Linear Algebra Done Right
Sheldon Axler
Preface to the Instructor
You are probably about to teach a course that will give students their
second exposure to linear algebra. During their first brush with the
subject, your students probably worked with Euclidean spaces and
matrices. In contrast, this course will emphasize abstract vector
spaces and linear maps.
The audacious title of this book deserves an explanation. Almost all linear
algebra books use determinants to prove that every linear operator on a
finite-dimensional complex vector space has an eigenvalue.
Determinants are difficult, nonintuitive, and often defined without
motivation. To prove the theorem about existence of eigenvalues on
complex vector spaces, most books must define determinants, prove that
a linear map is not invertible if and only if its determinant
equals 0, and then define the characteristic polynomial. This
tortuous (torturous?) path gives students little feeling for why
eigenvalues must exist.
In contrast, the simple determinant-free proofs presented here offer more insight.
Once determinants have been banished to the end of the book, a new route
opens to the main goal of linear algebra—understanding the structure
of linear operators.
This book starts at the beginning of the subject, with no prerequisites
other than the usual demand for suitable mathematical maturity. Even
if your students have already seen some of the material in the first
few chapters, they may be unaccustomed to working exercises of the
type presented here, most of which require an understanding of proofs.
- Vector spaces are defined in Chapter 1, and their basic properties
are developed.
- Linear independence, span, basis, and dimension are defined in
Chapter 2, which presents the basic theory of finite-dimensional
vector spaces.
- Linear maps are introduced in Chapter 3. The key result
here is that for a linear map T, the dimension of the null
space of T plus the dimension of the range of T equals the
dimension of the domain of T.
- The part of the theory of polynomials that will be needed to understand linear
operators is presented in Chapter 4. If you take class time going
through the proofs in this chapter (which contains no linear algebra), then you
probably will not have time to cover some important aspects of linear
algebra. Your students will already be familiar with the theorems
about polynomials in this chapter, so you can ask them to read the
statements of the results but not the proofs. The
curious students will read some of the proofs anyway, which is why they are
included in the text.
- The idea of studying a linear operator by restricting it to small subspaces
leads in Chapter 5 to eigenvectors. The highlight of the chapter is a
simple proof that on complex vector spaces, eigenvalues always exist.
This result is then used to show that each linear operator on a
complex vector space has an upper-triangular matrix with respect to
some basis. Similar techniques are used to
show that every linear operator on a real vector space has an invariant
subspace of dimension 1 or 2. This result is used to prove that every linear
operator on an odd-dimensional real vector space has an eigenvalue. All this
is done without defining determinants or characteristic polynomials!
- Inner-product spaces are defined in Chapter 6, and their basic
properties are developed along with standard tools such as orthonormal bases,
the Gram-Schmidt procedure, and adjoints. This chapter also shows
how orthogonal projections can be used to solve certain minimization problems.
- The spectral theorem, which characterizes the linear operators for
which there exists an orthonormal basis consisting of eigenvectors, is
the highlight of Chapter 7. The work in earlier chapters pays
off here with especially simple proofs. This chapter also deals with
positive operators, linear isometries, the polar decomposition, and
the singular-value decomposition.
- The minimal polynomial, characteristic polynomial, and generalized
eigenvectors are introduced in Chapter 8. The
main achievement of this chapter is the description of a
linear operator on a complex vector space in terms of
its generalized eigenvectors. This description
enables one to prove almost all the results usually
proved using Jordan form. For example, these tools are
used to prove that every invertible linear operator on a
complex vector space has a square root. The chapter concludes with a proof
that every linear operator on a complex vector space can be put into Jordan
form.
- Linear operators on real vector spaces occupy center stage in
Chapter 9. Here two-dimensional invariant subspaces make up for the
possible lack of eigenvalues, leading to results analogous to those obtained on
complex vector spaces.
- The trace and determinant are defined in Chapter 10 in terms of
the characteristic polynomial (defined earlier without determinants).
On complex vector spaces, these definitions can be restated: the trace
is the sum of the eigenvalues and the determinant is the product of
the eigenvalues (both counting multiplicity). These easy-to-remember
definitions would not be possible with the traditional approach to
eigenvalues because that method uses determinants to prove that
eigenvalues exist. The standard theorems about determinants now
become much clearer. The polar decomposition and the characterization
of self-adjoint operators are used to derive the change of variables
formula for multi-variable integrals in a fashion that makes the
appearance of the determinant there seem natural.
This book usually develops linear algebra simultaneously for real and
complex vector spaces by letting F denote either the real or the
complex numbers. Abstract fields could be used instead, but to do so
would introduce extra abstraction without leading to any new linear
algebra. Another reason for restricting attention to the real and
complex numbers is that polynomials can then be thought of as genuine
functions instead of the more formal objects needed for polynomials
with coefficients in finite fields. Finally, even if the beginning
part of the theory were developed with arbitrary fields, inner-product
spaces would push consideration back to just real and complex vector
spaces.
Even in a book as short as this one (251 pages), you cannot expect to cover
everything. Going through the first eight chapters is an ambitious
goal for a one-semester course. If you must reach Chapter 10,
then I suggest covering Chapters 1, 2, and 4
quickly (students may have seen this material in earlier courses) and
skipping Chapter 9 (in which case you should discuss
trace and determinants only on complex vector spaces).
A more important goal than teaching any particular set of theorems is to develop
in students the ability to understand and manipulate the objects of linear
algebra. Mathematics can be learned only by doing; fortunately linear
algebra has many good homework problems. When teaching this course, I
usually assign two or three of the exercises each class, due the next
class. Going over the homework might take up a third or even half of a typical class.
A solutions manual for all the exercises is available (without charge)
only to instructors who are using this book as a textbook. To obtain the solutions
manual, instructors should send an e-mail request to me (axler@sfsu.edu) or contact
Springer if I am no longer around.
Have fun!