Linear Algebra

1. Front Matter

   Pages i-xiii

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2. Matrices and Matrix Operations

   • Belkacem Said-Houari

   Pages 1-68

3. Determinants

   • Belkacem Said-Houari

   Pages 69-120

4. Euclidean Vector Spaces

   • Belkacem Said-Houari

   Pages 121-158

5. General Vector Spaces

   • Belkacem Said-Houari

   Pages 159-198

6. Linear Transformations

   • Belkacem Said-Houari

   Pages 199-225

7. Linear Transformations and Matrices

   • Belkacem Said-Houari

   Pages 227-268

8. Eigenvalues and Eigenvectors

   • Belkacem Said-Houari

   Pages 269-321

9. Orthogonal Matrices and Quadratic Forms

   • Belkacem Said-Houari

   Pages 323-376

10. Back Matter

    Pages 377-384

    PDF

This self-contained, clearly written textbook on linear algebra is easily accessible for students. It begins with the simple linear equation and generalizes several notions from this equation for the system of linear equations and introduces the main ideas using matrices. It then offers a detailed chapter on determinants and introduces the main ideas with detailed proofs. The third chapter introduces the Euclidean spaces using very simple geometric ideas and discusses various major inequalities and identities. These ideas offer a solid basis for understanding general Hilbert spaces in functional analysis. The following two chapters address general vector spaces, including some rigorous proofs to all the main results, and linear transformation: areas that are ignored or are poorly explained in many textbooks. Chapter 6 introduces the idea of matrices using linear transformation, which is easier to understand than the usual theory of matrices approach. The final two chapters are more advanced, introducing the necessary concepts of eigenvalues and eigenvectors, as well as the theory of symmetric and orthogonal matrices. Each idea presented is followed by examples.  

The book includes a set of exercises at the end of each chapter, which have been carefully chosen to illustrate the main ideas. Some of them were taken (with some modifications) from recently published papers, and appear in a textbook for the first time. Detailed solutions are provided for every exercise, and these refer to the main theorems in the text when necessary, so students can see the tools used in the solution.

• matrix algebra
• linear algebra
• eigenvalues
• orthogonal matrices
• symmetric matrices
• matrix theory

“This book can be used as a textbook for a first course in linear algebra for undergraduate students in all disciplines. It can be also used as a booklet for graduate students, allowing to acquire some concepts, examples, and basic results. It is also suitable for those students who are looking for a simple, easy, and clear textbook that summarizes the main ideas of linear algebra.” (Mihail Voicu, zbMATH 1381.15001, 2018)

• Department of Mathematics, College of Sciences, University of Sharjah, Sharjah, United Arab Emirates

  Belkacem Said-Houari

Belkacem Said-Houari is Professor at the Mathematics Department, University of Sharjah, Sharjah and author of the book on Differential Equation (ISBN 978-3-319-25734-1) in the same series.

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