Linear algebra is a branch of higher algebra. We know that linear equations are called linear equations, and the algebra that discusses linear equations and linear operations is called linear algebra. Determinants and matrices are the most important contents in linear algebra. Determinants and matrices received great attention in the 19th century, and thousands of articles were written on these two topics. From a mathematical point of view, the concept of vector is just a set of ordered ternary arrays. But it takes force or speed as its direct physical meaning, and can be used to write what is said in physics immediately in mathematics. The vectors used for gradient, divergence and curl are more convincing. Similarly, determinant and matrix are just like derivatives (although dy/dx is only a symbol in mathematics, which represents a long formula including the limit of △y/△x, derivative itself is a powerful concept, which enables us to directly and creatively imagine what happens in physics). Therefore, although on the surface, determinant and matrix are just a kind of language or shorthand, most of its vivid concepts can provide the key to a new field of thought. However, it turns out that these two concepts are very useful tools in mathematical physics.

The disciplines of linear algebra and matrix theory were introduced and developed with the study of equation coefficients of linear systems. The concept of determinant was first put forward by Japanese mathematician Guan Xiaohe in17th century. 1683 He wrote a book called Method of Solving Problems, which means "Method of Solving Determinant Problems". The concept of determinant and its development have been clearly stated in the book. The concept of determinant was first put forward in Europe by the German mathematician Leibniz (1693), one of the founders of calculus. 1750, Clem published an important basic formula for solving linear system equations (called Clem's rule) in his introduction D L 'analysis des lignes courbes Alge 'Briques. In 1764, Bezout systematizes the process of determining the symbols of determinant items. Given n homogeneous linear equations with n unknowns, Bezout proved that the coefficient determinant is equal to zero, which is the condition that this equation has a non-zero solution. Vandermonde was the first person to systematically expound the determinant theory (that is, to separate the row and column theory from the solution of linear equations). The rules of expanding determinant with second-order sub-formula and its complementary sub-formula are given. As far as determinant itself is concerned, he is the founder of this theory. 1772, Laplace proved some rules of Vandermonde, and extended his determinant expansion method. The determinant is expanded by the set of subclasses contained in row R and their complementary subclasses, and this method is still named after him. Jacobi, a German mathematician, also summarized and put forward the system theory of determinant in 184 1. Another mathematician who studies determinant is Cauchy, the greatest mathematician in France. He greatly developed the theory of determinant. In the notation of determinant, he arranged the elements into a square matrix and adopted the new notation of bipedal notation for the first time. At the same time, he found the formula of multiplying two determinants, improved and proved Laplace's expansion theorem. Relatively speaking, the concept of matrix was first used by Lagrange in bilinear work after 1700 years. Lagrange expects to know the maximum and minimum values of multivariate functions, and its method is called Lagrange iteration method. In order to accomplish this, he first needs the condition that the first-order partial derivative is 0 and the second-order partial derivative matrix. This condition is the so-called positive and negative definition today. Although Lagrange did not explicitly propose using matrices.

Gauss elimination was put forward by Gauss in about 1800, which was used to solve the least square problem in celestial body calculation and later earth surface measurement calculation. This branch of applied mathematics involves measuring and finding the shape or exact local position of the earth, which is called geodesy. ) Although Gauss is famous for successfully eliminating the variables of linear equations by this technique, as early as several centuries ago, in China's manuscript, there was an explanation of how to solve a ternary three-equation system by Gauss elimination method. In those years, Gaussian elimination method has been considered as a part of the development of geodesy, not mathematics. Gauss-Jordan elimination rule first appeared in the Handbook of Geodesy written by William Jordan. Many people mistake the famous mathematician Camille Jordan for Jordan in Gauss-Jordan elimination.

With the rich development of matrix algebra, people need to have appropriate symbols and definitions for matrix multiplication. These two people should meet at about the same time and place. 1848, J.J. Sylvester of Britain first put forward the word matrix, which comes from Latin and stands for a row of numbers. 1855 matrix algebra was cultivated by Arthur Cayley. Gloria studied the synthesis of linear transformation and put forward the definition of matrix multiplication, which made the coefficient matrix of synthetic transformation ST become the product of matrix S and matrix T. He further studied those algebraic problems including matrix inversion. The famous Cayley- Hamilton theory asserts that the square of a matrix is the root of its characteristic polynomial, which was put forward by Cayley in his Collection of Matrix Theory 1858. It is very important for the development of matrix algebra to represent the matrix with a single letter A. At the early stage of development, the formula det( AB) = det( A )det( B) provides the connection between matrix algebra and determinant. Cauchy, a mathematician, first gave the term of the characteristic equation, and proved that the matrix with order greater than 3 has the eigenvalue and the real symmetric determinant with any order has the real eigenvalue. The concept of similar matrix is given, and it is proved that similar matrices have the same eigenvalue. Studied the substitution theory,

Mathematicians try to study vector algebra, but there is no natural definition of the product of two vectors in any dimension. The first vector algebra involving noncommutative cross products (that is, v×w is not equal to w×v) was proposed by Herman grassmann in his book Linear Algebra. ( 1844) 。 His viewpoint was also introduced into the product of a column matrix and a row matrix, and this result is now called a matrix with rank 1, or a simple matrix. /kloc-at the end of 0/9, American mathematical physicist Willard Gibbs published a famous exposition on the elements of vector analysis. Later, physicist P.A.M. Dirac proposed that the product of row vector and column vector is scalar. The column matrices and vectors we are used to are given by physicists in the 20th century.

The development of matrix is closely related to linear transformation. In the19th century, it only occupied a limited space in the formation of linear transformation theory. The definition of modern vector space was put forward by Peano in 1888. With the development of post-modern digital computer in World War II, matrix has a new meaning, especially in the numerical analysis of matrix. Due to the rapid development and wide application of computers, many practical problems can be solved quantitatively by discrete numerical calculation. Therefore, as a linear algebra dealing with discrete problems, it has become an indispensable mathematical basis for scientific and technical personnel engaged in scientific research and engineering design.