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Linear Algebra - From the Beginnings to the Jordan Normal Forms (Hardcover, 1st ed. 2022)
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Linear Algebra - From the Beginnings to the Jordan Normal Forms (Hardcover, 1st ed. 2022)
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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.
General
Imprint: |
Springer Verlag, Singapore
|
Country of origin: |
Singapore |
Release date: |
September 2022 |
First published: |
2022 |
Authors: |
Toshitsune Miyake
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Dimensions: |
235 x 155 x 30mm (L x W x T) |
Format: |
Hardcover
|
Pages: |
362 |
Edition: |
1st ed. 2022 |
ISBN-13: |
978-981-16-6993-4 |
Categories: |
Books >
Science & Mathematics >
Mathematics >
Algebra >
General
|
LSN: |
981-16-6993-7 |
Barcode: |
9789811669934 |
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