# Linear Algebra Done Right

## 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!