Linear algebra should learn this way

Source: Internet
Author: User

On the Internet to see an article, looked after the feeling quite deep. He tells the nature of linear algebra, and intuitively describes linear spaces, vectors, and matrices.

The linear algebra course, whether you start from the determinant or directly from the matrix, is riddled with inexplicable beginnings.

For example, in the national General Engineering Department of teaching the most widely used in Tongji linear algebra textbook (now to the fourth edition), the first to introduce the odd concept of reverse order number, and then the number of reverse order to give the determinant of a

Very non-intuitive definition, followed by some simply silly determinant of the nature and exercise-the line by one factor to add to another row, and then to reduce the column, tossing it to call a lively, but it is not see

What's the use of this thing?

Most of the mediocre students like me here are a little dizzy: even this is a vague thing, began to drill the fire circle performance, it is too unreasonable! So people started skipping classes, more

People began to copy their homework. This is the middle of the move, because the subsequent development can be described with a twist, followed by this nonsense determinant, is an equally unreasonable but great to the extreme of the guy's out

Field--The matrix came! Years later, I realized that when the teacher foolishly used the brackets to put a bunch of silly, and slowly said: "This thing is called matrix" when my math career

What a solemn and tragic scene! Since then, in almost everything with the word "learning" a little bit of the edge of things, the matrix this guy never absent. For me, this one's not going to fix linear algebra.

The idiot, the Matrix boss's uninvited often make me disgraced, badly beaten. For a long time, I read in the matrix, just as Ah Q saw false foreign devil, rubbing forehead on detour.

In fact, I'm not an exception. General engineering students often find it difficult to beginner linear algebra. This situation is at home and abroad. Swedish mathematician Lars Garding in its masterpiece encounter with

Mathematics said: "If you are not familiar with the concept of linear algebra, to learn the natural sciences, now seems to be similar to illiteracy." However, according to the current international standards, linear algebra is expressed by axiomatic,

It is a second-generation mathematical model, which brings about difficulties in teaching. "In fact, when we began to learn linear algebra, unconsciously entered the" second generation mathematical model "category, which means that the mathematical

The way of expression and abstraction has a comprehensive evolution, for we have been in the "first generation mathematical model", that is, a practical-oriented, concrete mathematical model of learning, in the absence and clearly informed of the situation

It is strange to have such a drastic paradigm shift without feeling the difficulty.

Most engineering students tend to learn a number of follow-up courses, such as numerical analysis, mathematical programming, and matrix theory, before they are able to understand and skillfully use linear algebra. Even so, a lot of people, even if they are familiar

The practice of using linear algebra as a tool for scientific research and application, but it is not clear to many of the subjects of this course that appear to be very basic questions. For example:

1. What is a matrix?

2, the vector can be considered to have n independent nature (dimension) of the object's representation, what is the matrix?

3, if we think that the matrix is a set of columns (row) vector composed of a new composite vector expansion, then why this expansion has such a wide range of applications? In particular, why is the two-dimensional expansion so useful?

4, if each element of the matrix is a vector, then we expand once again, become three-dimensional square, is not more useful?

5. Why is the multiplication rule of matrix so stipulated? Why would such a bizarre multiplication rule have such a huge effect in practice? Many seem to be completely unrelated issues, and finally

It all boils down to the multiplication of matrices, isn't it a wonderful thing? Is there some fundamental law of the world under the rule of matrix multiplication that seems inexplicable? If so, these essential laws are

What the?

6. What is a determinant? Why is there such a strange rule of calculation? What is the relationship between determinant and its corresponding phalanx in nature? Why only the Phalanx has the corresponding determinant, and the general matrix is

No (don't think this is stupid, if necessary, the definition of determinant for the MXN matrix is not to be done, the reason why not, because it is not necessary, but why is not necessary)? And

The determinant of the calculation rules, it seems that the matrix of any calculation rules are not intuitive contact, why in many ways determine the nature of the matrix? Is this all just a coincidence?

