Basic introduction
Linearity refers to the proportion and linear relationship between quantities, which can be mathematically understood as a function with a constant first derivative.
Nonlinear refers to the non-proportional nonlinear relationship, and the first derivative is not constant.
Linear algebra originated from the study of two-dimensional and three-dimensional rectangular coordinate systems. Here, the vector is a directed linear algebra of line segments, which is expressed by length and direction. Such a vector can be used to represent a physical quantity, such as a force, or it can be added and multiplied by a scalar. This is the first example of real vector space.
Modern linear algebra has been extended to study arbitrary or infinite dimensional space. A vector space with dimension n is called an n-dimensional space. Most useful conclusions in 2D and 3D spaces can be extended to these high-dimensional spaces. Although it is difficult for many people to imagine a vector in N-dimensional space, such a vector (that is, an N-tuple) is very effective for representing data. Because as an n-tuple, vector is an "ordered" list of n elements, most people can effectively summarize and manipulate data under this framework. For example, eight-dimensional vectors can be used in economics to represent the gross national product (GNP) of eight countries. When arranging the order of all countries, such as (China, America, Britain, France, Germany, Spain, India and Australia), you can use vectors (V 1, V2, V3, V4, V5, V6, V7 and V8) to show their respective GNP in a certain year. Every country's GNP here is in its own position.
Vector space (linear space), as a purely abstract concept used to prove theorems, is a part of abstract algebra and has been well integrated into this field. Some famous examples are: irreversible linear mapping or matrix group, and linear mapping ring in vector space. Linear algebra also plays an important role in mathematical analysis, especially in describing higher-order derivatives in vector analysis and studying tensor products and commutative mappings.
Vector space is defined on a field, such as a real number field or a complex number field. Linear operators map the elements of one linear space to another linear space (or the same linear space), and keep the consistency of addition and scalar multiplication in vector space. The set of all these transformations is itself a vector space. If the basis of a linear space is determined, all linear transformations can be expressed as a table called a matrix. The in-depth study of matrix properties and matrix algorithms (including determinant and eigenvector) is also considered as a part of linear algebra.
We can simply say that linear problems in mathematics-those that show linearity-are the easiest to solve. For example, differential calculus studies many linear approximation problems of functions. In practice, the difference from nonlinear problems is very important.
Linear algebraic method refers to the method of looking at the problem from a linear point of view and describing and solving it in the language of linear algebra (matrix operation can be used if necessary). This is one of the most important applications in mathematics and engineering.