Advanced algebra is a general term for the development of algebra to an advanced stage, including many branches. Higher algebra offered by universities now generally includes linear algebra and polynomial algebra.
On the basis of elementary algebra, the research object of advanced algebra is further expanded, and the great mother-Yibo Xi is introduced. Mu Jun? Is it hot? Mirror scratches? ⑾⑼⑼⑼⑼⑼⑼⑼⑼⑼⑼⑼⑼⑼⑼⑼⑼⑼⑼⑼? What's the matter with you? Or? What is the core of Kangmujiao? What is your wife's background? span lang = EN-US & gt；
A set is the sum of things with certain attributes; Vector is a quantity with both direction and value; Vector space, also called linear space, is a collection of many vectors, which conforms to the rules of some specific operations. The object of operation in vector space is not just a number, but a vector, and its operation properties are also very different.
A brief history of the development of higher algebra
The history of algebra tells us that many mathematicians have gone through a rather uneven road and made painstaking efforts in solving higher-order equations.
People have long known the solution methods of linear and quadratic equations in one variable. As for the cubic equation, in the 7th century, China also got a general approximate solution, which was described in the Classic of Ancient Calculations compiled by the mathematician Wang Xiaotong in the Tang Dynasty. /kloc-in the 3rd century, Qin, a mathematician in the Song Dynasty, fully studied the method of finding the positive roots of digital higher-order equations in his book Shu Shu Jiu Zhang, that is to say, Qin got the general solution of the higher-order equations at that time.
In the west, it was not until the Renaissance at the beginning of16th century that Italian mathematicians discovered the Catan formula for the solution of cubic equation with one variable.
In the history of mathematics, it is said that this formula was first obtained by Italian mathematician Nicolo Tartaglia, and later deceived by Milan mathematician cardano (150 1 ~ 1576) and published in his own works. So now people still call this formula cardano formula (or Cardan formula), in fact, it should be called Nicolo Tartaglia formula.
After solving the cubic equation, the general quartic equation was quickly solved by Italian Ferrari (1522 ~ 1560). This naturally urges mathematicians to continue their efforts to find solutions to higher-order equations with five or more degrees. Unfortunately, although this problem has consumed many mathematicians' time and energy, it has lasted for more than three centuries and has not been solved.
1At the beginning of the 9th century, Abel, a young Norwegian mathematician (1802 ~ 1829), proved that an equation with five or more degrees cannot have algebraic solutions. Even the roots of these equations cannot be expressed by algebraic operations such as addition, subtraction, multiplication, division, multiplication and roots. Abel's proof is not only difficult, but also does not answer the question whether every specific equation can be solved by algebraic method.
Later, the problem that an equation with five or more degrees cannot have algebraic solution was completely solved by a young French mathematician, Galois. At the age of 20, Galois was arrested and imprisoned twice for taking an active part in the French bourgeois revolutionary movement. 1April, 832, died in a private duel shortly after he was released from prison, at the age of 2 1.
Before dying, Galois predicted that he could not get rid of the fate of death, so he wrote a letter to his friends overnight, sketched out his life's mathematical research experience, and attached a manuscript. In a letter to his friend Chevalier Ye, he said, "I have made some new discoveries in my analysis. Some are about equation theory; Some are about the overall function. The public demand for Jacobian or Gauss is not about the correctness of these theorems, but about the importance of these theorems. I hope that some people will find it beneficial for them to eliminate all this confusion in the future. "
After Galois died, according to his last wish, Chevalier Ye published his letter in the Encyclopedia Review. It took 14 years for his manuscript to be edited and published by joseph liouville (1809 ~ 1882) and recommended to the mathematical community.
With the passage of time, the significance of Galois Galois' research results is more and more recognized by people. Although Galois is very young, his contribution to the history of mathematics is not only to solve the algebraic solution problem of higher-order equations that has not been solved for centuries, but more importantly, he put forward the concept of "group" when solving this problem, and thus developed a whole set of theories about groups and fields, opening up a brand-new algebraic world and directly affecting the reform of algebraic research methods. Since then, algebra is no longer centered on equation theory, but instead studies the structural properties of algebra, which promotes the further development of algebra. Among the classic works of mathematical masters, Galois's paper is the thinnest, but his mathematical thought is brilliant.