7, why the matrix can be divided into block calculation? Chunking calculation This thing seems so random, why is it feasible?

8, for Matrix transpose operation at, there is (AB) T=btat, for the matrix to seek inverse A-1, there is (AB) -1=b-1a-1. Why is there a similar nature to the two operations that seem to have nothing to do with it?

Is this just a coincidence?

9, why say P?1ap get matrix and a matrix "similar"? What do you mean by "similarity" here?

10. What is the nature of eigenvalues and eigenvectors? Their definition is very surprising, because Ax=λx, a large matrix of the effect, but the equivalent of a small number λ, is indeed a bit strange. But what

What is the definition of "characteristic" or even "intrinsic"? What exactly are they carved out of?

Such a problem often makes it difficult for people who have been using linear algebra for many years. It is as if adults face the children's inquisitive, and in the end there will be forced to say, "This is it, that's it,"

In the face of such problems, many veteran end can only use: "It is so stipulated, you accept and remember the good" to stall.

However, if the question is not answered, linear algebra is a rude, unreasonable, inexplicable set of rules for us, and we will feel that we are not learning a science

Asked, but was without any explanation "thrown into" a mandatory world, only in the test of the whip waving forced to travel, completely unable to appreciate the beauty, harmony and unity. Until many years later, we have

Find this knowledge so useful, but still very confused: why so coincidence? I think this is the result of intuitive loss in our linear algebra teaching. These relate to "How to", "how can"

Question, only by pure mathematical proof to answer, can not make the questioner satisfied. For example, if you demonstrate that the matrix block operation is feasible through the general proof method, it is not possible to

The doubts of the question were resolved. Their real confusion is: Why is the matrix chunking operation feasible? Is it just coincidence, or is it necessarily determined by the nature of the object of the matrix? If

is the latter, then what are these essence of the matrix? Just a little consideration of the above questions, we will find that all these problems are not purely based on mathematical proof can be solved. Like our Department of Education.

In the book, all things mathematically prove that the students who are trained in the end can only use the tools skillfully, but lack the real sense of understanding.

Since the rise of the French Bourbaki school in the 1930 's, the axiomatic and systematic description of mathematics has been a great success, which has greatly improved the rigor of the mathematics education we receive. However the mathematical public

A controversial side effect of physics and chemistry is the loss of intuition in general mathematics education. Mathematicians seem to think that intuition and abstraction are contradictory, so do not hesitate to sacrifice the former. including me, though.

Many of us are skeptical, and we don't think that intuition and abstraction are necessarily contradictory, especially in math education and math textbooks, to help students build intuition and help them understand that

Some abstract concepts to understand the nature of mathematics. Conversely, if you pay attention to formal rigor, students as if forced to drill the fire ring in the performance of the mice, like the dry rules of slavery.

For the linear algebra similar to some of the above mentioned intuitive problems, for more than two years I have repeatedly thought of 四、五次, for this read several domestic and foreign linear algebra, numerical analysis, algebra and number

Learn the general theory of books, such as the former Soviet Classics "Mathematics: its content, methods and significance", Professor Gong Sheng's "linear algebra Five", the aforementioned encounter with mathematics

("A Mathematical Overview") and Thomas A. Garrity's "math supplements" gave me great inspiration. But even so, my understanding of the subject has gone through several self-denial. Like thinking before.

Some of the conclusions have been written in their own blog, but now it seems that these conclusions are basically wrong. So I intend to make a complete record of my present understanding, on the one hand, because I think

Now the understanding is more mature, can take out with others to discuss, to others to consult. On the other hand, if there is a further understanding of the future, the present understanding of the overthrow, and now write this snapshot

is also very meaningful.

Linear space

Let's talk about the understanding of several core concepts of linear space and matrices today. Most of these things are written in their own understanding, basically not transcription, there may be mistakes in the place, hoping to be pointed out. But I

Hope to be intuitive, that is to say the real problem behind the math.

