Chinese Name: History of Linear Algebra: Unit of Mathematical Calculation: Vector (Group), Matrix, Determinant Concept: Relevance is the basic introduction of linearity, determinant, matrix, equation, quadratic form, from solving equations to group theory. Basic introduction needs to examine multivariate functions, because studying quantities related to multiple factors will cause problems. If the correlation studied is linear, then the problem is called linear. The first problem of linear algebra in history is about solving linear equations. The development of linear equations theory has contributed to the establishment and development of matrix theory and determinant theory as tools, and has become the main part of our linear algebra teaching materials. The initial problems of linear equations mostly come from life practice, and it is these practical problems that lead to the birth and development of linear algebra. In addition, the requirements of modern mathematical analysis and geometry also promote the further development of linear algebra. Linear algebra has three basic calculation units: vector (group), matrix and determinant. By studying their properties and related theorems, we can solve linear equations, realize determinant and matrix calculation and linear transformation, and construct vector space and Euclidean space. The two basic methods of linear algebra are construction (decomposition) and algebraic method, and the basic ideas are simplification (degradation) and isomorphic transformation. Determinants in solving linear equations. Originally a shorthand expression, it is now a very useful tool in mathematics. Determinant was invented by Leibniz and Japanese mathematician Guan Xiaohe. 1693 In April, Leibniz used and gave the determinant in a letter to Robida, and gave the condition that the coefficient determinant of the equations was zero. Guan Xiaohe, a contemporary Japanese mathematician, also put forward the concept and algorithm of determinant in his book Method of Solving Problems Elements. 1750, the Swiss mathematician G Cramer (1704- 1752) gave a relatively complete and clear explanation of the definition and expansion rules of determinant in his book Introduction to Linear Algebra Analysis, and gave what we now call Cramer's rule for solving linear equations. Later, the mathematician E. Bezout (1730-1783) systematized the method of judging the symbols of determinant, and pointed out how to judge that a homogeneous linear equation group has a non-zero solution by using the concept of coefficient determinant. In short, for a long time, determinant was only used as a tool to solve linear equations, and no one realized that it could form a theory to study independently of linear equations. In the history of determinant development, the first person to make a coherent logical exposition of determinant theory, that is, to separate determinant theory from solving linear equations, is the French mathematician Vandermonde (A-T, 1735- 1796). Vandermonde, studied music under the guidance of his father since childhood, but became interested in mathematics and eventually became an academician of the French Academy of Sciences. In particular, he gave the rule of expanding determinant with second-order polynomial and its complement polynomial. As far as determinant itself is concerned, he is the founder of this theory. 1772 Laplace proved some rules put forward by Vandermonde in a paper, and extended his determinant expansion method. Cauchy, another great French mathematician after Vandermonde,, made outstanding contributions to determinant theory. In 18 15, Cauchy systematically and almost modestly dealt with determinant for the first time in a paper. One of the main results is the multiplication theorem of determinant. In addition, he was the first to arrange the elements of determinant into a square matrix and adopted the bipedal marking method; The concept of determinant characteristic equation is introduced. The concept of similar determinant is given. The expansion theorem of Laplace determinant is improved and proved. For more than half a century from 65438 to 2009, one of the authors who has been studying determinant theory is J. Sylvester (1814-1894). He is a lively, sensitive, excited, enthusiastic and even excitable person. However, because he is a Jew, he is treated unfairly by Cambridge University. Sylvester enthusiastically introduced his academic thoughts. One of his important achievements is to improve the method of eliminating x by linear polynomials and linear polynomials, which he calls collocation method. The necessary and sufficient conditions for these two polynomial equations to have a common root when the determinant is zero are given, but no proof is given. After Cauchy, the most prolific person in determinant theory is the German mathematician Jacobi (J. Jacobi, 1804- 185 1). He introduced the function determinant, namely "Jacobian determinant", pointed out the function of the function determinant in the substitution of multiple integral variables, and gave the derivative formula of the function determinant. Jacobi's famous paper "On the Formation and Properties of Determinants" marks the completion of the determinant system theory. Due to the application of determinant in mathematical analysis, geometry, linear equations theory and quadratic form theory, determinant theory itself has also made great development in 19 century. New results of determinant have appeared in the whole19th century. In addition to a large number of general determinant theorems, many other theorems about special determinants have been obtained one after another. Matrix matrix is an important basic concept in mathematics, the main research object of algebra, and an important tool for mathematical