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 ff and only if its determinant equals O, 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.

Preface to the Instructor

Preface to the Student

Acknowledgments

CHAPTER 1

Vector Spaces

  Complex Numbers

  Definition of Vector Space

  Properties of Vector Spaces

  Subspaces

  Sums and Direct Sums

  Exercises

CHAPTER 2

Finite-Dimenslonal Vector Spaces

  Span and Linear Independence

  Bases

  Dimension

  Exercises

CHAPTER 3

Linear Maps

  Definitions and Examples

  Null Spaces and Ranges

  The Matrix of a Linear Map

  Invertibility

  Exercises

CHAPTER 4

Potynomiags

  Degree

  Complex Coefficients

  Real Coefflcients

  Exercises

CHAPTER 5

Eigenvalues and Eigenvectors

  lnvariant Subspaces

  Polynomials Applied to Operators

  Upper-Triangular Matrices

  Diagonal Matrices

  Invariant Subspaces on Real Vector Spaces

  Exercises

CHAPTER 6

Inner-Product spaces

  Inner Products

  Norms

  Orthonormal Bases

  Orthogonal Projections and Minimization Problems

  Linear Functionals and Adjoints

  Exercises

CHAPTER 7

Operators on Inner-Product Spaces

  Self-Adjoint and Normal Operators

  The Spectral Theorem

  Normal Operators on Real Inner-Product Spaces

  Positive Operators

  Isometries

  Polar and Singular-Value Decompositions

  Exercises

CHAPTER 8

Operators on Complex Vector Spaces

  Generalized Eigenvectors

  The Characteristic Polynomial

  Decomposition of an Operator

  Square Roots

  The Minimal Polynomial

  Jordan Form 

  Exercises 

CHAPTER 9

Operators on Real Vector Spaces

  Eigenvalues of Square Matrices

  Block Upper-Triangular Matrices

  The Characteristic Polynomial 

  Exercises

CHAPTER 10

Trace and Determinant

  Change of Basis

  Trace

  Determinant of an Operator

  Determinant of a Matrix

  Volume

  Exercises

Symbol Index

Index