First of all talk about space, this concept is one of the lifeblood of modern mathematics, starting from the topological space, step by step up the definition, can form a lot of space. Linear space is actually relatively elementary, if the

The norm is defined in the inside, and it becomes the normed linear space. The normed linear space satisfies the completeness, it becomes the Banah space, and the defined angle in the normed linear space has the inner product space and the inner product space satisfies the completeness.

You get the Hilbert space. In short, there are many kinds of space. If you're going to see a mathematical definition of a space, it's basically: there's a set that defines a certain concept on this set, and then satisfies some nature,

It can be called space. This is a bit strange, why use "space" to call some of these collections? As you will see, this is actually very reasonable. The most familiar spaces of our people, no doubt

Is the three-dimensional space that we live in (according to Newton's absolute space-time view), mathematically speaking, this is a three-dimensional Euclidean space, we do not care so much, first look at our familiar such an empty

Some of the most basic characteristics, and then we live in this three-dimensional space to extend to other space. Think about it and we'll know that this three-dimensional space:

1. Consists of many (in fact, 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 movement, and here we call the movement from one point to another point of movement (transformation), rather than the "continuous" movement of the meaning of calculus.

Of these properties above, the most critical is the 4th article. 1th, 2 can only be said to be the basis of space, not the characteristics of space-specific, all the discussion of mathematical problems, there must be a set, most still have in this set

Define some structures (relationships), not that they are space. And the 3rd is too special, the other space does not need to have, is not the key nature. Only the 4th is the nature of space, that is to say,

accommodating motion is the essential feature of space . Recognizing this, we can extend our understanding of three-dimensional space to other spaces. In fact, whatever the space, it must accommodate and support the occurrence in which

Rules-compliant motion (transformation). You will find that there is often a relative transformation in some space, such as topological transformations in topological spaces, linear transformations in linear spaces, and affine changes in affine spaces.

In fact, these transformations are just the permissible form of motion in the corresponding space. so as long as it is known that "space" is the set of objects that hold motion, and the transformation specifies the motion of the corresponding space . Below we

Take a look at the linear space. The definition of a linear space is available in any book, but since we recognize that the linear space is a space, then there are two basic questions that must be solved first:

1. Space is a collection of objects, a linear space is also a space, so it is also a collection of objects (a collection). So what is the linear space of a collection of objects? Or, what do objects in linear space have in common?

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

We first answer the first question, the answer to this question is not to beat around the bush, can be straightforward to give the answer: any object in the linear space, by selecting the base and coordinates of the method, can be

in the form of expressions as vectors . The usual vector space I won't say, give two less mundane examples:

1, L1 is the highest of the polynomial is not greater than n times the whole composition of a linear space, that is, each object in this linear space is a polynomial. If we are based on x0,x1,..., xn, then any

Any such polynomial can be expressed as a set of n+1-dimensional vectors, in which each component AI is actually the coefficients of the xi?1 term in the polynomial. It is worth noting that there are many ways to select a base, as long as the selected

That group of bases is linearly irrelevant. This is going to use the concept mentioned later, so let's not say it, just mention it.

2, L2 is a closed interval [a, b] on the N-Order continuous micro-function of the whole, constitute a linear space. In other words, each object in this linear space is a continuous function. For any one of the continuous functions,

According to the Weierstrass theorem, it is possible to find a polynomial function with the highest number of items not greater than n, so that the difference from the continuous function is 0, that is, exactly equal. In this way, the problem boils down to L1. It's not back.

Repeat it again.

So, vectors are very powerful, so long as you find the right base, you can use vectors to represent any object in a linear space. Here the head great article, because the vector surface is only a column number, but in fact because of its

Order, so in addition to the information that these numbers carry, you can also carry information in the corresponding location of each number. Why are arrays the simplest and most powerful in programming? This is the root cause.

This is another question, and we won't say it here.

