Mathematics and Algorithms

A function is an infinite vector. Many definitions, theorems, equations, and inequalities in plane ry are applicable in function spaces. For example, the well-known schwaz-Coq inequality is just the extension of the plane triangle | cos (t) | <= 1.

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Calculus is used to solve the problems of "seeing the Shadows" and "looking into the shadows. By studying local simple problems, we can grasp the global complex problems. The bridge between them is the formula of Niu-Lai. In the external differential form, the ogao formula and the forkes formula are both the niukelai formula, that is, the 1, 2, and 3-dimensional niukelai formula, A simple Green formula can be used to summarize all the contents of the university calculus course.

A differential equation is an equation describing the "simple local problem". In essence, it is a description of a local rule. Because many linear assumptions can be made, this local rule is relatively easy to find. Therefore, many scientific disciplines can list differential equations. However, only the differential equations can grasp the overall nature, but it is not easy to solve the differential equations.

Academician Lin Qun said that each subject corresponds to a differential equation.

Local problems are well solved, while a large number of local problems are solved. After the results are accumulated, the global goal can be achieved. The algorithm is like this. Especially for recursive and iterative algorithms, a recursive/iterative process itself is a local rule, and its meaning is the same as that of a differential equation. Therefore, many theorems originally found in the theory of differential equations, such as the fixed point theorem, are also used in computational theory. The complexity of recursive algorithms may also be calculated using the theory of differential equations.

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The dot product of two vectors is equal to the projection length of one vector on the other and the algebraic sum of the product of the coordinate component of the two vectors. This is a wonderful thing. Even if it is easy to prove, I still think it is amazing. How can it be of such a wonderful nature?

A vector corresponds to a directed line segment, and a group of vectors corresponds to a group of directed line segments. Does a non-singular matrix correspond to a group of vectors in an n-dimensional space, and these vectors constitute a coordinate system. A vector multiplied by a matrix is a new vector formed by the projection of the vector on each axis of the new coordinate system represented by that matrix. It can also be said that the matrix is a vector converter. For a non-singular matrix, some vectors are particularly interesting. They form a new vector in the projection of the coordinate system, which is exactly twice the lambda of the original vector. That is to say, after the transformation of the matrix vector converter, there is no difference between the original vector and the real number lambda. So Lambda depicts a feature of the matrix, called the feature value of the matrix.

A matrix multiplication matrix is the projection value of a group of vectors in the coordinate system formed by another group of vectors. Orthogonal matrix is such a matrix that is projected on itself and the result is a matrix I. When will this happen? Of course, only in the vector group represented by this matrix, all vectors are perpendicular to each other. Therefore, the name is called an orthogonal matrix.