Understanding matrix (1)

Not long ago, for some ulterior motives, chensh wanted to act as a teacher and teach others linear algebra. As a result, I was held down and discussed with him several times about the virtual nature of linear algebra. Obviously, it is difficult for chensh to consider linear algebra as a mental disease rather than a strong student.

Poor chensh, who asked you to take a trip to this record ?! The color is so intelligent!

Linear Algebra courses, whether you start with the determinant or directly from the matrix, are full of inexplicable from the very beginning. For example, the Tongji Linear Algebra Teaching Material (now in the fourth edition), which is the most widely used in the Teaching of general engineering departments in China, first introduced the number of backward orders, then, we use the reverse order number to give an unintuitive definition of the determinant, then there were some silly deciding factors and exercises -- adding this line to another row by a factor, and then dropping that column, which was a hot topic, but I can't see why this is useful. Most of the students with mediocre qualifications like me are a little dizzy when they come here: even if this is something that is vague, they will start to perform in the Fire Circle. This is too "nonsense! As a result, some people skipped classes and more people began to copy homework. This is just a move, because the future development can be described with a return to the peak, followed by the deciding factor of this unsung head, it's the appearance of a great guy with no choice but no choice-Matrix! After many years, I realized that when the teacher put a bunch of silly data in brackets and said, "This is called a matrix, how miserable and miserable I was in my mathematical career! Since then, matrix has never been absent in almost all things that touch the word "learning. For the dumb I have never been able to deal with linear algebra at one time, the boss of matrix doesn't need to make me grayed out. For a long time, I have read a matrix, just as Q has seen a fake foreign devil, And he gritted his forehead and made a detour.

In fact, I am not a special case. It is usually difficult for engineering students to learn linear algebra. This situation exists at home and abroad. Swedish mathematician Lars garding said in his masterpiece encounter with mathematics:"If you are not familiar with the concept of linear algebra and want to learn natural science, now it seems like illiterate.", However"According to the current international standards, linear algebra is expressed through the Internet, and it is the second generation of mathematical model..., which brings about difficulties in teaching ."As a matter of fact, when we began to learn linear algebra, we unconsciously entered the scope of the "second generation mathematical model", which means that the expression and abstraction of mathematics had a comprehensive evolution, for those who have been learning the "First Generation Mathematical Model" since childhood, that is, the practical-oriented and specific mathematical model, it is strange not to feel difficult to perform such intense paradigm shift without explicit notice.

Most engineering students are often able to understand and skillfully use linear algebra after learning some subsequent courses, such as numerical analysis, mathematical planning, and matrix theory. Even so, even if many people are skilled in using linear algebra as a tool for scientific research and application, however, the questions raised by beginners of many courses seem to be very basic, but they are not clear. For example:

* What is a matrix? A vector can be considered as a representation of N mutually independent objects (dimensions). What is a matrix? If we think that a matrix is a new composite vector expansion composed of a group of columns (rows) vectors, why is this expansion so widely used? In particular, why is the two-dimensional expansion so useful? If every element in the matrix is a vector, then we expand it again and turn it into a three-dimensional cubic array, isn't it more useful?

* Why are the matrix multiplication rules defined in this way? Why can such a weird multiplication rule play such a huge role in practice? A lot of seemingly unrelated problems are actually attributed to the multiplication of matrices. Isn't that a wonderful thing? Is matrix multiplication an inexplicable rule that contains some essential laws of the world? If so, what are these essential rules?

* What is the determining factor? Why is there such a weird computing rule? What is the essentially relationship between a determining factor and its corresponding square matrix? Why is there a corresponding determinant only for the square matrix, but not for the general matrix? (do not think this problem is stupid. If necessary, it is not impossible to define the determinant for the m x n matrix, because there is no need for this, but why not )? What's more, it seems that the calculation rule of the determinant is not directly related to any calculation rule of the matrix. Why does it determine the nature of the matrix in many aspects? Is it just a coincidence?

* Why can a matrix be computed in blocks? Block computing seems so casual. Why is it feasible?

* For matrix transpose operation at, there is (AB) t = btat, for Matrix Inverse Operation A-1, there is (AB)-1 = B-1A-1. Two seemingly unrelated operations. Why are there similar operations? Is this just a coincidence?

