Table Of ContentLinear Algebra II (MAT 3141)
Course notes written by Damien Roy
with the assistance of Pierre Bel
and financial support from the University of Ottawa
for the development of pedagogical material in French
(Translated into English by Alistair Savage)
Fall 2012
Department of Mathematics and Statistics
University of Ottawa
Contents
Preface v
1 Review of vector spaces 1
1.1 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Vector subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Direct sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Review of linear maps 13
2.1 Linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The vector space L (V,W) . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
K
2.3 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Matrices associated to linear maps . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Change of coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Endomorphisms and invariant subspaces . . . . . . . . . . . . . . . . . . . . 26
3 Review of diagonalization 33
3.1 Determinants and similar matrices . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Diagonalization of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Diagonalization of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Polynomials, linear operators and matrices 47
4.1 The ring of linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Evaluation at a linear operator or matrix . . . . . . . . . . . . . . . . . . . . 54
5 Unique factorization in euclidean domains 59
5.1 Divisibility in integral domains . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Divisibility in terms of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 Euclidean division of polynomials . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4 Euclidean domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.5 The Unique Factorization Theorem . . . . . . . . . . . . . . . . . . . . . . . 69
5.6 The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . 72
i
ii CONTENTS
6 Modules 75
6.1 The notion of a module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2 Submodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.3 Free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.4 Direct sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.5 Homomorphisms of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7 The structure theorem 91
7.1 Annihilators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.2 Modules over a euclidean domain . . . . . . . . . . . . . . . . . . . . . . . . 99
7.3 Primary decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.4 Jordan canonical form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8 The proof of the structure theorem 119
8.1 Submodules of free modules, Part I . . . . . . . . . . . . . . . . . . . . . . . 119
8.2 The Cayley-Hamilton Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.3 Submodules of free modules, Part II . . . . . . . . . . . . . . . . . . . . . . . 127
8.4 The column module of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.5 Smith normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.6 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9 Duality and the tensor product 155
9.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
9.2 Bilinear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
9.3 Tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
9.4 The Kronecker product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
9.5 Multiple tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
10 Inner product spaces 179
10.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
10.2 Orthogonal operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
10.3 The adjoint operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
10.4 Spectral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.5 Polar decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Appendices 205
A Review: groups, rings and fields 207
A.1 Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
A.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
A.3 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
A.4 Group homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
A.5 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
A.6 Subrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
A.7 Ring homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
A.8 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
CONTENTS iii
B The determinant 223
B.1 Multilinear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
B.2 The determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
B.3 The adjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
iv CONTENTS
Preface
These notes are aimed at students in the course Linear Algebra II (MAT 3141) at the
University of Ottawa. The first three chapters contain a revision of basic notions covered
in the prerequisite course Linear Algebra I (MAT 2141): vector spaces, linear maps and
diagonalization. The rest of the course is divided into three parts. Chapters 4 to 8 make up
the main part of the course. They culminate in the structure theorem for finite type modules
over a euclidean ring, a result which leads to the Jordan canonical form of operators on a
finite dimensional complex vector space, as well as to the structure theorem for abelian
groups of finite type. Chapter 9 involves duality and the tensor product. Then Chapter 10
treats the spectral theorems for operators on an inner product space (after a review of some
notions seen in the prerequisite course MAT 2141). With respect to the initial plan, they
still lack an eleventh chapter on hermitian spaces, which will be added later. The notes are
completed by two appendices. Appendix A provides a review of basic algebra that are used
in the course: groups, rings and fields (and the morphisms betwen these objects). Appendix
B covers the determinants of matrices with entries in an arbitrary commutative ring and
their properties. Often the determinant is simply defined over a field and the goal of this
appendix is to show that this notion easily generalizes to an arbitrary commutative ring.
This course provides students the chance to learn the theory of modules that is not
covered, due to lack of time, in other undergraduate algebra courses. To simplify matters
andmakethecourseeasiertofollow, Ihaveavoidedintroducingquotientmodules. Thistopic
could be added through exercises (for more motivated students). According to comments I
have received, students appreciate the richness of the concepts presented in this course and
that they open new horizons to them.
