Some intuitive insights on Linear Space

Part 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.

Part 2

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.

Then we understand the matrix.

In the previous article, "the matrix is the description of motion." So far, it seems that everyone has no opinions yet. But I believe that sooner or later there will be a netizen from the mathematics department. Because the concept of motion is linked to calculus in mathematics and physics. When we study calculus, some people will follow the instructions to tell you that elementary mathematics is a constant mathematics, a static mathematics, a variable mathematics, and a motion mathematics. It is widely said that almost everyone knows this sentence. But there are not many people who really know what this sentence means. In short, in our human experience, motion is a continuous process. from point A to point B, even if the fastest light goes, it takes timePoint by pointThrough the path between AB, this brings about the concept of continuity. However, if we do not define the limit, we can't explain it. The ancient Greek mathematics was very strong, but they lacked the Limit Concept, so they could not explain the movements. The famous paradox of zhino (four paradox s, such as flying arrows and flying legs, Achilles, and turtles) let's get it done. This article is not about calculus, so I will not talk about it much. If you are interested, you can go to Professor Qi minyou's "review calculus". I just read the beginning of this book to understand the truth of the sentence "Advanced Mathematics is a mathematical study of motion.

However, in my article "Understanding matrix", the concept of "motion" is not a continuous movement in calculus, but an instantaneous change. For example, at, after a "movement","Transition"To B, which does not need to go through any point between A and B. Such "motion" or "transition" is against our daily experience. However, those who have some knowledge about quantum physics will immediately point out that quantum (such as electronics) jumps on different levels of energy in an instant and has such a kind of transition behavior. Therefore, this kind of movement does not exist in nature, but we cannot observe it at a macro level. However, in any case, the word "motion" is often used here to produce ambiguity. To be more precise, it should be "transition ". Therefore, this sentence can be changed:

"Matrix is the description of transition in Linear Space ".

This is too physical, that is, too specific, not mathematical, that is, not abstract. So we finally use a correct math term --Transform. In this case, everyone should understand,The so-called transformation is actually the transition from one point (element/object) to another point (element/object) in the space.. For example, a topological transformation is a transition from one point to another in a topological space. For example, an affine transformation is the transition from one point to another in an affined space. Additionally, this affine space and vector space are siblings. All the friends who do computer graphics know that, although a 3D object only needs 3D vectors to be described, all the computer graphics transformation matrices are 4x4. For the reason, many books have written "for convenience", which seems to me to be an attempt to pass through. The real reason is that the graphic transformation applied in computer graphics is actually carried out in the affine space rather than the vector space. Think about it. In vector space, when a vector is moved in parallel, it is still the same vector. However, the real world and other long parallel line segments cannot be considered the same thing, therefore, the living space of computer graphics is actually an affine space. The Matrix Representation of the affine transformation is 4x4 at all. If you are interested, you can go to computer graphics- ry tool algorithm details.

Once we understand the concept of "transformation", the definition of the matrix becomes:

"A matrix is the description of the transformation in a linear space ."

So far, we finally got a definition that looks more mathematical. But I have to say a few more words. This is generally the case in teaching materials. A linear transformation T in a linear space V can be expressed as a matrix after being elected as a group of bases. Therefore, we need to clarify what linear transformation is, what is base, and what is a group of base. The definition of linear transformation is very simple. It has a transformation T, so that for any two different objects x and y in the linear space V, as well as any real numbers A and B, there are:
T (AX + by) = at (x) + Bt (Y ),
T is called linear transformation.

The definition is written in this way, but the definition cannot be intuitively understood. What kind of transformation is linear transformation? As we have just said, transformation is a point transition from space to another point, while linear transformation, it is the movement from one point of a linear space V to another point of another linear space w. This sentence contains a layer of meaning, that is, a point can be transformed not only to another point in the same linear space, but also to another point in another linear space. No matter how you change it, as long as the objects in the linear space are before and after the transformation, this transformation must be a linear transformation, and it must be described using a non-singular matrix. A transformation you describe using a non-singular matrix must be a linear transformation. Some may ask, why do we emphasize non-singular matrices? The so-called non-singular, only meaningful to the square matrix, so what is the situation of non-square matrix? This will be lengthy. At last, we need to take linear transformation as a ing and discuss its ing nature and the concepts of kernel and image of linear transformation. I don't think this is important. If you have time, write it later.Below we will only discuss the most common and useful transformation, that is, linear transformation within the same linear space. That is to say, if the matrix mentioned below is not described, it is a square matrix, and it is a non-singular square matrix. To learn a subject, the most important thing is to grasp the main content and quickly establish the overall concept of this subject. You don't have to consider all the details and special circumstances from the very beginning.

