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Introduction to Linear Algebra Serge Lang [PDF]

Introduction to Linear Algebra (2E) by Serge Lang

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Introduction to Linear Algebra (2E) written by Serge Lang
This book is meant as a short text in linear algebra for a one-term course. Except for an occasional example or exercise the text is logically independent of calculus, and could be taught early. In practice, Serge Lang expect it to be used mostly for students who have had two or three terms of calculus. The course could also be given simultaneously with, or immediately after, the first course in calculus.
Serge Lang have included some examples concerning vector spaces of functions, but these could be omitted throughout without impairing the understanding of the rest of the book, for those who wish to concentrate exclusively on euclidean space. Furthermore, the reader who does not like n = n can always assume that n = 1, 2, or 3 and omit other interpretations.
However, such a reader should note that using n = n simplifies some formulas, say by making them shorter, and should get used to this as rapidly as possible. Furthermore, since one does want to cover both the case n = 2 and n = 3 at the very least, using n to denote either number avoids very tedious repetitions.
(Serge Lang)

Book Detail :-
Title: Introduction to Linear Algebra
Edition: 2nd
Author(s): Serge Lang
Publisher: Springer
Year: 1985
Pages: 303
Type: PDF
Language: English
ISBN: 9780387962054,0387962050
Country: US
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Author Serge Lang is Professor of Mathematics from Department of Mathematics, Yale University, New Haven, CT 06520, US.
Serge Lang was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the influential Algebra.
Lang was born in Saint-Germain-en-Laye, close to Paris, in 1927. Lang moved with his family to California as a teenager, where he graduated in 1943 from Beverly Hills High School. He subsequently graduated from the California Institute of Technology in 1946, and received a doctorate from Princeton University in 1951. He held faculty positions at the University of Chicago, Columbia University, and Yale University.
Lang studied under Emil Artin at Princeton University, writing his thesis on quasi-algebraic closure, and then worked on the geometric analogues of class field theory and diophantine geometry. Later he moved into diophantine approximation and transcendental number theory, proving the Schneider–Lang theorem. A break in research while he was involved in trying to meet 1960s student activism halfway caused him difficulties in picking up the threads afterwards. He wrote on modular forms and modular units, the idea of a 'distribution' on a profinite group, and value distribution theory. He made a number of conjectures in diophantine geometry: Mordell–Lang conjecture, Bombieri–Lang conjecture, Lang–Trotter conjecture, and the Lang conjecture on analytically hyperbolic varieties. He introduced the Lang map, the Katz–Lang finiteness theorem, and the Lang–Steinberg theorem (cf. Lang's theorem) in algebraic groups.
Lang was a prolific writer of mathematical texts, often completing one on his summer vacation. Most are at the graduate level. He wrote calculus texts and also prepared a book on group cohomology for Bourbaki. Lang's Algebra, a graduate-level introduction to abstract algebra, was a highly influential text that ran through numerous updated editions. His Steele prize citation stated, "Lang's Algebra changed the way graduate algebra is taught...It has affected all subsequent graduate-level algebra books." It contained ideas of his teacher, Artin; some of the most interesting passages in Algebraic Number Theory also reflect Artin's influence and ideas that might otherwise not have been published in that or any form.

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Book Contents :-
Introduction to Linear Algebra (2E) written by Serge Lang cover the following topics.
1. Vectors
Definition of Points in Space, Located Vectors, Scalar Prod uct, The Norm of a Vector, Parametric Lines, Planes
2. Matrices and Linear Equations
Matrices, Multiplication of Matrices, Homogeneous Linear Equations and Elimination, Row Operations and Gauss Elimination, Row Operations and Elementary Matrices, Linear Combinations
3. Vector Spaces
Definitions, Linear Combinations, Convex Sets, Linear Independence, Dimension, The Rank of a Matrix
4. Linear Mappings
Mappings, Linear Mappings, The Kernel and Image of a Linear Map, The Rank and Linear Equations Again, The Matrix Associated with a Linear Map, Appendix: Change of Bases
5. Composition and Inverse Mappings
Composition of Linear Maps, Inverses
6. Scalar Products and Orthogonality
Scalar Products, Orthogonal Bases, Bilinear Maps and Matrices
7. Determinants
Determinants of Order 2, 3 x 3 and n x n Determinants, The Rank of a Matrix and Subdeterminants, Cramer's Rule, Inverse of a Matrix, Determinants as Area and Volume
8. Eigenvectors and Eigenvalues
Eigenvectors and Eigenvalues, The Characteristic Polynomial, Eigenvalues and Eigenvectors of Symmetric Matrices, Diagonalization of a Symmetric Linear Map

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