Linear Algebra Notes
Chapter 19
KERNEL AND IMAGE OF A MATRIX
Take an
n
×
m
matrix
A
=
a
11
a
12
· · ·
a
1
m
a
21
a
22
· · ·
a
2
m
.
.
.
.
.
.
.
.
.
a
n
1
a
n
2
· · ·
a
nm
and think of it as a function
A
:
R
m
→
R
n
.
The
kernel
of
A
is defined as
ker
A
= set of all
x
in
R
m
such that
A
x
=
0
.
Note that ker
A
lives in
R
m
.
The
image
of
A
is
im
A
= set of all vectors in
R
n
which are
A
x
for some
x
∈
R
m
.
Note that im
A
lives in
R
n
. Many calculations in linear algebra boil down to the
computation of kernels and images of matrices.
Here are some different ways of
thinking about ker
A
and im
A
.
In terms of equations, ker
A
is the set of solution vectors
x
= (
x
1
, . . . , x
m
) in
R
m
of the
n
equations
a
11
x
1
+
a
12
x
2
+
· · ·
+
a
1
m
x
m
= 0
a
21
x
1
+
a
22
x
2
+
· · ·
+
a
2
m
x
m
= 0
.
.
.
a
n
1
x
1
+
a
n
2
x
2
+
· · ·
+
a
nm
x
m
= 0
,
(19a)
and im
A
consists of those vectors
y
= (
y
1
, . . . y
n
) in
R
n
for which the system
a
11
x
1
+
a
12
x
2
+
· · ·
+
a
1
m
x
m
=
y
1
a
21
x
1
+
a
22
x
2
+
· · ·
+
a
2
m
x
m
=
y
2
.
.
.
a
n
1
x
1
+
a
n
2
x
2
+
· · ·
+
a
nm
x
m
=
y
n
,
(19b)
1
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2
has a solution
x
= (
x
1
, . . . , x
m
).
A single equation
a
i
1
x
1
+
a
i
2
x
2
+
· · ·
+
a
im
x
m
=
y
i
is called a
hyperplane
in
R
m
. (So a line is a hyperplane in
R
2
, and a plane is a
hyperplane in
R
3
.)
Geometrically, ker
A
is the intersection of hyperplanes (19a),
and im
A
is the set of vectors (
y
1
, . . . , y
n
)
∈
R
n
for which the hyperplanes (19b)
intersect in at least one point.
If
x
and
x
are two solutions of (19b) for the same
y
, then
A
(
x

x
) =
A
x

A
x
=
y

y
=
0
,
so
x

x
belongs to the kernel of
A
.
If ker
A
= 0 (i.e., consists just of the zero
vector) then there can be at most one solution. In general, the bigger the kernel,
the more solutions there are to a given equation that has at least one solution.
Thus, if
x
is one solution, then all other solutions are obtained from
x
by adding a
vector from ker
A
.
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 Winter '07
 xu
 Linear Algebra, Algebra, Vector Space, Ker, dim ker

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