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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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**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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