Linear Algebra Done Right

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Springer Science & Business Media, Jul 18, 1997 - Mathematics - 251 pages
This text for a second course in linear algebra is aimed at math majors and graduate students. The novel approach taken here banishes determinants to the end of the book and focuses on the central goal of linear algebra: understanding the structure of linear operators on vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. For example, the book presents--without having defined determinants--a clean proof that every linear operator on a finite-dimensional complex vector space (or an odd-dimensional real vector space) has an eigenvalue. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus, the text starts by discussing vector spaces, linear independence, span, basis, and dimension. Students are introduced to inner-product spaces in the first half of the book and shortly thereafter to the finite-dimensional spectral theorem. This second edition includes a new section on orthogonal projections and minimization problems. The sections on self-adjoint operators, normal operators, and the spectral theorem have been rewritten. New examples and new exercises have been added, several proofs have been simplified, and hundreds of minor improvements have been made throughout the text.
 

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For an analytic approach to Linear Algebra

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Contents

Vector Spaces
1
Complex Numbers
2
Definition of Vector Space
4
Properties of Vector Spaces
11
Subspaces
13
Sums and Direct Sums
14
Exercises
19
FiniteDimensional Vector Spaces
21
Orthogonal Projections and Minimization Problems
111
Linear Functionals and Adjoints
117
Exercises
122
Operators on InnerProduct Spaces
127
SelfAdjoint and Normal Operators
128
The Spectral Theorem
132
Normal Operators on Real InnerProduct Spaces
138
Positive Operators
144

Span and Linear Independence
22
Bases
27
Dimension
31
Exercises
35
Linear Maps
37
Definitions and Examples
38
Null Spaces and Ranges
41
The Matrix of a Linear Map
48
Invertibility
53
Exercises
59
Polynomials
63
Degree
64
Complex Coefficients
67
Real Coefficients
69
Exercises
73
Eigenvalues and Eigenvectors
75
Invariant Subspaces
76
Polynomials Applied to Operators
80
UpperTriangular Matrices
81
Diagonal Matrices
87
Invariant Subspaces on Real Vector Spaces
91
Exercises
94
InnerProduct Spaces
97
Inner Products
98
Norms
102
Orthonormal Bases
106
Isometries
147
Polar and SingularValue Decompositions
152
Exercises
158
Operators on Complex Sector Spaces
163
Generalized Eigenvectors
164
The Characteristic Polynomial
168
Decomposition of an Operator
173
Square Roots
177
The Minimal Polynomial
179
Jordan Form
183
Exercises
188
Operators on Real Vector Spaces
193
Eigenvalues of Square Matrices
194
Block UpperTriangular Matrices
195
The Characteristic Polynomial
198
Exercises
210
Trace and Determinant
213
Change of Basis
214
Trace
216
Determinant of an Operator
222
Determinant of a Matrix
225
Volume
236
Exercises
244
Symbol Index
247
Index
249
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