This preview shows page 1. Sign up to view the full content.
IMAGE AND KERNEL
Math 21b, O. Knill
Homework: Section 3.1: 10,22,34,44,54,38*,48*
IMAGE. If
T
:
R
n
→
R
m
is a linear transformation, then
{
T
(
~x
)

~x
∈
R
n
}
is called the
image
of
T
.
If
T
(
~x
) =
A~x
, then the image of
T
is also called the image of
A
. We write im(
A
) or im(
T
).
EXAMPLES.
1) If
T
(
x, y, z
) = (
x, y,
0), then
T
(
~x
) =
A
x
y
z
=
1
0
0
0
1
0
0
0
0
x
y
z
. The image of
T
is the
x

y
plane.
2) If
T
(
x, y
) = (cos(
φ
)
x

sin(
φ
)
y,
sin(
φ
)
x
+ cos(
φ
)
y
) is a rotation in the plane, then the image of
T
is the whole
plane.
3) If
T
(
x, y, z
) =
x
+
y
+
z
, then the image of
T
is
R
.
SPAN. The
span
of vectors
~v
1
, . . . ,
~v
k
in
R
n
is the set of all combinations
c
1
~v
1
+
. . . c
k
~v
k
, where
c
i
are real
numbers.
PROPERTIES.
The image of a linear transformation
~x
7→
A~x
is the span of the column vectors of
A
.
The image of a linear transformation contains 0 and is closed under addition and scalar multiplication.
KERNEL. If
T
:
R
n
→
R
m
is a linear transformation, then the set
{
x

T
(
x
) = 0
}
is called the
kernel
of
T
.
If
T
(
~x
) =
A~x
, then the kernel of
T
is also called the kernel of
A
. We write ker(
A
) or ker(
T
).
EXAMPLES. (The same examples as above)
1) The kernel is the
z
axes. Every vector (0
,
0
, z
) is mapped to 0.
2) The kernel consists only of the point (0
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
 Math, Linear Algebra, Algebra

Click to edit the document details