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Contents 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-Dimensional 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v ix xiii xv 1 2 4 11 13 14 19 21 22 27 31 35 37 38 41 48 53 59

vi Contents Chapter 4 Polynomials Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Coefﬁcients Real Coefﬁcients . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 5 Eigenvalues and Eigenvectors Invariant Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . Polynomials Applied to Operators . . . . . . . . . . . . . . . . . Upper-Triangular Matrices . . . . . . . . . . . . . . . . . . . . . Diagonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . Invariant Subspaces on Real Vector Spaces . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 64 67 69 73 75 76 80 81 87 91 94 Chapter 6 Inner-Product Spaces 97 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Norms Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Orthogonal Projections and Minimization Problems . . . . . . 111 Linear Functionals and Adjoints . . . . . . . . . . . . . . . . . . 117 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Chapter 7 Operators on Inner-Product Spaces 127 Self-Adjoint and Normal Operators . . . . . . . . . . . . . . . . 128 The Spectral Theorem . . . . . . . . . . . . . . . . . . . . . . . . 132 Normal Operators on Real Inner-Product Spaces . . . . . . . . 138 Positive Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Polar and Singular-Value Decompositions . . . . . . . . . . . . 152 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Chapter 8 Operators on Complex Vector Spaces 163 Generalized Eigenvectors . . . . . . . . . . . . . . . . . . . . . . 164 . . . . . . . . . . . . . . . . . . . 168 The Characteristic Polynomial Decomposition of an Operator . . . . . . . . . . . . . . . . . . . 173

Contents vii Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 The Minimal Polynomial . . . . . . . . . . . . . . . . . . . . . . . 179 Jordan Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Chapter 9 Operators on Real Vector Spaces 193 Eigenvalues of Square Matrices . . . . . . . . . . . . . . . . . . . 194 Block Upper-Triangular Matrices . . . . . . . . . . . . . . . . . . 195 . . . . . . . . . . . . . . . . . . . 198 The Characteristic Polynomial Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Chapter 10 Trace and Determinant 213 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 . . . . . . . . . . . . . . . . . . . . 222 Determinant of an Operator Determinant of a Matrix . . . . . . . . . . . . . . . . . . . . . . . 225 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Symbol Index Index 247 249

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 op- erator on a ﬁnite-dimensional complex vector space has an eigenvalue. Determinants are difﬁcult, nonintuitive, and often deﬁned without mo- tivation. To prove the theorem about existence of eigenvalues on com- plex 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 of- fer 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 prerequi- sites 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. ix

x Preface to the Instructor • 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 un- derstand linear operators is presented in Chapter 4. If you take class time going through the proofs in this chapter (which con- tains 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, eigenval- ues always exist. This result is then used to show that each linear operator on a complex vector space has an upper-triangular ma- trix with respect to some basis. Similar techniques are used to show that every linear operator on a real vector space has an in- variant 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 ortho- normal 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 eigenvec- tors, 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 decom- position, and the singular-value decomposition.

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