Reading Notes-linear algebra in the procedural perspective

The core of Linear Algebra: Vector Model

Is linear algebra an objective natural law or a human design?

The commonality of all programming languages is that a set of models are established, a set of syntaxes are defined, and each syntax is mapped to specific semantics. Compliance between programmers and language implementersLanguage contract: the programmer ensures that the Code conforms to the language syntax, And the compiler/Interpreter ensures that the result of code execution conforms to the syntax-related semantics..

Program compilation and interpretation are essentially semantic mappings between different models.

From the perspective of application, linear algebra is a domain-specific language (DSL) designed by humans. It establishes a set of models and maps the syntax and semantics through the symbol system.

Creating a new model depends on the existing model, but this is a modeling method rather than the purpose. The purpose of any new model is to analyze and solve a certain type of problem more simply. When linear algebra is established, its various concepts and operation rules depend on elementary mathematics knowledge. However, once this abstract model is established, we willWe should be used to directly using high-level abstract models to analyze and solve problems..

The role of a model is to hide some of its complexity into the model.To make it easier for model users to solve problems.

The introduction of the dot product concept is a design, and there is room to make full use of it. Once the design is complete, the specific formula cannot be used freely. It must comply with the logic to ensure that it is mapped to the correctness of the elementary mathematical model. This is like a high-level language that can define many concepts, such as high-order functions and closures. However, it must ensure that the results produced during implementation mapped to the underlying implementation comply with the defined specifications.

The coordinate system we learned in Elementary Mathematics belongs to the analytical model proposed by Descartes. This model is very useful, but it also has great disadvantages. The coordinate system is a virtual reference system, but the problems we need to solve, such as area calculation, image rotation, stretching, and other applications are irrelevant to the coordinate system, creating a virtual coordinate system often does not help solve the problem.

The vector model overcomes the disadvantages of the analytical model. If the analytical model represents a "absolute" world view, then the vector model represents a "relative" world view, I recommend that you think of the vector model and the resolution model as two opposite models.

Linear Algebra defines the multiplication of Matrices and vectors, matrices, and matrices. The calculation rules are complex and unclear about what to do. Many beginners cannot understand it well, it can be said that matrix is the obstacle to learning Linear Algebra Well. When encountering complicated things, we often need to avoid getting into details first and grasp it as a whole. In fact, from the perspective of the program, no matter how strange the form is, it is nothing more than a syntax, And the syntax must correspond to the semantics, so the focus of the understanding of a matrix is to understand its semantics. There are more than one semantics of a matrix. There are different semantics in different environments and different interpretations can be made in the same environment. The most common examples include: 1) representing a linear transformation; 2) represents a set of column vectors or row vectors; 3) represents a set of child matrices.

This article puts forward a point of view: From the Perspective of application, we can regard linear algebra as a programming language in a specific field. Based on Elementary Mathematics, linear algebra establishes a vector model, defines a set of syntax and semantics, and conforms to the language contract of programming languages. Vector Model has coordinate system independence and linearity. It is the core of linear algebra and the best model for solving linear space problems.

Reading Notes-linear algebra in the procedural perspective