* Why is the matrix obtained by P-1AP "similar" to a matrix "? What does "similarity" mean?

* What is the essence of feature values and feature vectors? Their definition is surprising, because AX = λ x, the effect of a non-large matrix, is actually equivalent to a small number λ, it is indeed a bit amazing. But why is it defined by "Features" or "characteristics? What exactly are they?

This type of problem often makes it difficult for people who have been using linear algebra for many years. It's like an adult is always forced to say "that's the case, so far" in the end. In the face of such problems, many veterans can only use it at the end: "You accept and remember this. However, if such a question cannot be answered, linear algebra is a rough, irrational, and inexplicable set of rules for us. We will feel that, I am not learning a course, but being "throttled" into a forced world. I am forced to drive away with the whip of the exam, there is no way to appreciate the beauty, harmony, and unity. After many years, we have discovered that this learning is so useful, but we are still confused: how can we make it happen so well?

In my opinion, this is the consequence of the loss of intuition in our Linear Algebra Teaching. The above questions related to "How can" and "How can" cannot satisfy the questioner if they are answered by pure mathematical proof. For example, if you prove through the general proof method that the matrix block operation is indeed feasible, this cannot solve the problem of the questioner. Their real confusion is: why is Matrix partitioning feasible? Is it just a coincidence, or is this determined by the nature of a matrix object? If it is the latter, what are these elements of the Matrix? As long as we think about the above problems, we will find that none of these problems can be solved simply by mathematical proof. As in our textbooks, students who use mathematics to prove everything and finally cultivate them can only use tools skillfully, but lack a real understanding.

Since the rise of the burbaki School in the 1930 s, the atomicity and systematic descriptions of mathematics have achieved great success, which has greatly improved the rigor of our mathematics education. However, one of the controversial side effects of math Atomicity is the loss of intuition in general math education. Mathematicians seem to think that intuition is in conflict with abstraction, so they do not hesitate to sacrifice the former. However, many people, including myself, are skeptical about this. We do not think that intuition and abstraction are in conflict, especially in Mathematics Education and mathematics teaching materials, it helps them understand those abstract concepts and then understand the nature of mathematics. On the contrary, if you focus on formal rigor, the students will become boring rules slaves like mice forced to perform the drill-in-fire performances.

I have repeatedly thought about linear algebra for four or five times over the past two years, for this reason, I have read several books on linear algebra, numerical analysis, algebra, and general mathematics at home and abroad: its content, methods, and meaning, Professor Zheng Yu's linear algebra 5 lecture, the aforementioned encounter with Mathematics (mathematical Overview), and Thomas. garrity's mathematical heritage has inspired me a lot. Even so, my understanding of this topic has gone through several self-denial times. For example, some of the conclusions I have thought about have been written in my own blog, but now it seems that these conclusions are basically incorrect. Therefore, I plan to record my current understanding completely. On the one hand, I think my current understanding is more mature. I can discuss it with others and ask for advice from others. On the other hand, if you have further understanding and overwrite your current understanding, the snapshot is also meaningful.

Because I plan to write more data, I will write it several times. I don't know if I have time to complete it, but whether it will interrupt or not. Just read it.

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Today, let's talk about the core concepts of Linear Space and matrix. Most of these things are written based on your own understanding. Basically, they do not copy books, and there may be errors, hoping to be pointed out. But I want to be intuitive, that is to say, I can talk about the real problems behind mathematics.

First, let's talk about space. This concept is one of the lifeblood of modern mathematics. From the perspective of topological space, we can add definitions step by step to form a lot of space. Linear Space is actually quite elementary. If the norm is defined in it, it becomes a norm linear space. When the linear space of the norm satisfies the completeness, it becomes the banah space. When the defined angle in the linear space of the norm has the inner product space, and the inner product space satisfies the completeness, the Hilbert space is obtained.

In short, there are many types of space. If you look at the mathematical definitions of a certain space, you can refer to space as "there is a set, a certain concept is defined on this set, and certain properties are satisfied. This is a bit strange. Why do we use space to call such a set? We will see that this is actually quite reasonable.

