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: Paperback / softback
Length: 362 pages
Publication date: 05 September 2023
Publisher: Springer Verlag, Singapore

The aim 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. The book begins by defining matrices, emphasizing their connection to tables and operations. It demonstrates how matrices and matrix operations can be viewed as abstractions of tables and their corresponding operations. The author further asserts 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 significant 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 a solid foundation for further discussions. The concept of resultant polynomials is introduced, showcasing the application of determinants in solving linear equations. Abstract vector spaces over a field, such as the real number field or the complex number field, are defined. However, the book primarily focuses on vector spaces over these fields, assuming that readers may not have prior familiarity with abstract fields. Linear mappings and linear transformations of vector spaces, along with representation matrices of linear mappings, are defined. The characteristic polynomials and minimal polynomials of linear mappings are discussed, 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 emphasis on quadratic forms. Hermitian inner products are also defined, adding another layer of complexity to the mathematical concepts explored in the book.

Weight: 581g
Dimension: 235 x 155 (mm)
ISBN-13: 9789811669965
Edition number: 1st ed. 2022