research and application. The word "matrix" was first used by Sylvester, who invented this predicate to distinguish rectangular arrays from determinants. In fact, the subject of matrix has developed very well before it was born. It is obvious from a lot of work on determinant that for many purposes, whether the value of determinant is related to the problem or not, the square matrix itself can be studied and utilized, and many basic properties of matrix are also established in the development of determinant. Logically, the concept of matrix should precede the concept of determinant, but in history, the order is just the opposite. A. Cayley (182 1- 1895), a British mathematician, is recognized as the founder of matrix theory, because he first put forward matrix as an independent mathematical concept and published a series of articles on this subject. Gloria combined with the study of invariants under linear transformation, first introduced matrix to simplify notation. From 65438 to 0858, he published the first paper on the subject, Research Report on Matrix Theory, which systematically expounded the theory of matrix. In this paper, he defined a series of basic concepts such as matrix equality, matrix operation rule, matrix transposition and matrix inversion, and pointed out the interchangeability and combinability of matrix addition. In addition, Gloria also gives the characteristic equation and characteristic root (eigenvalue) of the square matrix and some basic results about the matrix. Gloria was born in an old and talented English family. After graduating from Trinity College, Cambridge University, he stayed to teach mathematics. Three years later, he switched to the profession of lawyer, and his work was fruitful. He studied mathematics in his spare time and published a large number of mathematical papers. In 1855, emmett (C. Hermite, 1822- 190 1) proved the special properties of the characteristic roots of some matrix classes discovered by other mathematicians, such as the characteristic roots of emmett matrices now. Later, Klebsch (A A. Clebsch, 183 1- 1872) and A.Buchheim proved the characteristic root property of symmetric matrices. H.Taber introduced the concept of trace of matrix and gave some related conclusions. In the history of matrix theory, the contribution of G. Frobenius (1849-1917) is indelible. He discussed the minimum polynomial problem, introduced the concepts of matrix rank, invariant factor and elementary factor, orthogonal matrix, similar transformation of matrix and contraction matrix, arranged the theories of invariant factor and elementary factor in logical form, and discussed some important properties of orthogonal matrix and contraction matrix. In 1854, Jordan studied the problem of transforming a matrix into a standard form. 1892, Metzler introduced the concept of matrix transcendental function and wrote it in the form of matrix power series. In the works of Fourier, Searle and Poincare, the problem of infinite order matrix is also discussed, which is mainly to meet the needs of equation development. The properties of the matrix itself depend on the properties of the elements. After more than two centuries of development, matrix has become an independent branch of mathematics-matrix theory. Matrix theory can be divided into matrix equation theory, matrix decomposition theory and generalized inverse matrix theory. Matrix and its theory have been widely used in various fields of modern science and technology. The solution of linear equations has been comprehensively discussed in the ancient Chinese mathematical book "Nine Chapters of Arithmetic Equations". Among them, the method is essentially equivalent to the modern method of eliminating unknown quantities by elementary row transformation of the augmented matrix of equations, that is, Gaussian elimination method. In the west, the study of linear equations was initiated by Leibniz in the late17th century. He once studied a system of equations composed of three binary linear equations. Maclaurin studied binary, ternary and quaternary linear equations in the first half of the18th century, and obtained the results now known as Cramer's Law. Clem soon announced this rule. /kloc-In the second half of the 8th century, the French mathematician Bezu made a series of studies on the theory of linear equations, and proved that the condition for homogeneous linear equations to have non-zero solutions is that the determinant of coefficients is equal to zero. /kloc-in the 9th century, British mathematicians H. Smith and C. L. Dodgson continued to study the theory of linear equations. The former introduces the concepts of augmented matrix and non-augmented matrix of equations, and the latter proves that the necessary and sufficient condition for the compatibility of equations with unknown numbers is that the rank of coefficient matrix and augmented matrix is the same. This is one of the important achievements of modern equation theory. A large number of scientific and technical problems often boil down to solving linear equations. Therefore, while developing the numerical solution of linear equations, the theoretical work on the structure of linear equations has also made satisfactory progress. Now, the numerical solution of linear equations plays an important role in computational mathematics. Quadratic quadratic form is also called "quadratic form", and the n-ary quadratic homogeneous polynomial in number field P is called the n-ary quadratic form in number field P, and quadratic form is the follow-up content of our linear algebra textbook. For our later study, here is also a brief introduction to the development history of quadratic form. The systematic study of quadratic forms began in the18th century, and originated from the discussion on the classification of quadric curves and quadric surfaces. /kloc-in the 8th century, the equation of conic and quadric surface was transformed, and the axis in the principal axis direction was selected as the coordinate axis to simplify the shape of the equation. Cauchy concluded in his book that when the equation is standard, quadrics are classified by the sign of quadratic terms. However, at that time, it was not clear why we always got the same number of positive and negative terms when simplifying to the standard form. Sylvester answered this question. He gave a binary quadratic law of inertia, but he did not prove it. This law was later rediscovered and proved by jacoby. 