
×
n
matrix, is
a linear transformation.
Example 1.3.
The map
T
:
R
n
→
R
n
, defined by
T
(
x
) =
λ
x
, where
λ
is a constant, is a linear
transformation, and is called the
dilation
by
λ
.
1