To answer the second question, the answer to this question relates to one of the most fundamental questions of linear algebra. The motion in a linear space is called a linear transformation. In other words, you move from one point in the linear space

To any other point, it can be done by a linear change. So what does a linear transformation mean? Very interesting, in a linear space, when you select a group of bases, you can not only use a vector to trace

Any one of the objects in the space, and a matrix can be used to describe any motion (transformation) in that space. The method of making an object correspond to motion is to use the matrix representing that movement, multiplied by the representation

The vector of that object. In short, after selecting a base in a linear space, the vector depicts the object, and the matrix depicts the motion of the object, which is applied by the multiplication of matrices and vectors. Yes, the essence of the matrix is the description of motion.

If someone later asks you what the matrix is, then you can tell him loudly that the essence of the matrix is the description of the motion.

But how interesting is it that the vectors themselves can also be viewed as n x 1 matrices? It is really fascinating that the object and movement in a space can be expressed in a similar way . Can you say it's a coincidence? If

It's a coincidence, that's a lucky coincidence! It can be said that most of the wonderful properties of linear algebra are directly related to this coincidence.

Then understand the matrix, which says "matrix is the description of motion", so far, it seems that everyone has no opinion. But I believe in the morning and evening there will be a mathematical department of the Netizen to make the decision turn. Because the concept of movement, in the number

Learning and physics are associated with calculus. When we learn calculus, there will always be someone to tell you, elementary mathematics is the study of constant mathematics, is the study of static mathematics, advanced mathematics is a variable

Mathematics, is the study of the movement of mathematics. Everyone is word of mouth, and almost everyone knows it. But people who really know what the meaning of this saying is, don't seem to have much.

Because this article is not about calculus, so I will not say more. Interested readers can take a look at the "relive calculus" written by the professor of civil friends. I just read the beginning of the book to understand that "advanced mathematics is

Study the Mathematics of the Movement "the truth of this sentence. However, in my article "Understanding the Matrix", the concept of "movement" is not a continuous movement in calculus, but a change that occurs instantaneously. Like this moment at point A,

After a "movement", suddenly " jump " to the b point, which does not need to go through a point and B point between any point. Such "movement", or "jump", is a violation of our daily experience. But know a little.

The knowledge of quantum physics will immediately point out that quantum (for example, electrons) jumps in different energy-level orbits, which happen instantaneously and have such a transition behavior. So, in nature, there is no such

Movement phenomenon, but macroscopic we cannot observe. But anyway, the word "movement" is used here, it is easy to produce ambiguity, more precisely, should be "jump". So this sentence can be changed to:

" matrices are the description of transitions in linear spaces ." But this is too physical, that is, too specific, but not enough mathematics, that is, not enough abstraction. So we end up with a genuine mathematical term--the transformation,

The matter. In this case, we should understand that the so-called transformation is actually the transition from one point (Element/object) to another (element/object) in space. An affine transformation, for example, is an affine

A transition from one point to another in space.

Incidentally, this affine space is a brother with vector space. As a friend of computer graphics knows, although describing a three-dimensional object requires only three-dimensional vectors, all computer graphics transformation matrices are 4x4

Of For the reasons, many of the books are written "for the convenience of use", which in my opinion is simply an attempt to muddle through. The real reason is that the graphic transformations applied in computer graphics are actually in the affine void

And not in the vector space. Think of it, in vector space, a vector parallel to the same vector, but the real world, such as two parallel segments of course can not be considered the same thing,

The living space of computer graphics is actually affine space. The matrix representation of affine transformations is essentially 4x4. Interested readers can go to the "computer graphics-geometric tools algorithm detailed".

Linear transformation in linear space

Once we understand the concept of "transformation," the definition of the matrix becomes: the matrix is a description of the transformations in the linear space . So far, we have finally got a definition of what looks like comparative mathematics. But I have to say a few more.

