Group Theory and Linear Algebra
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last updated: 21 September 2014
Lecture 10: Eigenvectors and annihilators
Let be a vector space over Let
be a linear transformation.
The defined by is the
vector space with given by
if and
has basis
and the matrix
defines Then
An subspace, or
of
is a subspace such that
if then
Let The of
is
An eigenvector with eigenvalue is a vector
In our previous example,
So
So
So
The eigenspace of is
In our previous example,
and
if
So
The annihilator of is
The minimal polynomial of is
such that
where
In our previous example,
and
So
Let be a linear transformation.
(a) |
Let Then
is an subspace of
|
(b) |
Let Then
is an subspace of
|
(c) |
If
then
|
(d) |
If and
then
|
![](plus.gif) |
![](minus.gif) |
Proof. |
|
(a) |
To show: |
is an subspace of
|
To show: |
(aa) |
If
then
|
(ab) |
If and
then
|
(ac) |
If then
|
|
(aa) |
Assume
To show:
We know:
So we know:
|
(ab) |
Assume and
To show:
To show:
We know:
|
(ac) |
Assume
To show:
We know:
To show:
So is an subspace of
|
|
|
(d) |
Assume and
To show:
We know:
To show: If then
Assume
To show:
|
(c) |
Assume
To show:
To show: If then
Assume
To show:
|
|
Notes and References
These are a typed copy of Lecture 10 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on August 16, 2011.
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