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**About this book :- **
**Serge Lang's Linear Algebra (Solution) ** written by
** Rami Shakarchi **

The present volume contains all the exercises and their solutions of Lang's' Linear Algebra. Solving problems being an essential part of the learning process, my goal is to provide those learning and teaching linear algebra with a large number of worked out exercises. Lang's textbook covers all the topics in linear algebra that are usually taught at the undergraduate level: vector spaces, matrices and linear maps including eigenvectors and eigenvalues, determinants, diagonalization of symmetric and hermitian maps, unitary maps and matrices, triangulation, Jordan canonical form, and convex sets. Therefore this solutions manual can be helpful to anyone learning or teaching linear algebra at the college level.

As the understanding of the first chapters is essential to the comprehension of the later, more involved chapters, I encourage the reader to work through all of the problems of Chapters I, II, III and IV. Often earlier exercises are useful in solving later problems. (For example, Exercise 35, §3 of Chapter II shows that a strictly upper triangular matrix is nilpotent and this result is then used in Exercise 7, §1 of Chapter X.) To make the solutions concise, I have included only the necessary arguments; the reader may have to fill in the details to get complete proofs.

(Rami Shakarchi)

**Book Detail :- **
** Title: ** Introduction to Linear Algebra
** Edition: **
** Author(s): ** Rami Shakarchi
** Publisher: ** Springer-Verlag New York
** Series: **
** Year: ** 1996
** Pages: ** 192
** Type: ** PDF
** Language: ** English
** ISBN: ** 978-0-387-94760-0,978-1-4612-0755-9
** Country: ** US

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**About Author :- **

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.

**All Famous Books of this Author :- **

Here is list all books/editions avaliable of this author, We recomended you to download all.

** • Download PDF Undergraduate Analysis (2E) by Serge Lang **

** • Download PDF Undergraduate Analysis (2E Solution) by Rami Shakarchi, Serge Lang **** **

** • Download PDF Introduction to Linear Algebra (2E) by Serge Lang **** **

** • Download PDF Serge Lang's Linear Algebra (Solution) by Rami Shakarchi **** **

** • Download PDF Commentary on Lang's Linear Algebra by Henry Pinkham **** **

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**Book Contents :- **
**Serge Lang's Linear Algebra (Solution) ** written by
** Rami Shakarchi **
cover the following topics.

1. Vector Spaces

2. Matrices

3. Linear Mappings

4. Linear Maps and Matrices

5. Scalar Products and Orthogonality

6. Determinants

7. Symmetric, Hermitian and Unitary Operators

8. Eigenvectors and Eigenvalues

9. Polynomials and Matrices

10. Triangulation of Matrices and Linear Maps

11. Polynomials and Primary Decomposition

12. Convex Sets

Appendix

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