Linear Algebra.pdf

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Linear Algebra
1 2
3 1
x
·
1 2
x
·
3 1
6
8
2
1
Jim Hefferon
Notation
R
N
C
{.
. . . . .
}
...
V, W, U
v, w
0, 0
V
B, D
E
n
=
e
1
, . . . , e
n
β, δ
Rep
B
(v)
P
n
M
n×m
[S]
M
⊕
N
V
∼
W
=
h, g
H, G
t, s
T, S
Rep
B,D
(h)
h
i,j
|T |
R(h), N
(h)
R
∞
(h),
N
∞
(h)
real numbers
natural numbers:
{0,
1, 2,
. . .
}
complex numbers
set of
. . .
such that
. . .
sequence; like a set but order matters
vector spaces
vectors
zero vector, zero vector of
V
bases
standard basis for
R
n
basis vectors
matrix representing the vector
set of
n-th
degree polynomials
set of
n×m
matrices
span of the set
S
direct sum of subspaces
isomorphic spaces
homomorphisms
matrices
transformations; maps from a space to itself
square matrices
matrix representing the map
h
matrix entry from row
i,
column
j
determinant of the matrix
T
rangespace and nullspace of the map
h
generalized rangespace and nullspace
Lower case Greek alphabet
name
alpha
beta
gamma
delta
epsilon
zeta
eta
theta
symbol
α
β
γ
δ
ζ
η
θ
name
iota
kappa
lambda
mu
nu
xi
omicron
pi
symbol
ι
κ
λ
µ
�½
ξ
o
π
name
rho
sigma
tau
upsilon
phi
chi
psi
omega
symbol
ρ
σ
τ
υ
φ
χ
ψ
ω
Cover.
This is Cramer’s Rule applied to the system
x
+ 2y = 6, 3x +
y
= 8. The area
of the first box is the determinant shown. The area of the second box is
x
times that,
and equals the area of the final box. Hence,
x
is the final determinant divided by the
first determinant.
Preface
In most mathematics programs linear algebra is taken in the first or second
year, following or along with at least one course in calculus. While the location
of this course is stable, lately the content has been under discussion. Some in-
structors have experimented with varying the traditional topics, trying courses
focused on applications, or on the computer. Despite this (entirely healthy)
debate, most instructors are still convinced, I think, that the right core material
is vector spaces, linear maps, determinants, and eigenvalues and eigenvectors.
Applications and computations certainly can have a part to play but most math-
ematicians agree that the themes of the course should remain unchanged.
Not that all is fine with the traditional course. Most of us do think that
the standard text type for this course needs to be reexamined. Elementary
texts have traditionally started with extensive computations of linear reduction,
matrix multiplication, and determinants. These take up half of the course.
Finally, when vector spaces and linear maps appear, and definitions and proofs
start, the nature of the course takes a sudden turn. In the past, the computation
drill was there because, as future practitioners, students needed to be fast and
accurate with these. But that has changed. Being a whiz at 5×5 determinants
just isn’t important anymore. Instead, the availability of computers gives us an
opportunity to move toward a focus on concepts.
This is an opportunity that we should seize. The courses at the start of
most mathematics programs work at having students correctly apply formulas
and algorithms, and imitate examples. Later courses require some mathematical
maturity: reasoning skills that are developed enough to follow different types
of proofs, a familiarity with the themes that underly many mathematical in-
vestigations like elementary set and function facts, and an ability to do some
independent reading and thinking, Where do we work on the transition?
Linear algebra is an ideal spot. It comes early in a program so that progress
made here pays off later. The material is straightforward, elegant, and acces-
sible. The students are serious about mathematics, often majors and minors.
There are a variety of argument styles—proofs by contradiction, if and only if
statements, and proofs by induction, for instance—and examples are plentiful.
The goal of this text is, along with the development of undergraduate linear
algebra, to help an instructor raise the students’ level of mathematical sophis-
tication. Most of the differences between this book and others follow straight
from that goal.
One consequence of this goal of development is that, unlike in many compu-
tational texts, all of the results here are proved. On the other hand, in contrast
with more abstract texts, many examples are given, and they are often quite
detailed.
Another consequence of the goal is that while we start with a computational
topic, linear reduction, from the first we do more than just compute. The
solution of linear systems is done quickly but it is also done completely, proving
i
everything (really these proofs are just verifications), all the way through the
uniqueness of reduced echelon form. In particular, in this first chapter, the
opportunity is taken to present a few induction proofs, where the arguments
just go over bookkeeping details, so that when induction is needed later (e.g., to
prove that all bases of a finite dimensional vector space have the same number
of members), it will be familiar.
