线性代数（linear algebra）
Linear algebra (Linear Algebra) is a branch of mathematics. Its rearch objects are vectors, vector spaces (or linear spaces), linear transformations and finite dimensional linear equations. Vector space is an important subject in modern mathematics. Therefore, linear algebra is widely ud in abstract algebra and functional analysis. Linear algebra can be expresd concretely by analytic geometry. The theory of linear algebra has been generalized to operator theory. Since nonlinear models in scientific rearch can often be approximated as linear models, linear algebra has been widely applied to natural and social sciences.
The development of linear algebra
Becau the work of Descartes and Fermat, linear algebra basically appeared in venteenth Century. Until the late eighteenth Century, the field of linear algebra was confined to planes and spaces. The first half of nineteenth Century to complete the transition matrix to the n-dimensional vector space theory begins with Kailai in the cond half of nineteenth Century, becau if when work reached its culmination in.1888, Peano axiomatically defined finite or infinite dimensional vector space. Toeplitz will be the main theorem is generalized to arbitrary body linear algebra on the general vector space. The c
oncept of linear mapping can in most cas get rid of matrix computation directed to the inherent reasoning, that is not dependent on the lection of the ba. Do not exchange and exchange or not with the ring as the operator domain, this concept to die, this concept very significantly extended vector space theory and re organize the nineteenth Century Institute
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The word "algebra" appeared relatively late in China, in the Qing Dynasty when the incoming China, it was translated into "Alj Bala", until 1859, the Qing Dynasty famous mathematician, translator Li Shanlan translated it as "algebra", still in u.bios是什么意思
世界末日英文The status of linear algebra
关于朋友的英语作文Linear algebra is a subject that discuss matrix theory and finite dimensional vector spaces combined with matrices and their linear transformation theory.
The main theory is mature in nineteenth Century, and the first cornerstone (the solution of two or three Yuan linear equations) appeared as early as two thousand years ago (e in our ancient mathematical masterpiece "nine chapters arithmetic").
sunshineThe linear algebra has many important applications in mathematics, mechanics, physics and technology, so it has important place in various branches of algebra;
In the computer today, computer graphics, computer aided design, cryptography, virtual reality and so on are all part of the theory and algorithm of linear algebra;.
Between geometric and algebraic methods embodied in the concept of the subject of the connection from the axiomatic method on the abstract concept and rigorous logic reasoning, cleverly summed up, to strengthen people's training in mathematics, science and intelligent gain is very uful;
And with the development of science, we should not only study the relationship between the individual variables, but also further study the relationship between multiple variables, all kinds of practical problems in most cas can be linearized, and becau of the development of the computer, the linearized problem can be calculated, linear algebra is a powerful tool to solve the problems.
Basic introduction to linear algebra
Linear algebra originated from the study of two-dimensional and three-dimensional Cartesian coordi
nate systems. Here, a vector is a line gment with a direction that is reprented by both length and direction. Thus vectors can be ud to reprent physical quantities, such as force, or to add and multiply scalar quantities. This is the first example of a real vector space.
Modern linear algebra has been extended to study arbitrary or infinite dimensional spaces. A vector space of dimension n is called n-dimensional space. In two-dimensional and
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three-dimensional space, most uful conclusions can be extended to the high-dimensional spaces. Although many people do not easily imagine vectors in n-dimensional space, such vectors (i.e., n tuples) are very uful for reprenting data. Since n is a tuple, and the vector is an ordered list of n elements, most people can effectively generalize and manipulate data in this framework. For example, in economics, 8 dimensional vectors can be ud to reprent the gross national product (GNP) of 8 countries. When all the national
order (such as scheduled, China, the United States, Britain, France, Germany, Spain, India, Australia), you can u the vector (V1, V2, V3, V4, V5, V6, V7, V8) showed that the countries a year each GNP. Here, each country's GNP are in their respective positions.mua是什么意思
As a purely abstract concept ud in proving theorems, vector spaces (linear spaces) are part of abs
tract algebra and have been well integrated into this field. Some notable examples are: irreversible linear maps or groups of matrices, rings of linear mappings in vector spaces. Linear algebra also plays an important role in mathematical analysis,
keepthefaithEspecially in vector analysis, higher order derivatives are described, and tensor product and commutative mapping are studied.
A vector space is defined on a domain, such as a real or complex domain. Linear operators map the elements of a linear space into another linear space (or in the same linear space), and maintain the consistency of addition and scalar multiplication in the vector space. The t of all such transformations is itlf a vector space. If a basis of linear space is determined, all linear transformations can be expresd as a table, called matrix. Further studies of matrix properties and matrix algorithms (including determinants and eigenvectors) are also considered part of linear algebra.
We can simply say that the linear problems in Mathematics - tho that exhibit linear problems - are most likely to be solved. For example, differential calculus studies the problempinan
of linear approximation of functions. In practice, the difference between a nonlinear problem and a n
onlinear one is very important.
The linear algebra method refers to the problem of using a linear viewpoint to describe it and to describe it in the language of linear algebra and to solve it (when necessary) by using matrix operations. This is one of the most important applications in mathematics and engineering.
Some uful theorems
Every linear space has a ba.
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The nonzero matrix n for a row of N rows A, if there is a matrix B that makes AB = BA = I (I is the unit matrix), then A is nonsingular matrix.
A matrix is nonsingular if and only if its determinant is not zero.
A matrix is nonsingular if and only if the linear transformation it reprents is a automorphism.
A matrix is mi positive if and only if each of its eigenvalues is greater than or equal to zero.
A matrix is positive if and only if each of its eigenvalues is greater than zero.
Generalizations and related topics
