The Linear Algebra a Beginning Graduate Student Ought to Know

1. Front Matter

   Pages I-XI

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2. Notation and Terminology

   • Jonathan S. Golan

   Pages 1-3

3. Fields

   • Jonathan S. Golan

   Pages 5-19

4. Vector Spaces Over a Field

   • Jonathan S. Golan

   Pages 21-38

5. Algebras Over a Field

   • Jonathan S. Golan

   Pages 39-56

6. Linear Independence and Dimension

   • Jonathan S. Golan

   Pages 57-88

7. Linear Transformations

   • Jonathan S. Golan

   Pages 89-111

8. The Endomorphism Algebra of a Vector Space

   • Jonathan S. Golan

   Pages 113-131

9. Representation of Linear Transformations by Matrices

   • Jonathan S. Golan

   Pages 133-146

10. The Algebra of Square Matrices

    • Jonathan S. Golan

    Pages 147-188

11. Systems of Linear Equations

    • Jonathan S. Golan

    Pages 189-220

12. Determinants

    • Jonathan S. Golan

    Pages 221-253

13. Eigenvalues and Eigenvectors

    • Jonathan S. Golan

    Pages 255-296

14. Krylov Subspaces

    • Jonathan S. Golan

    Pages 297-316

15. The Dual Space

    • Jonathan S. Golan

    Pages 317-332

16. Inner Product Spaces

    • Jonathan S. Golan

    Pages 333-367

17. Orthogonality

    • Jonathan S. Golan

    Pages 369-394

18. Selfadjoint Endomorphisms

    • Jonathan S. Golan

    Pages 395-418

19. Unitary and Normal Endomorphisms

    • Jonathan S. Golan

    Pages 419-440

20. Moore–Penrose Pseudoinverses

    • Jonathan S. Golan

    Pages 441-452

Linear algebra is a living, active branch of mathematics which is central to almost all other areas of mathematics, both pure and applied, as well as to computer science, to the physical, biological, and social sciences, and to engineering. It encompasses an extensive corpus of theoretical results as well as a large and rapidly-growing body of computational techniques. Unfortunately, in the past decade, the content of linear algebra courses required to complete an undergraduate degree in mathematics has been depleted to the extent that they fail to provide a sufficient theoretical or computational background. Students are not only less able to formulate or even follow mathematical proofs, they are also less able to understand the mathematics of the numerical algorithms they need for applications. Certainly, the material presented in the average undergraduate course is insufficient for graduate study. This book is intended to fill the gap which has developed by providing enough theoretical and computational material to allow the advanced undergraduate or beginning graduate student to overcome this deficiency and be able to work independently or in advanced courses. The book is intended to be used either as a self-study guide, a textbook for a course in advanced linear algebra, or as a reference book. It is also designed to prepare a student for the linear algebra portion of prelim exams or PhD qualifying exams. The volume is self-contained to the extent that it does not assume any previous formal knowledge of linear algebra, though the reader is assumed to have been exposed, at least informally, to some of the basic ideas and techniques, such as manipulation of small matrices and the solution of small systems of linear equations over the real numbers. More importantly, it assumes a seriousness of purpose, considerable motivation, and a modicum of mathematical sophistication on the part of the reader. In the latest edition, new major theorems have been added, as wellas many new examples. There are over 130 additional exercises and many of the previous exercises have been revised or rewritten. In addition, a large number of additional biographical notes and thumbnail portraits of mathematicians have been included.

• linear algebra
• matrix theory

From the reviews of the third edition:

“This edition is enhanced by the inclusion of further results and many new examples. There are also over 130 additional exercises and many of the earlier ones have been revised. Furthermore, many more biographical notes and thumbnail portraits of mathematicians connected in some way with linear algebra have been added. This book continues to be very successful and useful, not only as a textbook for advanced linear algebra courses, but also for self-study and reference purposes.” (Rabe von Randow, Zentralblatt MATH, Vol. 1237, 2012)

• Dept. Math & Computer Science, University of Haifa, Haifa, Israel

  Jonathan S. Golan

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