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: 20 September 2014
Lecture 7: Vector spaces and linear transformations
Let be a field. A vector space over is an abelian group with
a function
such that
(a) |
If and
then
|
(b) |
If then
|
(c) |
If and
then
|
(d) |
If and
then
|
Column vectors of length 3, is a vector space over
with
Vectors in is a vector space over
Let be a vector space over A subspace of is a
subset such that
(a) |
If then
|
(b) |
If then
|
(c) |
|
(d) |
If and then
|
Let be a vector space over Let and
be subspaces of Then
are subspaces of
Let be a vector space over Let be subspaces of
The subspaces are complementary if they satisfy
Write
if
and
are complementary
subspaces of
Linear transformations are for comparing vector spaces.
Let be a field and let and be vector spaces over
A linear transformation from to is a function
such that
(a) |
If then
|
(b) |
If and then
|
Let be a linear transformation. Then
(a) |
and
|
(b) |
if then
|
![](plus.gif) |
![](minus.gif) |
Proof. |
|
Assume is a linear transformation.
|
To show: |
(a) |
|
(b) |
If then
|
|
(a) |
To show:
Add to each side.
So
|
(b) |
Assume
To show: |
|
To show: |
(ba) |
|
(bb) |
|
|
(ba) |
|
(bb) |
This follows from (a) since is a commutative group.
|
|
|
|
Let be a linear transformation. The kernel of
or null space, of is
The image of is
Let be a linear transformation.
(a) |
is a subspace of
|
(b) |
is a subspace of
|
(c) |
if and only if
is injective.
|
(d) |
if and only if is surjective.
|
![](plus.gif) |
![](minus.gif) |
Proof. |
|
(a) |
To show: |
is a subspace of
|
To show: |
(aa) |
If
then
|
(ab) |
|
(ac) |
If then
|
(ad) |
If and
then
|
|
(aa) |
Assume
To show:
To show:
We know: and
So
|
(ab) |
To show:
To show:
Add to each side.
So
|
(ac) |
To show: If then
Assume
To show:
To show:
We know:
|
(ad) |
To show: if and then
Assume and
To show:
To show:
We know:
So is a subspace of
|
|
|
(c) |
To show: |
if and only if
is injective.
|
To show: |
(ca) |
If then
is injective.
|
(cb) |
If is injective then
|
|
(ca) |
Assume
To show: is injective.
To show: If and
then
Assume and
To show:
To show:
Since
then
Since
then
So is injective.
|
(cb) |
To show: If is injective then
Assume is injective.
To show:
To show: If and then
Assume and
To show:
We know:
Since is injective,
|
|
|
|
Notes and References
These are a typed copy of Lecture 7 from a series of handwritten lecture notes for the class Group Theory and Linear Algebra given on August 9, 2011.
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