Linear Algebra

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Springer Science & Business Media, Jan 26, 1987 - Mathematics - 285 pages
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"Linear Algebra" is intended for a one-term course at the junior or senior level. It begins with an exposition of the basic theory of vector spaces and proceeds to explain the fundamental structure theorem for linear maps, including eigenvectors and eigenvalues, quadratic and hermitian forms, diagnolization of symmetric, hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form. The book also includes a useful chapter on convex sets and the finite-dimensional Krein-Milman theorem. The presentation is aimed at the student who has already had some exposure to the elementary theory of matrices, determinants and linear maps. However the book is logically self-contained. In this new edition, many parts of the book have been rewritten and reorganized, and new exercises have been added.
 

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Contents

Vector Spaces
1
1 DEFINITIONS
2
2 BASES
10
3 DIMENSION OF A VECTOR SPACE
15
4 SUMS AND DIRECT SUMS
19
Matrices
23
2 LINEAR EQUATIONS
29
3 MULTIPLICATION OF MATRICES
31
8 INVERSE OF A MATRIX
174
9 THE RANK OF A MATRIX AND SUBDETERMINANTS
177
Symmetric Hermitian and Unitary Operators
180
2 HERMITIAN OPERATORS
184
3 UNITARY OPERATORS
188
Eigenvectors and Eigenvalues
194
2 THE CHARACTERISTIC POLYNOMIAL
200
3 EIGENVALUES AND EIGENVECTORS OF SYMMETRIC MATRICES
213

Linear Mappings
43
2 LINEAR MAPPINGS
51
3 THE KERNEL AND IMAGE OF A LINEAR MAP
59
4 COMPOSITION AND INVERSE OF LINEAR MAPPINGS
66
5 GEOMETRIC APPLICATIONS
72
Linear Maps and Matrices
81
2 THE MATRIX ASSOCIATED WITH A LINEAR MAP
82
3 BASES MATRICES AND LINEAR MAPS
87
Scalar Products and Orthogonality
95
2 ORTHOGONAL BASES POSITIVE DEFINITE CASE
103
3 APPLICATION TO LINEAR EQUATIONS THE RANK
113
4 BILINEAR MAPS AND MATRICES
118
5 GENERAL ORTHOGONAL BASES
123
6 THE DUAL SPACE AND SCALAR PRODUCTS
125
7 QUADRATIC FORMS
132
8 SYLVESTERS THEOREM
135
Determinants
140
2 EXISTENCE OF DETERMINANTS
143
3 ADDITIONAL PROPERTIES OF DETERMINANTS
150
4 CRAMERS RULE
157
5 TRIANGULATION OF A MATRIX BY COLUMN OPERATIONS
161
6 PERMUTATIONS
163
7 EXPANSION FORMULA AND UNIQUENESS OF DETERMINANTS
168
4 DIAGONALIZATION OF A SYMMETRIC LINEAR MAP
218
5 THE HERMITIAN CASE
225
6 UNITARY OPERATORS
227
Polynomials and Matrices
231
2 POLYNOMIALS OF MATRICES AND LINEAR MAPS
233
Triangulation of Matrices and Linear Maps
237
2 THEOREM OF HAMILTONCAYLEY
241
3 DIAGONALIZATION OF UNITARY MAPS
243
Polynomials and Primary Decomposition
245
2 GREATEST COMMON DIVISOR
248
3 UNIQUE FACTORIZATION
251
4 APPLICATION TO THE DECOMPOSITION OF A VECTOR SPACE
255
5 SCHURS LEMMA
260
6 THE JORDAN NORMAL FORM
262
Convex Sets
268
2 SEPARATING HYPERPLANES
270
3 EXTREME POINTS AND SUPPORTING HYPERPLANES
272
4 THE KREINMILMAN THEOREM
274
Complex Numbers
277
Iwasawa Decomposition and Others
283
Index
293
Copyright

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References to this book

Algebra
Serge Lang
Limited preview - 2005
Algebra
Serge Lang
Limited preview - 2005

About the author (1987)

Professeur a Yale University

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