I thank Pierre Bel who greatly helped with the typing and updating of these notes. I also
´
thank the Rectorat aux Etudes of the University of Ottawa for its financial support of this
project via the funds for the development of French teaching materials. This project would
not have been possible without their help. Finally, I thank my wife Laura Dutto for her help
in the revision of preliminary versions of these notes and the students who took this course
with me in the Winter and Fall of 2009 for their participation and their comments.
Damien Roy (as translated by Alistair Savage) Ottawa, January 2010.
v
Chapter 1
Review of vector spaces
This chapter, like the two that follow it, is devoted to a review of fundamental concepts
of linear algebra from the prerequisite course MAT 2141. This first chapter concerns the
main object of study in linear algebra: vector spaces. We recall here the notions of a vector
space, vector subspace, basis, dimension, coordinates, and direct sum. The reader should
pay special attention to the notion of direct sum, since it will play a vital role later in the
course.
1.1 Vector spaces
Definition 1.1.1 (Vector space). A vector space over a field K is a set V with operations of
addition and scalar multiplication:
V ×V −→ V K ×V −→ V
and
(u,v) (cid:55)−→ u+v (a,v) (cid:55)−→ av
that satisfies the following axioms:
(cid:27)
VS1. u+(v+w) = (u+v)+w
for all u,v,w ∈ V.
VS2. u+v = v+u
VS3. There exists 0 ∈ V such that v+0 = v for all v ∈ V.
VS4. For all v ∈ V, there exists −v ∈ V such that v+(−v) = 0.
VS5. 1v = v
VS6. a(bv) = (ab)v
for all a,b ∈ K and u,v ∈ V.
VS7. (a+b)v = av+bv
VS8. a(u+v) = au+av
These eight axioms can be easily interpreted. The first four imply that (V,+) is an
abelian group (cf. Appendix A). In particular, VS1 and VS2 imply that the order in which
1
2 CHAPTER 1. REVIEW OF VECTOR SPACES
we add the elements v ,...,v of V does not affect their sum, denoted
1 n
n
(cid:88)
v +···+v or v .
1 n i
i=1
Condition VS3 uniquely determines the element 0 of V. We call it the zero!vector of V,
the term vector being the generic name used to designate an element of a vector space. For
each v ∈ V, there exists one and only one vector −v ∈ V that satisfies VS4. We call it the
additive inverse of v. The existence of the additive inverse allows us to define subtraction
in V by
u−v := u+(−v).
Axioms VS5 and VS6 involve only scalar multiplication, while the Axioms VS7 and
VS8 link the two operations. These last axioms require that multiplication by a scalar be
distributive over addition on the right and on the left (i.e. over addition in K and in V).
They imply general distributivity formulas:
(cid:32) (cid:33)
n n n n
(cid:88) (cid:88) (cid:88) (cid:88)
a v = a v and a v = av
i i i i
i=1 i=1 i=1 i=1
for all a,a ,...,a ∈ K and v,v ,...,v ∈ V.
1 n 1 n
We will return to these axioms later when we study the notion of a module over a
commutative ring, which generalizes the idea of a vector space over a field.
Example 1.1.2. Let n ∈ N . The set
>0
a
1
a
2
Kn = . ; a ,a ,...,a ∈ K
. 1 2 n
.
a
n
of n-tuples of elements of K is a vector space over K for the operations:
a b a +b a ca
1 1 1 1 1 1
a b a +b a ca
2 2 2 2 2 2
. + . = . and c . = . .
. . . . .
. . . . .
a b a +b a ca
n n n n n n
In particular, K1 = K is a vector space over K.
Example 1.1.3. More generally, let m,n ∈ N . The set
>0
a ··· a
11 1n
. .
Matm×n(K) = .. .. ; a11,...,amn ∈ K
a ... a
m1 mn
1.2. VECTOR SUBSPACES 3
of m×n matrices with entries in K is a vector space over K for the usual operations:
a ··· a b ··· b a +b ··· a +b
11 1n 11 1n 11 11 1n 1n
. . . . . .
