Lecture 7
Properties of incompatible systems
UCSD Math 171A: Numerical Optimization
Philip E. Gill
(
[email protected]
)
Wednesday, January 21, 2009
Question
Given
A
∈
R
m
×
n
, how do we characterize all vectors
w
∈
R
m
such
that
w
6∈
range(
A
)?
UCSD Center for Computational Mathematics
Slide 2/34, Wednesday, January 21, 2009
y
y
=
Ax
x
Set of all rhs vectors
R
n
R
m
unreachable part of
R
m
Notice that if
w
is not in range(
A
), then
w
+
y
is not in range(
A
)
either (even if
y
∈
range(
A
)!)
UCSD Center for Computational Mathematics
Slide 3/34, Wednesday, January 21, 2009
Example:
A
=
1
5
3

1

5

3
!
,
rank(
A
) = 1
range(
A
) =
±
y
:
y
=
α
a
1
=
α
²
1

1
³
for some
α
´
Observe that
z
=
1
1
!
6∈
range(
A
)
because there is no
α
such that
1
1
!
=
α
1

1
!
UCSD Center for Computational Mathematics
Slide 4/34, Wednesday, January 21, 2009
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View Full DocumentThe vectors
z
=
1
1
!
and
a
1
=
1

1
!
are independent.
Moreover, they are
orthogonal
z
T
a
1
=
±
1 1
²
1

1
!
= 0
But
a
1
spans all of range(
A
)
⇒
z
is orthogonal to
every
vector in range(
A
)
⇒
z
satisﬁes
A
T
z
= 0
UCSD Center for Computational Mathematics
Slide 5/34, Wednesday, January 21, 2009
Result
Let
z
be a
nonzero
vector that is orthogonal to every column of
A
=
±
a
1
a
2
···
a
n
²
i.e.,
z
T
a
j
= 0 for
j
= 1, 2, .
. . ,
n
. Then
z
6∈
range(
A
).
Proof: Let
z
∈
range(
A
) with
z
T
a
j
= 0 for
j
= 1, 2, .
. . ,
n
.
As
z
∈
range(
A
), there must be some
y
such that
z
=
Ay
, i.e.,
z
=
Ay
=
±
a
1
a
2
···
a
n
²
y
=
n
X
j
=1
a
j
y
j
If we form
z
T
z
, we get
z
T
z
=
z
T
Ay
=
n
X
j
=1
z
T
a
j
y
j
=
n
X
j
=1
(
z
T
a
j
)
y
j
= 0
UCSD Center for Computational Mathematics
Slide 6/34, Wednesday, January 21, 2009
⇒
z
T
z
= 0.
but
z
T
z
=
∑
n
j
=1
z
2
j
, so
z
= 0
⇒
z
= 0 is the only vector in range(
A
) with
z
T
a
j
= 0.
If we write the
n
identities
z
T
a
j
= 0 as components of a vector:
(
0 0
···
0
)
=
(
z
T
a
1
z
T
a
2
···
z
T
a
n
)
=
z
T
(
a
1
a
2
···
a
n
)
=
z
T
A
Taking the transpose does not change this identity, so
A
T
z
= 0
UCSD Center for Computational Mathematics
Slide 7/34, Wednesday, January 21, 2009
Deﬁnition (Nullspace of a matrix)
The set of
z
such that
A
T
z
= 0 is called the
null space
of
A
T
.
null(
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