The space most common people are familiar with is undoubtedly the three-dimensional space in which we live (according to Newton's absolute concept of Time and Space). In mathematics, This is a three-dimensional Euclidean space, let's take a look at the basic features of such a space that we are familiar. Think about it and we will know that the three-dimensional space: 1. composed of many (actually Infinitely multiple) location points; 2. there is a relative relationship between these points; 3. the length and angle can be defined in the space; 4.This space can accommodate motion. Here we refer to the movement from one point to another, rather than the continuous movement in the calculus sense,

Among the above properties, the most critical is 4th. Article 1 and Article 2 can only be said to be the basis of space, not the unique nature of space. Any discussion of mathematical problems requires a set, and most of them have to define some structures (Relationships) on this set ), this is not to say that it is space. The 3rd items are so special that other spaces do not need to be possessed. They are not critical in nature. Only 4th items are the essence of space, that is,Accommodating motion is an essential feature of space.

By recognizing this, we can extend our understanding of 3D space to other spaces.In fact, no matter what space, it must accommodate and support the regular motion (transformation) in it ). You will find that there is often a corresponding transformation in a certain space. For example, there are topological transformations in the topological space, linear transformations in the linear space, and affine transformations in the affinic space, in fact, these transformations are just the motion forms allowed in the corresponding space.

So as long as you know, "Space" is a set of objects that hold motion, and transformation specifies the motion of the corresponding space.

Let's take a look at linear space. Linear Space is defined in any book, but since we acknowledge that linear space is a space, there are two basic problems that must be solved first:

1. Space is a set of objects, and linear space is also a set of objects. So what kind of object set is linear space? Or, what do objects in a linear space have in common?

2. How is the motion in a linear space expressed? That is, how is linear transformation represented?

First, let's answer the first question. When answering this question, you don't have to turn around. You can give a straight answer.Any object in a linear space can be expressed as a vector by selecting the base and coordinates.I will not talk about the normal vector space. Here are two extraordinary examples:

L1. a linear space is formed for all polynomials whose maximum number is not greater than N. That is to say, every object in this linear space is a polynomial. If we use x0, X1 ,..., XN is the basis, so any such polynomial can be expressed as a group of N + 1 dimensional vectors, where each component AI is actually the coefficient of X (I-1) in the polynomial. It is worth noting that there are multiple ways to select the base, as long as the selected group of linear independence can be. This involves the concepts mentioned later.

L2. the whole of the n-order continuous microfunction on the closed interval [a, B] forms a linear space. That is to say, every object in this linear space is a continuous function. For any continuous function, according to The weierstras theorem, we can find a polynomial function with the maximum number of items not greater than N, so that the difference between it and the continuous function is 0, that is, completely equal. In this way, the problem is reduced to L1. You don't need to repeat it later.

Therefore, vectors are very powerful. As long as you find a suitable base, vectors can represent any object in a linear space. There are many articles in this article, because the vector surface is only a number of columns, but in fact, due to its orderliness, in addition to the information carried by these numbers, they can also carry information in the corresponding positions of each number. In programming, why is array the simplest but powerful? This is the root cause. This is another problem.

Next we will answer the second question. The answer to this question involves a fundamental question of linear algebra.

Motion in a linear space is called linear transformation. That is to say, you can perform linear changes from one point in a linear space to any other point. How is linear transformation expressed?Interestingly, in linear space, when you select a group of bases, you can not only use a vector to describe any object in the space, in addition, a matrix can be used to describe any motion (transformation) in the space ). The method for causing a corresponding motion of an object is to use a matrix representing that motion, multiplied by the vector representing that object.

In short,After a base is selected in a linear space, the vector depicts the object, the matrix depicts the motion of the object, and the multiplication of the matrix and the vector applies the motion.

Yes, the essence of a matrix is the description of motion. If someone asks you what the matrix is, then you can tell him loudly,The essence of a matrix is the description of motion.. (Chensh, let's talk about you !)

But how interesting is the vector itself? Can it be regarded as a matrix of N x 1? This is really amazing,Objects and motion in a space can be expressed in a similar way.Can this be a coincidence? If it's a coincidence, it's a lucky coincidence! It can be said that most of the wonderful properties in linear algebra are directly related to this coincidence.

(To be continued)