.. .. + .. .. = .. ..
a ... a b ... b a +b ... a +b
m1 mn m1 mn m1 m1 mn mn
a ··· a ca ··· ca
11 1n 11 1n
. . . .
and c .. .. = .. .. .
a ... a ca ... ca
m1 mn m1 mn
Example 1.1.4. Let X be an arbitrary set. The set F(X,K) of functions from X to K is a
vector space over K when we define the sum of two functions f: X → K and g: X → K to
be the function f +g: X → K given by
(f +g)(x) = f(x)+g(x) for all x ∈ X,
and the product of f: X → K and a scalar c ∈ K to be the function cf: X → K given by
(cf)(x) = cf(x) for all x ∈ X.
Example 1.1.5. Finally, if V ,...,V are vector spaces over K, their cartesian product
1 n
V ×···×V = {(v ,...,v ); v ∈ V ,...,v ∈ V }
1 n 1 n 1 1 n n
is a vector space over K for the operations
(v ,...,v )+(w ,...,w ) = (v +w ,...,v +w ),
1 n 1 n 1 1 n n
c(v ,...,v ) = (cv ,...,cv ).
1 n 1 n
1.2 Vector subspaces
Fix a vector space V over a field K.
Definition 1.2.1 (Vector subspace). A vector subspace of V is a subset U of V satisfying the
following conditions:
SUB1. 0 ∈ U.
SUB2. If u,v ∈ U, then u+v ∈ U.
SUB3. If u ∈ U and c ∈ K, then cu ∈ U.
Conditions SUB2 and SUB3 imply that U is stable under the addition in V and stable
under scalar multiplication by elements of K. We thus obtain the operations
U ×U −→ U K ×U −→ U
and
(u,v) (cid:55)−→ u+v (a,v) (cid:55)−→ av
on U and we see that, with these operations, U is itself a vector space. Condition SUB1
can be replaced by U (cid:54)= ∅ because, if U contains an element u, then SUB3 implies that
0u = 0 ∈ U. Nevertheless, it is generally just as easy to verify SUB1. We therefore have
the following proposition.
4 CHAPTER 1. REVIEW OF VECTOR SPACES
Proposition 1.2.2. A vector subspace U of V is a vector space over K for the addition and
scalar multiplication restricted from V to U.
We therefore see that all vector subspaces of V give new examples of vector spaces. If
U is a subspace of V, we can also consider subspaces of U. However, we see that these are
simply subspaces of V contained in U. Thus, this does not give new examples. In particular,
the notion of being a subspace is transitive.
Proposition 1.2.3. If U is a subspace of V and W is a subspace of U, then W is a subspace
of V.
We can also form the sum and intersection of subspaces of V:
Proposition 1.2.4. Let U ,...,U be subspaces of V. The sets
1 n
U +···+U = {u +···+u ; u ∈ U ,...,u ∈ U } and
1 n 1 n 1 1 n n
U ∩···∩U = {u; u ∈ U ,...,u ∈ U },
1 n 1 n
called, respectively, the sum and intersection of U ,...,U , are subspaces of V.
1 n
Example 1.2.5. Let V and V be vector spaces over K. Then
1 2
U = V ×{0} = {(v ,0); v ∈ V } and U = {0}×V = {(0,v ); v ∈ V }
1 1 1 1 1 2 2 2 2 2
are subspaces of V ×V . We see that
1 2
U +U = V ×V and U ∩U = {(0,0)}.
1 2 1 2 1 2
1.3 Bases
In this section, we fix a vector space V over a field K and elements v ,...,v of V.
1 n
Definition 1.3.1 (Linear combination). We say that an element v of V is a linear combination
of v ,...,v if there exist a ,...,a ∈ K such that
1 n 1 n
v = a v +···+a v .
1 1 n n
We denote by
(cid:104)v ,...,v (cid:105) = {a v +···+a v ; a ,...,a ∈ K}
1 n K 1 1 n n 1 n
thesetoflinearcombinationsofv ,...,v . ThisisalsosometimesdenotedSpan {v ,...,v }.
1 n K 1 n
The identities
0v +···+0v = 0
1 n
n n n
(cid:88) (cid:88) (cid:88)
a v + b v = (a +b )v
i i i i i i i
i=1 i=1 i=1
n n
(cid:88) (cid:88)
c a v = (ca )v
i i i i
i=1 i=1
