Linear Algebra Math 310

Spring 2017

**A Note To Math Majors**

In math courses up through calculus, most students focus primarily
on
*procedures*, and on using those procedures to solve problems.
In
contrast, more advanced courses are much more concerned with the theory
of
mathematics. In these advanced courses, students are expected to pay
much
more attention to *what* is true (meaning definitions and
theorems)
and *why* it is true (meaning proofs), than was the case in
introductory
courses. For example, in Calculus 1 you learn how to take derivatives
and
use them to solve max/min problems. But in advanced calculus, you will
be
expected to understand the precise meaning of derivatives, and the
proofs
that the techniques used in calculus are valid. In calculus you use
with
little thought the fact that local maxima and minima occur at points
where
the derivative of a function is 0. In advanced calculus the focus is on
the
assumptions that go into this result, and how these assumptions are
used
in the proof. The point is that being a mathematics major means
learning
to discover and develop new mathematical results and procedures, not
just
learning to use the results and techniques that have been discovered by
others.

For most math majors, somewhere between calculus and the more advanced courses there is a transitional period. Years ago, linear algebra was a course mainly for math majors, and was designed to assist in the transition to a greater emphasis on theory. In more recent years, linear algebra courses have attracted a much more diverse audience because applications to other fields have become so widespread. In response to the changing needs of the students, linear algebra has become more like calculus with an increasing emphasis on procedures. Unfortunately, that makes the course less effective in helping math majors prepare for more advanced courses.

Many of the students in this linear algebra course are not math majors. They are primarily interested in understanding and mastering the procedures linear algebra provides for solving a range of problems. They do not necessarily need to be concerned about definitions and theorems and proofs. But YOU do. You need to begin paying much closer attention to how terms are defined, and how those definitions are used in deriving new results. One way to do this is to change your reading and study habits. Don't read primarily to understand how to do the exercises. Instead, as you read, keep a written list of new terms and definitions and notation, as well as the statements of theorems. When proofs are provided, read them carefully, and try to understand them on two levels: questioning why each assertion made along the way is true, and also questioning whether the combination of all these statements really does provide an irrefutable demonstration of the desired final result.

A second important tactic is to pay close attention to the more theoretical questions in each homework assignment. Some problems are clearly drill on specific procedures. They usually appear in groups made up of very similar problems with slight variations. There may be several of these groups in each section of the text. Drill problems usually ask you to solve something, or compute something, or find something. The more theoretical problems are more likely to ask you to prove something, or to explain why something is true. They may involve letters instead of specific numbers. Some of these problems are marked with an *.

I am asking all math majors, potential math majors, and anyone who
expects
to take more advanced math courses to make a special effort to work on
these
theoretical questions, and especially the * problems. We will probably
not have time to fully cover these problems in
class,
so YOU take the initiative of asking me outside of class about your
solutions.

This may seem to place an extra burden on math majors. My justification is that the linear algebra course has a different goal and significance for math majors than for students from other disciplines. Any extra work you do now will be important for your success in more advanced math courses. Moreover, this work will give you a good opportunity to gain familiarity with the theoretical aspects of mathematics, and will provide a preview of more advanced mathematics courses.