180 1 year, Gauss introduced the terms of positive definite, negative definite, semi-positive definite and semi-negative definite of quadratic form in arithmetic research. The further study of quadratic simplification involves the concept of quadratic characteristic equation or determinant. The concept of characteristic equation appeared implicitly in Euler's works, and Lagrange explicitly gave this concept for the first time in his works on linear differential equations. The truth value of the eigenvalues of three-variable quadratic forms is established by J-N.P. Ashett, gaspard monge and Poisson (S.D. Poisson, 178 1- 1840). On the basis of others' work, Cauchy began to study the quadratic form of simplified variables and proved that the characteristic equation is invariant under any transformation in rectangular coordinate system. Later, he proved that two quadratic forms of a variable can be transformed into sum of squares simultaneously by the same linear transformation. 185 1 year, Sylvester needs to consider the classification of conic and conic bundle when studying the contact and intersection between conic and conic. He introduced the concepts of elementary factor and invariant factor into the classification method, but did not prove the conclusion that "invariant factor constitutes the complete set of two quadratic invariants". In 1858, Weierstrass gave a general method that the sum of two quadratic forms is square at the same time, and proved that if one quadratic form is positive definite, this simplification is possible even if some characteristic roots are equal. Wilstrass systematically completed the theory of quadratic form and extended it to bilinear form. The problem of finding roots from solving equations to group theory is a central topic in equation theory. /kloc-in the 6th century, mathematicians solved the root formulas of cubic and quartic equations, and whether the root formulas of higher-order equations existed became another problem discussed by mathematicians at that time. This problem has cost many mathematicians a lot of time and energy. I have experienced many failures, but I can't get out of trouble. /kloc-in the second half of the 8th century, Lagrange carefully summarized and analyzed the previous failure experience, deeply studied the relationship between the roots and permutation of higher-order equations, and put forward the concept of resolvent, foreseeing that resolvent is related to the form invariance of roots under permutation and permutation. But he finally failed to solve the problem of higher order equation. Lagrange's disciple Rufini (1765- 1862) also made a lot of efforts, but all ended in failure. The discussion of radical solutions of higher order equations has made great progress in Abel, an outstanding mathematician in Norway. Abel (N.K.Abel, 1802- 1829) lived only 27 years old. He was poor and ill all his life, but he left many creative jobs. In 1824, Abel proved that a general algebraic equation greater than quartic cannot have a radical solution. But the problem has not been completely solved, because some special equations can be solved by roots. Therefore, when there is no radical solution for algebraic equations higher than quartic is a problem that needs to be further solved. This problem was completely solved by French mathematician Galois. E Galois (181-1832) carefully studied the work of Lagrange and Abel, established the "allowed" permutation of the roots of the equation, and put forward the concept of permutation group, and obtained that the necessary and sufficient condition for the root solution of algebraic equation is that the automorphism group of permutation group is solvable. In this sense, we say that Galois is the founder of group theory. Galois was born in a wealthy family near Paris and received a good parenting education when he was young. Unfortunately, this talented mathematician died young. 1832 May, he was killed in a duel because of political and love disputes, only 2 1. The concept and conclusion of permutation group is the first main source of abstract group. The second main sources of abstract groups are Dai Dejin (R. Dedekind,1831-1916) and Kroneck (L. Kroneck,1823-189/kloc). In addition, Klein (F. Klein, 1849- 1925) and Poincare (J-H. Poincare, 1854- 19 12) gave infinite transformation groups and others. 65438. 10006 16666 In the 1980s, mathematicians finally successfully summarized the axiomatic system of abstract group theory. In the 1980s, the concept of group has been generally regarded as one of the most basic concepts in mathematics and its many applications. It not only plays an important role in many branches of mathematics such as geometry, algebraic topology, function theory and functional analysis, but also forms some new disciplines such as topological groups, Lie groups and algebraic groups. They also have other structures related to group structure, such as topology, analytic manifold, algebraic cluster and so on. , and plays an important role in crystallography, theoretical physics, quantum chemistry, coding and automata theory.