Nullspaces of commuting transformations
: LinearAlg
Jonathan L.F. King
University of Florida, Gainesville FL 326112082, USA
[email protected]
Webpage
http://www.math.ufl.edu/
∼
squash/
26 October, 2011
(at
10:25
)
Abstract:
Conditions under which the nullspace of a
composition
B
◦
A
is the span of the two nullspaces. This
is then applied to constantcoefficient linear differential
equations.
§
A
Entrance
The
setting
is
a
vectorspace
V
and
two
linear
transformations
A
,
B
:
V
.
Use
both
Nul(
A
)
and
A
◦
for
the
nullspace
of
trn
A
.
Use
C
for the composition
C
:=
BA
.
1
:
Fact.
Suppose
B
,
A
:
V
→
V
. Then
Dim(
B
◦
)+Dim(
A
◦
)
1
0
:
>
Dim Nul(
BA
)
2
:
>
Dim(
A
◦
)
.
♦
Now suppose that
A
B
(
the trns commute
) so
that
C
=
BA
=
AB
. Then, automatically,
C
◦
⊃
Spn(
B
◦
,
A
◦
)
.
3
:
We explore when we have equality in (1
0
), and
when in (3).
For two subspaces
W
,
W
0
⊂
V
, let
W
⊥
W
0
mean that these
{
W
,
W
0
}
is a (
linearly
)
independent set
, as in (
*
), below.
[
In particular,
W
and
W
0
only intersect in the singleton
{
0
}
.
]
More
generally,
⊥
K
j
=1
W
j
indicates mutual independence
of the subspaces in that the
only
solution to
w
1
+
· · ·
+
w
K
=
0
with each
w
j
∈
W
j
,
*
:
is to have every
w
j
be
0
.
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 Fall '07
 JURY
 Calculus, Linear Algebra, Transformations, Vector Space, Nul, A◦

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