Sentence It is generally said in the textbook that a linear transformation T in a linear space V can be expressed as a matrix when a set of bases is selected . So we're going to talk about what is a linear transformation, what is a base,

What do you mean by selecting a group of bases. The definition of a linear transformation is simple, with a transform T, which allows for any two different objects X and Y in the middle of a linear space V, and any real numbers a and B, as follows:

T (ax+by) =at (x) +bt (y), then called T is a linear transformation. Definitions are written in this way, but there is no intuitive understanding of the definition of light. What kind of transformation is a linear transformation? We just said, change.

Change is to move from one point in space to another, whereas a linear transformation is the movement of one point of a linear space V to another point of another linear space W. This sentence contains a layer of meaning, is

Say that a point can not only be transformed to another point in the same linear space, but can be transformed to another point in another linear space. No matter how you change, as long as the transformation is a linear space in the object, which

A transformation must be a linear transformation, it can be described with a non-singular matrix. And you use a non-singular matrix to describe a transformation, it must be a linear transformation.

Some people may ask, why do we emphasize the non-singular matrix here? The so-called non-singular, only to the square is meaningful, then the situation of non-phalanx how? This will be more verbose, and finally the linear transformation as a

mapping, and the concept of the kernel and image of linear transformations can be thoroughly understood.

Here we only explore one of the most common and useful transformations, that is, a linear transformation within the same linear space. In other words, the matrix as described below, if not illustrated, is the square (the non-square matrix seems to be able to transform across linear space). ), and is a non-singular square.

To learn a learning, the most important thing is to grasp the backbone of the content, the rapid establishment of the whole concept of this knowledge, do not have to start with all the details and special circumstances, from the chaos.

What is a base? The question is to be talked about later, as long as the base is considered a coordinate system in a linear space. Note that the coordinate system, not the coordinate value, is a "contradictory unity".

In this way, "Select a group of bases" means that a coordinate system is selected in the linear space. Well, finally we refine the definition of the matrix as follows:"matrices are a description of linear transformations in linear spaces. In a linear space

, as long as we select a group of bases, any linear transformation can be described with a definite matrix. " the key to understanding this sentence lies in the" linear transformation "and" a description of linear transformation "area

Don't drive. One is the object, the other is the expression of that object. Just as we are familiar with object-oriented programming, an object can have multiple references, each reference can be called a different name, but all refer to the same

An object. If the image is not yet, then a very vulgar analogy. For example, if you have a pig, you want to take a picture of it, as long as you have selected a camera position, you can shoot a picture of the pig.

This picture can be seen as a description of the pig, but only a one-sided description, because in a different lens position for the pig to take pictures, can get a separate picture, is another one-sided description of the pig.

All these pictures are described by the same pig, but they are not the pig itself. Similarly, for a linear transformation, as long as you select a set of bases, then you can find a matrix to describe this

linear transformations. To change a group of bases, you get a different matrix. All of these matrices are a description of the same linear transformation, but they are not linear transformations themselves .

But then, here's the problem. If you give me two pictures of a pig, how do I know that the same pig is on the two pictures? Similarly, you give me two matrices, how do I know that both matrices are described by the same

A linear transformation? If it is the same linear transformation of the different matrix description, that is the Clan brothers, meet do not know, not become a joke. Fortunately, we can find the same linear transformation of the Matrix Brothers

A property, that is: if the matrix A and B are the same linear transformation of the two different descriptions (the reason is different because the selection of different bases, that is, the selection of a different coordinate system), you will be able to find a

A non-singular matrix P, which enables a and b to satisfy such a relationship: A=P?1BP. Linear algebra A little bit more familiar to the reader to see, this is the definition of the similarity matrix. Yes, the so-called similarity matrix is the same

A different description matrix for a linear transformation. According to this definition, photos of different angles of the same pig can also be similar photos. Vulgar a little, but can make people understand. and in the upper equation, the matrix p, its

is a transformation relationship between the base of a matrix and the base of the B-matrix.

On this conclusion, I can prove it in a very intuitive way (not the formal proof in the general textbook), and if there is time, I will add this proof in my blog later. This discovery is too important.

