| Dewey Class |
512.5 |
| Title |
Linear Algebra ( EBook) : From the Beginnings to the Jordan Normal Forms / / by Toshitsune Miyake. |
| Author |
Miyake, Toshitsune |
| Other name(s) |
SpringerLink (Online service) |
| Edition statement |
1st ed. 2022. |
| Publication |
Singapore : : Springer Nature Singapore : : Imprint: Springer, , 2022. |
| Physical Details |
XVII, 362 p. 15 illus., 2 illus. in color. : online resource. |
| ISBN |
9789811669941 |
| Summary Note |
The purpose of this book is to explain linear algebra clearly for beginners. In doing so, the author states and explains somewhat advanced topics such as Hermitian products and Jordan normal forms. Starting from the definition of matrices, it is made clear with examples that matrices and matrix operation are abstractions of tables and operations of tables. The author also maintains that systems of linear equations are the starting point of linear algebra, and linear algebra and linear equations are closely connected. The solutions to systems of linear equations are found by solving matrix equations in the row-reduction of matrices, equivalent to the Gauss elimination method of solving systems of linear equations. The row-reductions play important roles in calculation in this book. To calculate row-reductions of matrices, the matrices are arranged vertically, which is seldom seen but is convenient for calculation. Regular matrices and determinants of matrices are defined and explained. Furthermore, the resultants of polynomials are discussed as an application of determinants. Next, abstract vector spaces over a field K are defined. In the book, however, mainly vector spaces are considered over the real number field and the complex number field, in case readers are not familiar with abstract fields. Linear mappings and linear transformations of vector spaces and representation matrices of linear mappings are defined, and the characteristic polynomials and minimal polynomials are explained. The diagonalizations of linear transformations and square matrices are discussed, and inner products are defined on vector spaces over the real number field. Real symmetric matrices are considered as well, with discussion of quadratic forms. Next, there are definitions of 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. It is shown that any transformations of vector spaces over the complex number field have matrices of Jordan normal forms as representation matrices.: |
| Contents note |
Preface -- 1. Matrices -- 2. Linear Equations -- 3. Determinants -- 4. Vector Spaces -- 5. Linear Mappings -- 6. Inner Product Spaces -- 7. Hermitian Inner Product Spaces -- 8. Jordan Normal Forms.-Notation -- Answers to Exercises -- References -- Index of Theorems -- Index. |
| Mode of acces to digital resource |
Digital reproduction.- |
| Cham : |
| Springer International Publishing, |
| 2022. - |
| Mode of access: World Wide Web. System requirements: Internet Explorer 6.0 (or higher) or Firefox 2.0 (or higher). Available as searchable text in PDF format. |
| System details note |
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users). |
| Internet Site |
https://doi.org/10.1007/978-981-16-6994-1 |
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