Next, what is the base? This question will be further discussed later.Consider the base as the coordinate system in a linear space.Note that it is a coordinate system, not a coordinate value, which is a "unity of opposites ". In this way, "selecting a group of bases" means selecting a coordinate system in a linear space. This means.

Well, the definition of the matrix is improved as follows:

"A matrix is a description of linear transformation in a linear space. In a linear space, as long as we select a group of bases, any linear transformation can be described using a definite matrix ."

The key to understanding this sentence is to distinguish "linear transformation" from "A description of linear transformation.One is the object, and the other is the expression of the object. It is like in the familiar object-oriented programming, an object can have multiple references. Each reference can be called by a different name, but it is the same object. If you do not have an image, you can simply make a vulgar analogy.

For example, if you want to take a picture of a pig, you can take a picture of the pig as long as you select a camera position. This picture can be seen as a description of this pig, but it is only a one-sided description. Because I can take a picture of this pig with another camera, I can get a different picture, this is another one-sided description of this pig. All the pictures taken in this way are described by the same pig, but they are not the same pig.

Similarly, for a linear transformation, as long as you select a group of bases, you can find a matrix to describe this linear transformation. If you change the base group, a different matrix is obtained. All these matrices are described in the same linear transformation, but they are not linear transformations themselves.

However, the problem arises. If you give me two pictures of pig, how do I know that the two pictures contain the same pig? Similarly, how do I know that these two matrices describe the same linear transformation? If it is a different matrix description of the same linear transformation, it is the brother of the family. If you don't know each other, it won't be a joke.

Fortunately, we can find a property of the matrix brothers of the same linear transformation, that is:

If matrix A and matrix B are two different descriptions of the same linear transformation (the reason is that different bases are selected, that is, different coordinate systems are selected ), then, a non-singular matrix P can be found to satisfy the following relationship between A and B:

A = P-1BP

If linear algebra is a bit familiar, you can see that this is the definition of the similarity matrix. Yes,Similar matrices are different descriptors of the same linear transformation.According to this definition, photos from different angles of the same pig can also be similar photos. It is a little vulgar, but it is understandable.

In the formula above, the matrix P is actually a transformation relationship between the base based on matrix A and the Base Based on matrix B. This conclusion can be proved in a very intuitive way (rather than the formal proof in general textbooks). If I have time, I will add this proof to my blog later.

This discovery is too important.The original family of similarity matrices are described in the same linear transformation!No wonder this is so important! Graduate courses in engineering include matrix theory, matrix analysis, and other courses, which talk about a variety of similar transformations, such as what is similar standard type, and what is the content of the right corner, the matrix obtained after transformation is required to be similar to the previous matrix. Why is this requirement? Only in this way can we ensure that the two matrices before and after the transformation describe the same linear transformation. Of course, the descriptions of different matrices of the same linear transformation do not distinguish the ring from the actual computing nature. Some descriptive matrices are much better than other matrices. It is easy to understand that photos of the same pig are also ugly. Therefore, the similarity transformation of a matrix can convert an ugly matrix into a more beautiful matrix, ensuring that both matrices describe the same linear transformation.

In this way, the matrix is basically clearly described as a side of linear transformation. However, it is not that simple, or linear algebra has a more amazing nature than that, that is,A matrix can be used not only as a description of linear transformation, but also as a set of base descriptions. As a matrix of transformation, we can not only transform one point in a linear space to another, but also change one coordinate system (base) table in a linear space to another coordinate system (base) go. In addition, the transformation point and the transformation coordinate system have the same effect. The most interesting mysteries of linear algebra are contained in them. With this understanding, many theorems and rules in linear algebra become clearer and more intuitive.