Friday January 23
−
Lecture 9 :
Nullspace, columnspace and range of a matrix.
(Refers to section 3,3)
Objectives: .
1.
Find Null
A
, Col
A
(Im
A
) of a matrix
A
. Similarly find Null
T
, Im
T
of a linear
transformation. .
2.
Define the
span
of a set of vectors {
v
1
,
v
2
,...,
v
m
}, span{
v
1
,
v
2
,...,
v
m
}. Define
spanning family.
3.
Find a spanning family for Null
A
and Col
A
.
9.1
Recall – Recall that, for the matrix
A
n x m
,
A
x
=
b
denotes a system of linear equations.
•
Viewed as
A
x
=
b
it is a matrix equation.
•
If
b
=
0
,
A
x
=
0
is called a
homogeneous system
.
•
The dimensions of the matrix
A
tell us that this homogeneous system must have
m
variables (or unknowns) and is made of
n
equations.
•
Furthermore,
A
can also be viewed as a function called a
linear transformation
(function) which maps points
x
in
R
m
to points
in
R
n
.
•
The solution set of
A
x
=
b
is the complete solution to the system.
•
The complete solution to the homogeneous system
A
x
=
0
is called the
Null space
of
A
and is denoted by
Null
A
.
9.1.1
Example
−
Find Null(
A
) for the following matrix
A
.
⎡
2
1
3
⎤
⏐
1
1/2
3/2
⏐
⎣
−
4
−
2
−
6
⎦
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•
We seek the solution to
A
x
=
0
where
x
= (
x
1
,
x
2
,
x
3
) .
•
For
A
in
RRE
F we have:
⎡
1
1/2
3/2
⎤
⏐
0
0
0
⏐
⎣
0
0
0
⎦
•
Hence the solution is
x
= (
x
1
,
x
2
,
x
3
) = (
(−
1/2)
x
2
−
(3/2)
x
3
,
x
2
,
x
3
)
=
(
(−
1/2)
x
2
,
x
2
, 0) + (
−
(3/2)
x
3
, 0,
x
3
)
=
x
2
(
−
1/2 , 1 ,
0) +
x
3
(
−
3/2 , 0,
1)
•
Thus Null(
A
) is the set
Null(
A
)
=
{
(
x
1
,
x
2
,
x
3
) :
(
x
1
,
x
2
,
x
3
) =
x
2
(
−
1/2 , 1 ,
0) +
x
3
(
−
3/2 , 0,
1),
x
2
,
x
3
in
R
}
.
•
We can also write:
Null(
A
)
=
{
x
in
R
3
:
x
=
α
(
−
1/2 , 1 ,
0) +
β
(
−
3/2 , 0,
1),
α
,
β
ranges over
R
}
.
9.1.2
Remarks
•
The
nullspace
of a matrix
A
is always a linear combination of vectors.
•
The nullspace of a matrix
A
is never empty since it always contains the vector
0
(called the
trivial solution
).
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 Winter '08
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 Linear Algebra, Algebra, Vector Space, ax, spanning family

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