Page 11 - Linear Algebra Done Right

Preface to the Instructor
• The minimal polynomial, characteristic polynomial, and general-
ized eigenvectors are introduced in Chapter 8. The main achieve-
ment of this chapter is the description of a linear operator on xi
a complex vector space in terms of its generalized eigenvectors.
This description enables one to prove almost all the results usu-
ally 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 determi-
nants). On complex vector spaces, these definitions can be re-
stated: the trace is the sum of the eigenvalues and the determi-
nant is the product of the eigenvalues (both counting multiplic-
ity). 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 po-
lar decomposition and the characterization of self-adjoint opera-
tors 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 al-
gebra. Another reason for restricting attention to the real and complex
numbers is that polynomials can then be thought of as genuine func-
tions instead of the more formal objects needed for polynomials with
coefficients in finite fields. Finally, even if the beginning part of the the-
ory were developed with arbitrary fields, inner-product spaces would
push consideration back to just real and complex vector spaces.