Linear Algebra: From the Beginnings to the Jordan Normal Forms

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.

這本書的目的是為初學者清楚解釋線性代數。在這過程中，作者提出並解釋了一些較高級的主題，如共軛內積和Jordan標準形。從矩陣的定義開始，通過例子清楚地表明矩陣和矩陣運算是表格和表格操作的抽象。作者還強調，線性方程組是線性代數的起點，線性代數和線性方程組密切相關。通過解矩陣的行簡化，可以找到線性方程組的解，這相當於高斯消元法解線性方程組。行簡化在本書的計算中起著重要作用。為了計算矩陣的行簡化，矩陣被垂直排列，這在實際中很少見，但對於計算很方便。定義和解釋了常規矩陣和矩陣的行列式。此外，還討論了多項式的結果作為行列式的應用。接下來，定義了在域K上的抽象向量空間。然而，在本書中，主要考慮的是在實數域和複數域上的向量空間，以防讀者對抽象域不熟悉。定義了向量空間的線性映射和線性變換，以及線性映射的表示矩陣，並解釋了特徵多項式和最小多項式。討論了線性變換和方陣的對角化，並在實數域上的向量空間上定義了內積。同樣考慮了實對稱矩陣，並討論了二次型。接下來，介紹了共軛內積的定義。解釋了共軛變換、酉變換、正規變換以及正規變換和矩陣的譜分解。書的最後討論了Jordan標準形。證明了任何在複數域上的向量空間的變換都有Jordan標準形的矩陣作為表示矩陣。

The author is currently Professor Emeritus at Hokkaido University. He is also the author of Modular Forms (published by Springer) in 1989.

該作者目前是北海道大學的名譽教授。他也是1989年由Springer出版的《Modular Forms》一書的作者。