"Linear algebra" 01-Ancient new disciplines

Linear algebra (Linear Algebra) This discipline everyone is not unfamiliar, if someone still feel a little rusty, then "determinant", "matrix" These concepts you should always have the impression. Each major of the university will study this course in different shades, the liberal arts generally put in the "University mathematics", the engineering students generally in the "linear algebra" or "higher mathematics" to see it, and our mathematical department is more on the "Advanced algebra" this course. The reason why linear algebra can squeeze into college mathematics with calculus is the simplicity of its model and the universality of its application. It is not only a basic tool of mathematical research, but also a theoretical basis for many business disciplines.

But interestingly, linear algebra became a system discipline, but it was the beginning of the 20th century. Before this, various related theories independently developed, then produced the intersection, finally in the axiom and the abstract algebra impetus, finally fruition, forms a complete set of disciplines. It is not surprising that its growth experience, like most mathematical disciplines, was first presented as a method in specific applications, and then gradually extracted from the universal theory and formed a discipline. Linear relation, as the simplest kind of relationship, its models and methods exist in many places, so initially they are only developed independently with different branches, and the axiomatic method finally captures their "linear" commonality, and uses the concept of "linear space" to concatenate them together.

Simply put, "linear" is the "one" and "positive proportional" relationship between variables, and the rate of change of variables is constant. Intuitively, there is a relationship between all elements in the equation (1), where \ (k_i\) is the coefficient. Such a relationship is too common, and even in the oldest civilizations, everyone is very familiar with it. The ancients began to solve some linear equations (groups) early, and formed a number of practical methods, China's "nine-chapter arithmetic" gives the general method of solving ternary one-time equations, which is the same as the "Gaussian elimination element method" of the multivariate linear equation group.

\[y=k_1x_1+k_2x_2+\cdots+k_nx_n\tag{1}\]

In the process of solving the multivariate linear equations, Leibniz (Leibniz) first used the determinant to denote the solution of the low-dimensional equation, and Clem (Cramer) gave the solution of the general equation Group. The determinant is only a notation, which is used only in the form of the solution of the equations. Thereafter, Vandermonde (Vandermonde) independently of the determinant, and studied the determinant of the sub-expansion, which marked the beginning of the determinant as a discipline. Cauchy (Cauchy) has systematically studied the determinant, fused the concept of the Matrix and drawn many important conclusions, so Cauchy is also regarded as the founder of the determinant in the modern sense.

Although we generally say "matrix determinant", but actually the matrix concept is produced later than the determinant. Initially, we were only formally using this structure (horizontal arrangement), Gloria (Cayley) first to the matrix with a single letter, which marks people began to accept the matrix as an independent entity, Gloria's achievements also make it become the founder of matrix theory. Determinant and matrix are mostly independent development, their content and method relationship is not close, but eventually each other achievement, combined together.

Cauchy (1789-1857) Cayley (1821-1895)

Another source of linear algebra is analytic geometry, spatial (planar) point vectorization, making it very convenient to use algebra to study geometry, space element vector definition, making it possible to discuss high-dimensional space. In fact, the generalized vector is not confined to the discussion of space, but it is essentially a representation of the linear relationship that is generally present. Even in non-linear situations, it can be approximated by linear relation in local, which is widely used in computational mathematics. It is also found that in order to discuss the transformation relationship of vector space, it will eventually lead to the discussion of equations and matrices, so I also decided to open our linear path from the vector space.

Influenced by the axiomatic trend of thought, many basic concepts of linear algebra can also be expressed in the language of abstract algebra. Now that we have studied abstract algebra, I will not shy away from references to concepts in writing, but as long as you know the basic concepts, you will not be able to influence understanding. Here we are talking about linear algebra as a branch of abstract algebra, but the structure of this study is more special.

There are some specific problems in the space vector, one is the classification of the two-time curved surface (line) equation, with the aid of the matrix theory, the curved surface (line) equation can be reduced to the simplest form, which is convenient for classification, and this part is the latter two-dimensional theory. The other is that the concept of length and angle in space geometry needs to be extended to the general vector space, which is briefly described in the last part. Polynomial theory is also the basic tool of linear algebra, which is no longer elaborated because it has been discussed in a higher perspective in abstract algebra. There are also some independent branches of the matrix, such as matrix theory, matrix analysis, which is actually not related to the "linear", I intend to open another topic introduction.

Abstract algebra of "pre-order disciplines"

Resources

[1] Advanced algebra, qiu-dimensional sound, 2013

The classical textbook of advanced algebra, the author tries to put all the concepts in series, the sequence is novel and reasonable, the course content is very detailed, that is, suitable for the introduction, but also to the deep content of the discussion.

[2] Advanced algebra, Cha Jianguo, 1988

A classic old textbook, a relatively high angle of view, the layout of reasonable structure, analytical and meticulous, the explanation is enlightening, suitable for advanced reading.

[3] Advanced Algebra (3rd), Wang Jifang, 2003

An excellent introductory textbook in China, starting low, gradual, detailed and smooth, in line with the basis of less people to read. Exercises are rich, it is difficult to add exercises.

[4] Concise tutorial on Advanced Algebra (2nd), Blue to Medium, 2007

The overall comparison is concise and clear, the order has its own characteristics, to apply as a point of entry, increased the intuitive impression, the book also has a large length of the expansion of content.

[5] Advanced Algebra (5th), Schouten, 2007

Very simple introductory textbook, suitable for almost 0 basic people to read, the overall is a regular textbook, the content is more practical.

[6] Advanced linear algebra, Zhang Xianke, 2012

The content is very compact, the amount of information is relatively large, but also contains a lot of expanded content, no ideological, but suitable for direct access to conclusions.

[7] Introduction to Linear Space (3rd), Chen, 2009

An ingenious textbook, the material selection is relatively novel, different from the general linear algebra, the discussion has the strong inspiration and the guidance, does not emphasize chatty.

[8] linear algebra and Matrix theory (2nd), Xu Super, 2008

Calculate a regular textbook, content enrichment introduction, plus some extended knowledge, the order has its own characteristics, more suitable for beginners.

[9] "Linear algebra" (5th), Department of Mathematics, Tongji University, 2007

More than one textbook for engineering majors, only the most basic content of linear algebra is extracted, which is suitable for learning with the purpose of rapid application.

[10] Advanced algebra method of Problem Solving (2nd), Zhang Xianke, 2005

The content is very comprehensive, the topic coverage is wide, and the summary is very organized, if you want to find a summary of knowledge and exercise books, I first this one.

[11] "Linear algebra Problem Sets", Ploskulikov, 1978

The former Soviet Union problem sets, that amount and difficulty, you know! Anyway, I did not do, a cursory glance, the quality should be very high.

[12] Advanced algebra techniques and methods of solving problems, Libertone, 2003

A book to study the topic is more difficult, but it is research and enlightening, it deserves careful pondering.

[13] The essence of advanced algebra, Money Jilin, 2002

The general examination problem, most of them are very simple, and research significance, can be used as test practiced hand.

[14] Synthesis of advanced algebra, Cai Jianfang, 1986

A problem sets attached to a textbook can be practiced along with the textbook.

[15] "Advanced Linear Algebra", N. loehr,2014

[16] "Advanced Linear Algebra", S. roman,2007

Two classic English textbooks, but are some of the high-level content, I have not seen, interested can learn.

"Linear algebra" 01-Ancient new disciplines