Still another consequence is that the second chapter immediately uses this
background as motivation for the definition of a real vector space. This typically
occurs by the end of the third week. We do not stop to introduce matrix
multiplication and determinants as rote computations. Instead, those topics
appear naturally in the development, after the definition of linear maps.
To help students make the transition from earlier courses, the presentation
here stresses motivation and naturalness. An example is the third chapter,
on linear maps. It does not start with the definition of homomorphism, as
is the case in other books, but with the definition of isomorphism. That’s
because this definition is easily motivated by the observation that some spaces
are just like each other. After that, the next section takes the reasonable step of
defining homomorphisms by isolating the operation-preservation idea. A little
mathematical slickness is lost, but it is in return for a large gain in sensibility
to students.
Having extensive motivation in the text helps with time pressures. I ask
students to, before each class, look ahead in the book, and they follow the
classwork better because they have some prior exposure to the material. For
example, I can start the linear independence class with the definition because I
know students have some idea of what it is about. No book can take the place
of an instructor, but a helpful book gives the instructor more class time for
examples and questions.
Much of a student’s progress takes place while doing the exercises; the exer-
cises here work with the rest of the text. Besides computations, there are many
proofs. These are spread over an approachability range, from simple checks
to some much more involved arguments. There are even a few exercises that
are reasonably challenging puzzles taken, with citation, from various journals,
competitions, or problems collections (as part of the fun of these, the original
wording has been retained as much as possible). In total, the questions are
aimed to both build an ability at, and help students experience the pleasure of,
doing
mathematics.
Applications, and Computers.
The point of view taken here, that linear
algebra is about vector spaces and linear maps, is not taken to the exclusion
of all other ideas. Applications, and the emerging role of the computer, are
interesting, important, and vital aspects of the subject. Consequently, every
chapter closes with a few application or computer-related topics. Some of the
topics are: network flows, the speed and accuracy of computer linear reductions,
Leontief Input/Output analysis, dimensional analysis, Markov chains, voting
paradoxes, analytic projective geometry, and solving difference equations.
These are brief enough to be done in a day’s class or to be given as indepen-
ii
dent projects for individuals or small groups. Most simply give a reader a feel
for the subject, discuss how linear algebra comes in, point to some accessible
further reading, and give a few exercises. I have kept the exposition lively and
given an overall sense of breadth of application. In short, these topics invite
readers to see for themselves that linear algebra is a tool that a professional
must have.
For people reading this book on their own.
The emphasis on motivation
and development make this book a good choice for self-study. While a pro-
fessional mathematician knows what pace and topics suit a class, perhaps an
independent student would find some advice helpful. Here are two timetables
for a semester. The first focuses on core material.
week
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Mon.
1.I.1
1.I.3
1.III.1,
2.I.2
2.III.1,
2.III.2,
3.I.2
3.II.2
3.III.1
3.IV.2,
3.IV.4,
4.I.3
4.III.1
5.II.2
Wed.
1.I.1, 2
1.II.1
1.III.2
2.II
2.III.2
2.III.3
3.II.1
3.II.2
3.III.2
3.IV.4
3.V.1, 2
4.II
5.I
5.II.3
Fri.
1.I.2, 3
1.II.2
2.I.1
2.III.1
exam
3.I.1
3.II.2
3.III.1
3.IV.1, 2
exam
4.I.1, 2
4.II
5.II.1
review
2
2
3
3, 4
3.V.1
The second timetable is more ambitious (it presupposes 1.II, the elements of
vectors, usually covered in third semester calculus).
week
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Mon.
1.I.1
1.I.3
2.I.1
2.III.1
2.III.4
3.I.2
3.III.1
3.IV.2
3.V.1
3.VI.2
4.I.2
4.II
5.II.1, 2
5.III.2
Wed.
1.I.2
1.III.1, 2
2.I.2
2.III.2
3.I.1
3.II.1
3.III.2
3.IV.3
3.V.2
4.I.1
4.I.3
4.II, 4.III.1
5.II.3
5.IV.1, 2
Fri.
1.I.3
1.III.2
2.II
2.III.3
exam
3.II.2
3.IV.1, 2
3.IV.4
3.VI.1
exam
4.I.4
4.III.2, 3
5.III.1
5.IV.2
See the table of contents for the titles of these subsections.
iii

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