The Originally a family of similar matrices are the same linear transformation of the description Ah! No wonder it's so important! In the graduate course of engineering, there are some courses, such as Matrix theory and matrix analysis, which tell a variety of similar transformations, such as what phase

Like the standard type, diagonalization and the like, all require the transformation of the matrix obtained after the same matrix as the previous one, why so required? Because this is the only requirement, the two matrices before and after the transformation can be guaranteed.

is to describe the same linear transformation.

First, let's summarize some of the main conclusions in the previous section:

1. First there is space, space can accommodate the object movement. A space corresponds to a class of objects.

2. There is a space called linear space, the linear space is to accommodate the vector object motion.

3. Motion is instantaneous and therefore also referred to as transformation.

4. A matrix is a description of motion (transformation) in a linear space.

5. Multiplication of matrices and vectors is the process of implementing motion (transformation).

6. The same transformation, in different coordinate systems behave as different matrices, but their nature is the same, so the eigenvalues (eigenvalues) are the same.


Of course, the different matrices of the same linear transformation are described, and the actual operation is not a good ring. Some descriptor matrices are much better than other matrix properties. It's easy to understand, and there's a picture of the same pig.

The beauty of the ugly. So the similarity transformation of the matrix can transform an ugly matrix into a more beautiful matrix, and the two matrices are guaranteed to describe the same linear transformation. As a result, the matrix acts as a linear transformation

the one side of the description is basically clear . However, it is not so simple, or linear algebra has a more wonderful nature, that is, the matrix can not only be used as a description of the linear transformation, but also can be used as

A set of base descriptions. As a matrix of transformations, not only can one point in a linear space be transformed to another point, but also a coordinate system (base) table in linear space may be swapped to another coordinate system (base).

To. Moreover, the transformation point and the transformation coordinate system have the same effect. The most interesting mysteries of linear algebra are contained therein. Understanding the content, many theorems and rules in linear algebra become clearer


Let's focus our eyesight on a bit to change the way we look at matrices in the past. We know that the basic object in a linear space is a vector.

This is what the vector says: [A1,a2,a3,..., an]. The matrix is that: A11,a12,a13,..., a1n,a21,a22,a23,..., a2n,..., an1,an2,an3,..., Ann

Don't be too clever, we can see that the matrix is a set of vectors. In particular, square matrices in n-dimensional linear spaces consist of n-dimensional vectors. We're only here to discuss this n-order, non-singular square,

Because understanding it is the key to understanding the matrix, it is the general situation, and other matrices are unexpected, are forced to deal with the nasty situation, big can be put on one side.

Here more than a mouth, learning to grasp the mainstream, do not dwell on the offshoot stub. Unfortunately, most of our textbook textbooks are buried in the details of the main line, so that we do not understand how the first to be knocked out.
For example, mathematical analysis, clearly the most important idea is that an object can be expressed as an infinite number of reasonable choices of the linear and the object, the concept is throughout, but also the essence of mathematical analysis.
But the textbook does not say this sentence, anyway is let you do Demidovich, master a lot of solution tricky skills, remember all kinds of special circumstances,
Two types of discontinuity, weird micro and integrable conditions (who remember Cauchy conditions, Durrichle conditions ...?) ), after the final exam, all forget the light.
To me, it is better to repeatedly emphasize this thing, it is deeply engraved in the brain, other things forget to forget, really encountered problems, and then check the Math Handbook, why pound foolish it?

