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**About this book :- **
**Commentary on Lang's Linear Algebra ** written by
** Henry Pinkham **

These notes were written to complement and supplement Lang’s Linear Algebra as a textbook in a Honors Linear Algebra class at Columbia University. The students in the class were gifted but had limited exposure to linear algebra. As Lang says in his introduction, his book is not meant as a substitute for an elementary text. The book is intended for students having had an elementary course in linear algebra. However, by spending a week on Gaussian elimination after covering the second chapter of [4], it was possible to make the book work in this class. Henry C. Pinkham had spent a fair amount of time looking for an appropriate textbook, and I could find nothing that seemed more appropriate for budding mathematics majors than Lang’s book. He restricts his ground field to the real and complex numbers, which is a reasonable compromise.

The book has many strength. No assumptions are made, everything is defined. The first chapter presents the rather tricky theory of dimension of vector spaces admirably. The next two chapters are remarkably clear and efficient in presently matrices and linear maps, so one has the two key ingredients of linear algebra: the allowable objects and the maps between them quite concisely, but with many examples of linear maps in the exercises. The presentation of determinants is good, and eigenvectors and eigenvalues is well handled. Hermitian forms and hermitian and unitary matrices are well covered, on a par with the corresponding concepts over the real numbers. Decomposition of a linear operator on a vector space is done using the fact that a polynomial ring over a field is a Euclidean domain, and therefore a principal ideal domain. These concepts are defined from scratch, and the proofs presented very concisely, again. The last chapter covers the elementary theory of convex sets: a beautiful topic if one has the time to cover it. Advanced students will enjoy reading Appendix II on the Iwasawa Decomposition.

(HENRY C. PINKHAM)

**Book Detail :- **
** Title: ** Commentary on Lang's Linear Algebra
** Edition: **
** Author(s): ** Henry Pinkham
** Publisher: **
** Series: ** Expository notes
** Year: ** 2013
** Pages: ** 94
** Type: ** PDF
** Language: ** English
** ISBN: **
** 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.

• Undergraduate Analysis (2E) by Serge Lang

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

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

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

• Commentary on Lang's Linear Algebra by Henry Pinkham

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**Book Contents :- **
**Commentary on Lang's Linear Algebra ** written by
** Henry Pinkham **
cover the following topics.

1. Gaussian Elimination

2. Matrix Inverses

3. Matrices

4. Representation of a Linear Transformation by a Matrix

5. Row Rank is Column Rank

6. Duality

7. Orthogonal Projection

8. Uniqueness of Determinants

9. The Key Lemma on Determinants

10. The Companion Matrix

11. Positive Definite Matrices

12. Homework Solutions

A. Symbols and Notational Conventions

References

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