"Operations Research" series of articles:

Let's first look at what operations research is? As Sima Qian wrote in Historical Records: Making correct deployments in a small military tent can determine Victory and defeat on the battlefield thousands of miles away. I think words such as "deployment and planning" can be used to explain the meaning of operations.

A very vivid metaphor is: the leader and his think tank are planning a specific project in a conference room to achieve the expected goals. Operations research is the study of how to better plan to achieve this goal. It can give a specific plan or some powerful information. Of course, for a leader, it is impossible to completely refer to the plan directly given by a model in operations research. In addition, sometimes a plan cannot be given directly, but some valuable conclusions can be consulted. . I think there is a sentence in the textbook that sums it up very well:

In fact, it is not just leaders, there are many examples in our daily lives that can be planned. For example, how do I go from Zhongke Building to the Bund? I can choose to take the subway, bus or even ride a small yellow car directly there. I can plan a plan based on the most economical goal. But because these are small things in life, we can generally use some common sense to judge.

Let’s take a closer look at the planning issue from Zhongke Building to the Bund.

Let’s only consider these factors first, and then we will classify the input variables:

These input elements have certain constraints, which we use g() to represent.

At the same time, it has evaluation criteria, which we use U=f() to represent here. In this example the evaluation criterion is "most economical".

The above professional terms are written on page 7 of the "Operations Research" textbook.

We divide these two problems into two sub-problems.

The latter question is more interesting. Let’s take the above one from Zhongke Building to the Bund as an example. I don’t need to know anything about linear programming or these theories full of mathematical symbols. To make it simple, I just take the subway, because the distance is relatively long, the subway is faster, and it is less prone to traffic jams. From this example, we can see that this is indeed the case. We do not need to use mathematics. From practical experience, taking the subway is also a good choice.

But let’s take a longer view, if you want to deal with the problem as mentioned in the textbook: "In 2008, formulate a new Dutch train timetable."

At this time, I think it is absolutely impossible for you to plan this problem without using mathematics. You can't look at a map and see the parameters of a train and just write down on paper how it should be arranged. Therefore, we should not stick to some small problems, but now we should enlarge our sights and take a long-term view. Some more examples are given in the textbook: for example, in the military and in marketing. At this time you have to use mathematics to solve these scientific planning problems.

In fact, "Linear Algebra" is not unpopular at all. There are roughly twelve subject categories in the subject setting in our country. The natural sciences are generally considered to refer to: The four major subject categories of science, engineering, agriculture and medicine: they are just the two major applied disciplines of biology: Agriculture and medicine pay more attention to experiments and generally do not need So much math. But for the majority of engineering majors, "Linear Algebra" is a very important course, and a lot of mathematics is used in many professional courses. When I was taking the course "Numerical Analysis" (a professional core course in the Department of Mathematics) as an undergraduate, the mathematics teacher at the time often said: Many engineering majors learn more deeply than our mathematics majors.

Let’s get to the point: Why are there so many linear algebras? First, let me make an argument that makes people feel a little confused: These theories of matrices are the second generation of mathematical modeling languages. In order to illustrate this argument, I divided it into two parts:

I don’t know if you have noticed the concept of language. It is the most important communication tool for human beings and the various expression symbols used by people to communicate. It is an important tool for people to preserve and pass on the achievements of human civilization. We are generally familiar with Mandarin and English, but in fact mathematics is a language, but it focuses on reading and writing, and is full of a large amount of logic, structure and symbolic formal systems. At the beginning of the 20th century, there were three major schools of mathematics: first, the formal school represented by Hilbert, which believed that mathematics was a form; second, the French school of structural mathematics, which believed that mathematics was a structure; Third, the logic school believes that mathematics is logic. But later we learned that we cannot simply think of mathematics as one of them, but should look at the three together.

Let’s look at it dialectically. Since the language of mathematics is full of a large amount of logic, structure and symbolic formal systems, it certainly does not have the same features as natural languages ??(Mandarin, English, etc.) Some features that can please people, such as writing novels, telling jokes, etc.

Yes, mathematics is the language of science. Galileo Galilei had a similar saying! There are two important concepts to distinguish here: the difference between mathematics and arithmetic! An obvious example is: We in ancient China have known "hook three strands, four strings and five" for a long time. If we need to use a right-angled triangle in actual production, if we know that the two right-angled sides are 3 and 4, then we can calculate the third The side is 5, that's arithmetic. But when it comes to the Pythagorean Theorem: a^2 + b^2 = c^2, it becomes abstract and becomes mathematics.

On this issue, I strongly recommend that you read Teacher Meng Yan’s two blog posts: Understanding Matrix.

Generally speaking, the most basic elements of language are vocabulary, collocation and grammar. Vectors are equivalent to words in mathematical language, and matrices are collocations, and combined with certain grammar, they form the matrix description we see (such as page 59 of the textbook: Matrix description of the simplex method).

Let’s talk about a little aside. The system used to imprison people in "The Matrix" is also called the Matrix. Matrix is ??a complex simulation system program, and its basic operating language must be matrix. You can read this blog post.

There is also the "Biostatistics" we will learn in the next semester. It can be written in matrix form, but I don't think it is done this way. When I was studying "Biostatistics and Field Experiments" in my undergraduate course I learned the matrix expression form.

Let’s look back at analytic geometry, which was a headache in high school. The one we dealt with most was Vedic’s theorem. Generally speaking, if you make a mistake in this analytic geometry, it will be very difficult to continue the calculation later. Later, when I entered university, I discovered that there was a course called "Advanced Algebra" (linear algebra is part of advanced algebra). The calculation method in the university is to use vectors and matrices to perform operations. Its most basic operation is no longer the operation of a real number, but the simultaneous operation of a group of real numbers.

Let’s look at what a mathematical model is: for a problem, we can explicitly take out an f(x) to solve the problem and give a solution. Then this f(x) is a mathematical model. It's just that x is no longer a real number, but a vector or matrix.

Using the matrix theory, we can better discuss the planning problems in "Operations Research", so the textbook has so much linear algebra content. Of course, we don’t actually need to study these things thoroughly.

As the textbook says in the introduction (see next question), we just need to know how to use it!