# Linear Algebra/Introduction

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book helps students to master the material of a standard undergraduate
linear algebra course.

The material is standard in that the topics covered are Gaussian
reduction, vector spaces, linear maps, determinants, and eigenvalues and
eigenvectors. The audience is also standard: sophomores or juniors,
usually with a background of at least one semester of calculus and
perhaps with as many as three semesters.

The help that it gives to students comes from taking a developmental
approach---this book\'s presentation emphasizes motivation and
naturalness, driven home by a wide variety of examples and extensive,
careful, exercises. The developmental approach is what sets this book
apart, so some expansion of the term is appropriate here.

Courses in the beginning of most mathematics programs reward students
less for understanding the theory and more for correctly applying
formulas and algorithms. Later courses ask for mathematical maturity:
the ability to follow different types of arguments, a familiarity with
the themes that underlie many mathematical investigations like
elementary set and function facts, and a capacity for some independent
reading and thinking. Linear algebra is an ideal spot to work on the
transition between the two kinds of courses. It comes early in a program
so that progress made here pays off later, but also comes late enough
that students are often majors and minors. The material is coherent,
accessible, and elegant. 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.

So, the aim of this book\'s exposition is to help students develop from
being successful at their present level, in classes where a majority of
the members are interested mainly in applications in science or
engineering, to being successful at the next level, that of serious
students of the subject of mathematics itself.

Helping students make this transition means taking the mathematics
seriously, so all of the results in this book are proved. On the other
hand, we cannot assume that students have already arrived, and so in
contrast with more abstract texts, we give many examples and they are
often quite detailed.

In the past, linear algebra texts commonly made this transition
abruptly. They began with extensive computations of linear systems,
matrix multiplications, and determinants. When the concepts---vector
spaces and linear maps---finally appeared, and definitions and proofs
started, often the change brought students to a stop. In this book,
while we start with a computational topic, linear reduction, from the
first we do more than compute. We do linear systems quickly but
completely, including the proofs needed to justify what we are
computing. Then, with the linear systems work as motivation and at a
point where the study of linear combinations seems natural, the second
chapter starts with the definition of a real vector space. This occurs
by the end of the third week.

Another example of our emphasis on motivation and naturalness is that
the third chapter on linear maps does not begin with the definition of
homomorphism, but with that of isomorphism. That\'s because this
definition is easily motivated by the observation that some spaces are
\"just like\" others. After that, the next section takes the reasonable
step of defining homomorphism by isolating the operation-preservation
idea. This approach loses mathematical slickness, but it is a good trade
because it comes in return for a large gain in sensibility to students.

One aim of a developmental approach is that students should feel
throughout the presentation that they can see how the ideas arise, and
perhaps picture themselves doing the same type of work.

The clearest example of the developmental approach taken here---and the
feature that most recommends this book---is the exercises. A student
progresses most while doing the exercises, so they have been selected
with great care. Each problem set ranges from simple checks to
reasonably involved proofs. Since an instructor usually assigns about a
dozen exercises after each lecture, each section ends with about twice
that many, thereby providing a selection. There are even a few problems
that are challenging puzzles taken from various journals, competitions,
or problems collections. (These are marked with a \"**?**\" and as part
of the fun, the original wording has been retained as much as possible.)
In total, the exercises 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 complete exclusion of others.
Applications and the role of the computer are important and vital
aspects of the subject. Consequently, each of this book\'s chapters
closes with a few application or computer-related topics. Some 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 difference
equations.

These topics are brief enough to be done in a day\'s class or to be
given as independent projects for individuals or small groups. Most
simply give the reader a taste of the subject, discuss how linear
algebra comes in, point to some further reading, and give a few
exercises. 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.

This book\'s emphasis on motivation and development make it a good
choice for self-study. But, while a professional instructor can judge
what pace and topics suit a class, if you are an independent student
then perhaps you would find some advice helpful.

Here are two timetables for a semester. The first focuses on core
material.

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One.II, the elements of vectors, usually covered in third semester
calculus).

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See the table of contents for the titles of these subsections.

To help you make time trade-offs, in the table of contents I have marked
subsections as optional if some instructors will pass over them in favor
of spending more time elsewhere. You might also try picking one or two
topics that appeal to you from the end of each chapter. You\'ll get more
from these if you have access to computer software that can do the big
calculations.

The most important advice is: do many exercises. The recommended
exercises are labeled throughout. (The answers are available.) You
should be aware, however, that few inexperienced people can write
correct proofs. Try to find a knowledgeable person to work with you on
this.

Finally, if I may, a caution for all students, independent or not: I
cannot overemphasize how much the statement that I sometimes hear, \"I
understand the material, but it\'s only that I have trouble with the
problems\" reveals a lack of understanding of what we are up to. Being
able to do things with the ideas is their point. The quotes below
express this sentiment admirably. They state what I believe is the key
to both the beauty and the power of mathematics and the sciences in
general, and of linear algebra in particular (I took the liberty of
formatting them as poems).

*I know of no better tactic*\
* than the illustration of exciting principles*\
*by well-chosen particulars.*\
        *\--Stephen Jay Gould*

*If you really wish to learn*\
* then you must mount the machine*\
* and become acquainted with its tricks*\
*by actual trial.*\
        *\--Wilbur Wright*

Jim Hefferon\
Mathematics, Saint Michael\'s College\
Colchester, Vermont USA 05439\
`http://joshua.smcvt.edu`\
2006-May-20

*Author\'s Note.* Inventing a good exercise, one that enlightens as well
as tests, is a creative act, and hard work.

The inventor deserves recognition. But for some reason texts have
traditionally not given attributions for questions. I have changed that
here where I was sure of the source. I would greatly appreciate hearing
from anyone who can help me to correctly attribute others of the
questions.



ja:線型代数学/序論
