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E-raamat: Advanced Linear Algebra

(University of California, Santa Cruz, USA)
  • Formaat: 622 pages
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 16-Dec-2015
  • Kirjastus: Chapman & Hall/CRC
  • Keel: eng
  • ISBN-13: 9781040055748
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Advanced Linear AlgebraSuurem pilt
  • Formaat: 622 pages
  • Sari: Textbooks in Mathematics
  • Ilmumisaeg: 16-Dec-2015
  • Kirjastus: Chapman & Hall/CRC
  • Keel: eng
  • ISBN-13: 9781040055748
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Advanced Linear Algebra, Second Edition takes a gentle approach that starts with familiar concepts and then gradually builds to deeper results. Each section begins with an outline of previously introduced concepts and results necessary for mastering the new material. By reviewing what students need to know before moving forward, the text builds a solid foundation upon which to progress.

The new edition of this successful text focuses on vector spaces and the maps between them that preserve their structure (linear transformations). Designed for advanced undergraduate and beginning graduate students, the book discusses the structure theory of an operator, various topics on inner product spaces, and the trace and determinant functions of a linear operator. It addresses bilinear forms with a full treatment of symplectic spaces and orthogonal spaces, as well as explains the construction of tensor, symmetric, and exterior algebras.

Featuring updates and revisions throughout, Advanced Linear Algebra, Second Edition:





Contains new chapters covering sesquilinear forms, linear groups and groups of isometries, matrices, and three important applications of linear algebra Adds sections on normed vector spaces, orthogonal spaces over perfect fields of characteristic two, and Clifford algebras Includes several new exercises and examples, with a solutions manual available upon qualifying course adoption

The book shows students the beauty of linear algebra while preparing them for further study in mathematics.

Arvustused

"This is the substantially extended second edition of a book comprising an advanced course in linear algebra " Zentralblatt MATH 1319

Praise for the First Edition:"The book is well written, and the examples are appropriate. Each section contains relevant problems at the end. The What You Need to Know feature at the beginning of each section outlining the knowledge required to grasp the material is useful. Summing Up: Recommended."

CHOICE, January 2011

"Pedagogically, a structural and general approach is taken, and topically, the material has been chosen in order to cover the material a beginning graduate student would be expected to know when taking a first course in group or field theory or functional analysis." SciTech Book News, February 2011

Preface to the Second Edition xiii
Preface to the First Edition xv
Acknowledgments xix
List of Figures
xxi
Symbol Description xxiii
1 Vector Spaces
1(46)
1.1 Fields
2(5)
1.2 The Space Fn
7(4)
1.3 Vector Spaces over an Arbitrary Field
11(4)
1.4 Subspaces of Vector Spaces
15(10)
1.5 Span and Independence
25(6)
1.6 Bases and Finite-Dimensional Vector Spaces
31(7)
1.7 Bases and Infinite-Dimensional Vector Spaces
38(4)
1.8 Coordinate Vectors
42(5)
2 Linear Transformations
47(40)
2.1 Introduction to Linear Transformations
48(8)
2.2 The Range and Kernel of a Linear Transformation
56(8)
2.3 The Correspondence and Isomorphism Theorems
64(4)
2.4 Matrix of a Linear Transformation
68(7)
2.5 The Algebra of L(V, W) and Mmn(F)
75(6)
2.6 Invertible Transformations and Matrices
81(6)
3 Polynomials
87(18)
3.1 The Algebra of Polynomials
88(11)
3.2 Roots of Polynomials
99(6)
4 Theory of a Single Linear Operator
105(46)
4.1 Invariant Subspaces of an Operator
106(8)
4.2 Cyclic Operators
114(5)
4.3 Maximal Vectors
119(4)
4.4 Indecomposable Linear Operators
123(7)
4.5 Invariant Factors and Elementary Divisors
130(9)
4.6 Canonical Forms
139(7)
4.7 Operators on Real and Complex Vector Spaces
146(5)
5 Normed and Inner Product Spaces
151(56)
5.1 Inner Products
152(4)
5.2 Geometry in Inner Product Spaces
156(8)
5.3 Orthonormal Sets and the Gram-Schmidt Process
164(8)
5.4 Orthogonal Complements and Projections
172(7)
5.5 Dual Spaces
179(5)
5.6 Adjoints
184(7)
5.7 Normed Vector Spaces
191(16)
6 Linear Operators on Inner Product Spaces
207(30)
6.1 Self-Adjoint and Normal Operators
208(4)
6.2 Spectral Theorems
212(5)
6.3 Normal Operators on Real Inner Product Spaces
217(6)
6.4 Unitary and Orthogonal Operators
223(7)
6.5 Polar and Singular Value Decomposition
230(7)
7 Trace and Determinant of a Linear Operator
237(34)
7.1 Trace of a Linear Operator
238(6)
7.2 Determinant of a Linear Operator and Matrix
244(18)
7.3 Uniqueness of the Determinant of a Linear Operator
262(9)
8 Bilinear Forms
271(52)
8.1 Basic Properties of Bilinear Maps
272(11)
8.2 Symplectic Spaces
283(10)
8.3 Quadratic Forms and Orthogonal Space
293(14)
8.4 Orthogonal Space, Characteristic Two
307(9)
8.5 Real Quadratic Forms
316(7)
9 Sesquilinear Forms and Unitary Geometry
323(20)
9.1 Basic Properties of Sesquilinear Forms
324(9)
9.2 Unitary Space
333(10)
10 Tensor Products
343(56)
10.1 Introduction to Tensor Products
345(10)
10.2 Properties of Tensor Products
355(9)
10.3 The Tensor Algebra
364(9)
10.4 The Symmetric Algebra
373(6)
10.5 The Exterior Algebra
379(8)
10.6 Clifford Algebras, char F ≠ 2
387(12)
11 Linear Groups and Groups of Isometries
399(62)
11.1 Linear Groups
400(8)
11.2 Symplectic Groups
408(14)
11.3 Orthogonal Groups, char F ≠ 2
422(18)
11.4 Unitary Groups
440(21)
12 Additional Topics in Linear Algebra
461(48)
12.1 Matrix Norms
462(10)
12.2 The Moore-Penrose Inverse of a Matrix
472(8)
12.3 Nonnegative Matrices
480(13)
12.4 The Location of Eigenvalues
493(8)
12.5 Functions of Matrices
501(8)
13 Applications of Linear Algebra
509(42)
13.1 Least Squares
510(16)
13.2 Error Correcting Codes
526(15)
13.3 Ranking Webpages for Search Engines
541(10)
Appendix A Concepts from Topology and Analysis 551(4)
Appendix B Concepts from Group Theory 555(8)
Appendix C Answers to Selected Exercises 563(10)
Appendix D Hints to Selected Problems 573(14)
Bibliography 587(2)
Index 589
Bruce Cooperstein is a professor of mathematics at the University of California, Santa Cruz, USA. He was a visiting scholar at the Carnegie Foundation for the Advancement of Teaching (spring 2007) and a recipient of the Kellogg National Fellowship (19821985) and the Pew National Fellowship for Carnegie Scholars (19992000). Dr. Cooperstein has authored numerous papers in refereed mathematics journals.
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