Basic contents of higher algebra
Algebra started from the general problems of higher algebra and developed into a large-scale mathematical discipline including many independent branches, such as polynomial algebra and linear algebra. The research object of algebra is not only numbers, but also the transformations of matrix, vector and vector space, which can be operated. Although it is also called addition or multiplication, the basic arithmetic of numbers is sometimes no longer effective. Therefore, the content of algebra can be summarized as studying some sets with operations, which are mathematically called algebraic systems. Such as groups, rings, domains, etc.
Polynomial is the most common and simplest function, and it is widely used. Polynomial theory is based on the calculation and distribution of roots of algebraic equations, which is also called equation theory. The study of polynomial theory mainly lies in discussing the properties of algebraic equations, so as to find a simple method to solve the equations.
The research contents of polynomial algebra include divisibility theory, greatest common factor, multiple factors and so on. These are basically the same as middle school algebra. The divisibility of polynomials is very useful for solving algebraic equations. Solving an algebraic equation is nothing more than finding the zero of the corresponding polynomial. When the zero point does not exist, the corresponding algebraic equation has no solution.
We know that a linear equation is called a linear equation, and the algebra that discusses linear equations is called a linear algebra. Determinants and matrices are the most important contents in linear algebra.
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, the title of which means "Determinant Method of Solving Problems". The concept of determinant and its development have been clearly stated in the book. The concept of determinant was first put forward by German mathematician Leibniz in Europe. Jacoby, a German mathematician, summarized and put forward the system theory of determinant in 184 1.
Determinant has certain calculation rules, and the solution of a linear system of equations can be expressed as a formula by determinant, so determinant is a tool for solving linear systems of equations. Determinant can represent the solution of linear equations as a formula, that is, determinant represents a number.
Because the determinant requires the same number of rows and columns, the arranged table is always square. Through the study of it, the theory of matrix is discovered. A matrix is also a table of numbers. The numbers are arranged in rows and columns. The number of rows and numbers can be equal or different.
Matrix and determinant are two completely different concepts. A determinant represents a number, while a matrix is just an ordered arrangement of some numbers. Using the tool of matrix, the coefficients in linear equations can be formed into vectors in vector space; In this way, a series of theoretical problems such as the solution of a multivariate linear equation system and the relationship between different solutions can be completely solved. Matrix is widely used in many aspects, not only in the field of mathematics, but also in the fields of mechanics, physics, science and technology.
Algebra studies not only numbers, but also the transformation of matrix, vector and vector space. For these objects, you can perform operations. Although it is also called addition or multiplication, the basic arithmetic of numbers is sometimes no longer effective. So the content of algebra can be summarized as some sets with operations. In mathematics, such sets are called algebraic systems. The more important algebraic systems are group theory, ring theory and domain theory. Group theory is a powerful tool to study the symmetry law of mathematical and physical phenomena. Now the concept of group has become the most important and universal mathematical concept in modern mathematics, and it has been widely used in other departments.
The relationship between advanced algebra and other disciplines
Algebra, geometry and analytical mathematics are the three basic disciplines of mathematics, and the occurrence and development of each branch of mathematics are basically around these three disciplines. So what's the difference between algebra and the other two subjects?
First of all, algebraic operations are limited and lack the concept of continuity, which means that algebra mainly focuses on discreteness. Although continuity and discontinuity are dialectical unity in reality, in order to understand reality, it is sometimes necessary to divide it into several parts, then study and understand it separately, and then combine them to have a general understanding of reality. This is a simple but important scientific means for us to know things, and it is also the basic idea and method of algebra. Algebra pays attention to discrete relations, but it cannot explain its shortcomings at this time. Time has proved that this characteristic of algebra is effective in many times and directions.
Secondly, algebra is not only of direct practical significance to physics, chemistry and other sciences, but also plays an important role in mathematics itself. Many new ideas and concepts in algebra have greatly enriched many branches of mathematics and become the common foundation of many disciplines. ? The branches of operational research include operational research, mathematical programming and linear programming.
nonlinear programming
integer programming
goal programming
Dynamic programming
parametric programming
stochastic programming
combinatorial optimization
graph theory
queuing theory
Inventory theory
Game theory (game theory)
theory of decision-making
Search theory
Master planning theory
Optimize
Heuristic algorithm
Computer simulation
data mining
Predictive science
Soft system method
Cognitive mapping