In fact, if a group of vectors are linearly independent of each other, then they can become a group of bases that measure this linear space, and thus become a system of coordinate systems in which each vector lies in one

axis, and becomes the basic unit of measure on that axis (length 1). Now it's a critical step. It seems that a matrix consists of a set of vectors, and if the matrix is not singular (I said,

consider only this, then the set of vectors that make up this matrix is linearly independent, and can be a coordinate system for measuring linear space. Conclusion: The matrix describes a coordinate system . "Wait a while!" ”,

You're yelling, "You liar!" Didn't you say that the matrix is motion? Why is this matrix again a coordinate system? "Well, so I'm talking about a key step. I did not cheat, the reason the matrix is movement,

is also the coordinate system, that is because-"motion is equivalent to the coordinate system transformation". I'm sorry, but it's not accurate, I just want to impress you. The exact phrase is: "The transformation of an object is equivalent to the transformation of a coordinate system." Or

By: The transformation of the next object in the fixed coordinate system is equivalent to the coordinate system transformation where the fixed object is located. "Plainly," the movement is relative. ”

Let's think about achieving the same transformation results, such as turning points (2,3) to a point where you can do it two ways. First, the coordinate system does not move, move, Move (the) point to (2,3) to go. Second, don't move, change.

coordinate system, so that the x-axis measurement (unit vector) into the original 1/2, so that the y-axis measurement (unit vector) into the original 1/3, so that point or that point, but the coordinates of the point becomes (2,3). different ways, knot

The same as the fruit. In the first way, The matrix is regarded as the motion description, and the matrix and vector multiplication is the process of making the vector (point) move. In this way, ma=b means: "Vector a passes the matrix M

The transformation described is transformed into a vector b. In the second way, The matrix M describes a coordinate system, which is called M. So: Ma=b means: "There is a vector, which is under the measure of the coordinate system m

The resulting vector of measure result is a, then it is measured in the coordinate system I, the result of this vector is B. "I here refers to the unit matrix, which is the matrix where the main diagonal is 1 and the other is zero." And these two ways are essentially

is equivalent to the above. I hope you understand this, because this is the key to this article. Because it is the key, I have to explain it again. In the sense that M is a coordinate system, if M is placed in front of a vector a, the shape

into a MA style, we can think of this as an environmental declaration on vector A. It's equivalent to saying, "Pay attention!" Here is a vector, which is measured in the coordinate system m, and the resulting measurement can be expressed as a. But

If it is measured in another coordinate system, it will get different results. To be clear, I put M in front of you to understand that this is the result of the vector being measured in the coordinate system M. ”

So let's look at the solitary vector b:b a few more times, don't you see? It is not actually B, it is: IB is also said: "In the unit coordinate system, that is, we usually say the Cartesian coordinate system I, there is a vector,

The result of the measure is B. "And Ma=ib means:" In the M coordinate system in the amount of vector a, with the I coordinate system in the amount of vector B, is actually a vector ah! "Where is this multiplication calculated,

It's all about identification.

In this sense we re-understand the vector. Vector this thing objectively exists, but to express it, it is necessary to put it in a coordinate system to measure it, and then the measurement of the

The result (the projected values of the vectors on each axis) is listed in a certain order, which is a vector representation of what we normally see. You choose a different coordinate system (base), the representation of the vector is different. Vector

Or that vector, the chosen coordinate system is different, and its representation is different. Therefore, it is reasonable to say that each expression of a vector should be declared in which coordinate system is measured. Table

Is the MA, that is, a vector that is measured in the coordinate system represented by the M-Matrix. We usually say that a vector is [2 3 5 7]t, which implies that the measurement of this vector in the I coordinate system is [2 3 5 7]t, so this form is a simplified special case.

