Linear Algebra: From the Beginnings to the Jordan Normal Forms

This book aims to explain linear algebra clearly for beginners, covering advanced topics such as Hermitian products and Jordan normal forms. It emphasizes the connection between linear algebra and linear equations, and uses row-reductions to solve matrix equations. Abstract vector spaces, linear mappings, and transformations are defined, and real symmetric matrices and quadratic forms are discussed. Hermitian inner products, Hermitian transformations, unitary transformations, normal transformations, and the spectral resolution of normal transformations and matrices are explained. The book ends with Jordan normal forms.

Format: Hardback
Length: 362 pages
Publication date: 11 July 2022
Publisher: Springer Verlag, Singapore

The purpose of this book is to provide a clear and comprehensive introduction to linear algebra for beginners. In pursuit of this goal, the author delves into somewhat advanced topics such as Hermitian products and Jordan normal forms, ensuring that the subject matter is accessible and understandable to novice readers.

Beginning with the definition of matrices, the author elucidates the concept of matrices and matrix operations as abstractions of tables and operations of tables. Through illustrative examples, it becomes evident that matrices and matrix operations represent a concise and efficient way to manipulate and analyze data.

The author further emphasizes that systems of linear equations serve as the foundation of linear algebra, and there is a close relationship between linear algebra and linear equations. The solutions to systems of linear equations are obtained by solving matrix equations through the row-reduction of matrices, which is equivalent to the Gauss elimination method used for solving systems of linear equations.

Row-reductions play a crucial role in the calculations presented in this book. To facilitate the calculation of row-reductions, matrices are arranged vertically, a less common arrangement but one that proves convenient for mathematical operations. Regular matrices and determinants of matrices are defined and explained, providing readers with a solid foundation in matrix theory.

Furthermore, the book discusses the application of determinants in the context of the resultant of polynomials. Abstract vector spaces over a field K are defined, with a focus on vector spaces over the real number field and the complex number field, to accommodate readers who may not be familiar with abstract fields. Linear mappings and linear transformations of vector spaces, as well as representation matrices of linear mappings, are defined. The characteristic polynomials and minimal polynomials of linear mappings are explained, along with the diagonalizations of linear transformations and square matrices. Inner products are defined on vector spaces over the real number field, and real symmetric matrices are considered, with an exploration of quadratic forms.

Lastly, definitions of Hermitian inner products are provided, adding another dimension to the study of linear algebra.

In conclusion, this book serves as an invaluable resource for beginners seeking to grasp the fundamentals of linear algebra. By presenting complex topics in a clear and concise manner, the author enables readers to build a strong foundation in the subject and apply linear algebra to various real-world scenarios. Whether you are a student, professional, or simply interested in expanding your mathematical knowledge, this book is a must-read.

Weight: 727g
Dimension: 235 x 155 (mm)
ISBN-13: 9789811669934
Edition number: 1st ed. 2022