Notice that the coordinate system represented by the M-matrix consists of a set of bases, which are also made up of vectors, and there is also the problem of what coordinate system the set of vectors are measured in. In other words, the general method of expressing a matrix should also indicate the datum coordinate system in which it is located. The so-called M, in fact, is IM, that is, m in the group of the basis of the measurement is in the I coordinate system to derive. From this point of view, the MXN is not a matrix multiplication, but rather the declaration of a measure in the M coordinate system N, where m itself is measured in the I coordinate system.
Back to the problem of transformation. I just said, "the transformation of the next object in the fixed coordinate system is equivalent to the coordinate system transformation of the fixed object", the "fixed object" we found, that is the vector. But what about the transformation of the coordinate system? Why didn't I see it?
Please see:
Ma = Ib
I want to change M to I now, how to change? By the M-1, the inverse matrix of M is multiplied by the previous one. In other words, do you have a coordinate system m, and now I multiply it by a M-1 and I, so that if a in the M coordinate system is a quantity in I, I get B.
I suggest you pick up a pen and paper at this moment and draw a picture to get an understanding of the matter. For example, if you draw a coordinate system, the unit of measure on the x-axis is 3, and in such a coordinate system, the coordinate is (2,y), which is actually the point in the Cartesian coordinate system (2, 3). The way to let it be true is to put the original coordinate system:
2 0
0 3
The X-direction metric shrinks to the original 1/2, and the Y-direction metric shrinks to the original 1/3, so that the coordinate system becomes the unit coordinate system I. Keep the point constant, and the vector now becomes (2, 3).
How can I reduce the X-direction metric to the original 1/2 and the Y-direction metric to the original 1/3? is to have the original coordinate system:
2 0
0 3
By Matrix:
1/2 0
0 1/3
Left multiply. And this matrix is the inverse matrix of the original matrix.
Let's come to an important conclusion: "The method of applying transformations to a coordinate system is to multiply the matrix representing that coordinate system with the matrix that represents that change." ”
Once again, the multiplication of matrices becomes the exertion of motion. However, the applied motion is no longer a vector, but another coordinate system.
If you think you can make it clear, think about the conclusion that the Matrix MXN, on the one hand, indicates that the transformation result of the coordinate system n under the motion m, on the other hand, regards m as the n prefix and as the environment description of N, then that is, in the M coordinate system, there is another coordinate system N. This coordinate system n if measured in the I coordinate system, the result is a coordinate system of MXN.
Here, I have actually answered the question that the average person is learning linear algebra is one of the most puzzling, and that is why the multiplication of matrices should be defined as such. To put it simply, it is because:
1. From the point of view of transformation, the M transform is applied to the coordinate system n, that is, the M transform is applied to each vector that makes up the coordinate system N.
2. From the point of view of the coordinate system, the second coordinate system of n is represented in the M coordinate system, which also boils down to finding the coordinates of the N coordinate system in the I coordinate system and then forming a new matrix.
3. As for the matrix multiplied by the vector, it is because a vector measured in M is a, and if you want to restore the true image in I, you have to do an inner product operation with each vector in M, respectively. I leave the derivation of this conclusion to the interested friend. It should be said that, in fact, to this step, has been very easy.
Synthesize above 1/2/3, the multiplication of the matrix must be so stipulation, all substantiated, is not which neuropathy is out of the imagination.
I can't say much more. The matrix is also the coordinate system and the transformation. In the end is the coordinate system, or transformation, has been said to be unclear, the movement and the entity here Unified, the boundary between material and consciousness has disappeared, all attributed to unspeakable, can not be defined. DAO can road, very way, name can name, very name. A matrix is something that is not the path of the word, not the name of the name. By this time, we have to admit that our great linear algebra textbook says that the matrix definition is incomparably correct:
A matrix is a mathematical object that consists of a number of rows and n columns of M. ”
Well, that's basically all I want to say. Also left a determinant of the problem. The determinant of the matrix M is actually the volume of the various vectors that make up m in accordance with the parallelogram principle into an n-dimensional cube. For this, I can only sigh at its subtlety, but can not uncover the mystery. Perhaps I have not mastered the mathematical tools, I hope someone can give us all to explain the reason.
I don't know if I can tell you enough, but this part will take some effort to figure it out. Also, please do not wait for the next part of this series. As far as my work is concerned, it is difficult to ensure that I continue to devote my brainpower to this field in the near future, although I am still interested in it. However, if there are (four), it may be some application-level considerations, such as the understanding of computer graphics-related algorithms. But I do not promise that these discussions will emerge in the near future.

Linear algebra should learn this way

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