In-depth understanding of matrices-matrix revolution (full version)

Matrix Revolution-Understanding 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), one to introduce the reverse number of this "unprecedented, no one," the eccentric concept, and then use the reverse number to give the determinant of a very non-intuitive definition, followed by some simply silly determinant of the nature and exercise-- Add this line to another row by a factor, and then reduce the column, tossing it to call a lively, but it is not at all to see what 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 ring performance, it is too "unreasonable" it! So people started skipping classes, and many more 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 the greatest of all the guy's appearance-The matrix came! Years later, I realized that when the teacher foolishly with the square brackets to a bunch of silly, and slowly said: "This thing is called The Matrix", my math career opened how tragic and bitter, the scene of a very tragic! 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 is not a time to deal with linear algebra of the idiot, The Matrix boss of the 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. "If you are not familiar with the concept of linear algebra and want to learn the natural sciences, it is now almost as illiterate," said Swedish mathematician Lars Garding in his famous encounter with mathematics. ", however," according to the current international standards, linear algebra is expressed by axiomatic, it is the second generation of mathematical models, ..., which brings difficulties in teaching. "In fact, when we
When we start to learn linear algebra, we unconsciously enter into the category of "second generation mathematical model",
This means that mathematical representations and abstractions have evolved in a comprehensive way, and have been "the first
Model ", that is, a practical-oriented, concrete mathematical model in which we learn, without
And clearly told the case to carry out such drastic paradigm shift, without feeling the difficulty is strange.
Most engineering students tend to learn a number of subsequent courses, such as numerical analysis, mathematical programming, matrix
After that, the linear algebra can be understood and skillfully applied gradually. 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 for many of the courses of beginners put forward
, which seems to be a very basic problem, is unclear. For example:
* What exactly is a matrix? Vectors can be thought of as having n mutually independent properties (dimensions)
What is the representation of the object and the matrix? If we think that the matrix is a set of columns (rows) of the vector composed of a new
The expansion of the composite vectors, then why this kind of expansion has such a wide range of applications? In particular, to
What is the two-dimensional expansion so useful? If each element in the matrix is another vector, then I
Do you think it would be more useful to start a three-dimensional square?
* Why is the multiplication rule of matrices so stipulated? Why such a strange multiplication rule can
Do you have such a great 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? In the matrix multiplication that looks like mo
Under the rules of the good name, contains some of the world's essential laws? If so, these essential laws are
What the?
* What exactly is a determinant? Why is there such a strange rule of calculation? Determinant and its
What is the nature of the corresponding phalanx? Why only the Phalanx has the corresponding determinant, and the general matrix is
No (don't think the problem is stupid, if necessary, defining the determinant for m x n matrices is not doing
Not, the reason why do not do, because there is no need, but why not this necessary? And
Determinant of the calculation rules, it seems that the matrix of any calculation rules are not intuitive contact, why again
Determines the nature of the matrix in many ways? Is this all just a coincidence?
* Why the matrix can be divided into block calculation? Chunking calculation This thing seems so random, why is it
Is it possible?
* For Matrix transpose operation at, there is (ab) T = Btat, for matrix 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? This is only
Is it a coincidence?
* Why does P-1ap get a matrix that is "similar" to a matrix? What do you mean by "similarity" here?
* What is the nature of eigenvalues and eigenvectors? Their definition is surprising because Ax =λx, a
The effect of a large matrix, but the equivalent of a small number λ, it is a bit strange indeed. But where to
In 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's like an adult.
In the face of the children's inquisitive, there will always be forced to say, "This is it, so far," the same
In the face of such problems, many veterans end up with the following: "This is the rule, you accept and remember
"Good Stay" However, if the question is not answered, the linear algebra for us
is a rough, unreasonable, inexplicable set of rules, we will feel that we are not
In the study of a learning, but is without any explanation "thrown into" a mandatory world, just in the exam
The whip was forced to travel, completely unable to appreciate the beauty, harmony and unity. Until many years later,
We have found that this knowledge is so useful, but still very confused: why so coincidence?
This, I think, is the consequence of the intuitive loss of our linear algebra teachings. These relate to "how to
Can "," How to "question, only through pure mathematical proof to answer, can not make the questioner satisfied
Of For example, if you demonstrate that the matrix block operation is feasible through the general proof method, then this
Not be able to get the doubts of the questioner solved. Their real confusion is: Why is the Matrix chunking operation
Is that feasible? Is it just coincidence, or is it necessarily determined by the nature of the object of the matrix?
Of If this is the latter, then what is the nature of the matrix? Just a little consideration of the above questions,
We will find that none of these problems can be solved by simply relying on mathematical proofs. Like ours.
Textbooks, in all things with mathematical proof, the last training students, can only skillfully use tools, but
Lack of a true sense of understanding.
Since the rise of the French Bourbaki school in the 1930 's, the axiomatic and systematic description of mathematics has been
Great success, which makes the mathematics education we accept greatly improved in rigor. However the mathematical axiom
A controversial side effect of this 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. However, including myself in
Many people are skeptical about this, we do not think that intuition and abstraction must contradict each other, especially
In math education and math textbooks, helping students to build intuition helps them understand the abstract
To understand the nature of mathematics. Conversely, if you pay attention to formal rigor, students seem to be
The little mouse, forced to perform the fire-ring, became a slave to the dull rules.
For a linear algebra similar to some of the above-mentioned intuitive problems, for more than two years I have been intermittently and
四、五次 and read several domestic and foreign linear algebra, numerical analysis, algebra and mathematics
A general book of books, such as the masterpieces of the former Soviet Union, Mathematics: its content, methods and meanings, Professor Gong Sheng
"Linear algebra Five", the aforementioned encounter with Mathematics ("Mathematics Overview")
and Thomas A. Garrity's "math supplements" gave me a lot of inspiration. But even so, I
The recognition of this subject has also experienced several self-denial. For example, some of the conclusions of previous thinking have been written in
Own blog, but now it seems that these conclusions are basically wrong. So I'm going to put myself
Now the relevant understanding of the relatively complete record, on the one hand because I think the present understanding is more mature
, can take out to discuss with others, consult others. On the other hand, if further recognition is
Knowledge, the current understanding of the overthrow, and now write this snapshot is also very meaningful.
Because of the intention to write more, so will be a few times slowly write. Do not know whether there is time to write the complete,
Will not interrupt, write to see it.
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Let's talk about the understanding of several core concepts of linear space and matrices today. Most of these things are self-
Own understanding of the written, basically not transcription, there may be errors 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, 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 completeness, it becomes the bar
To define the angle in the normed linear space, there is 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 look at a mathematical definition of a space, it's basically "there's a set
Define a concept on this set, and then satisfy certain properties ", it can be called space. This
It's a bit strange, why use "space" to call some of these collections? As you will see,
In fact, this is very reasonable.
There is no doubt that we are most familiar with the space in which we live (in accordance with Newton's absolute time
The three-dimensional space, mathematically speaking, this is a three-dimensional Euclidean space, we first regardless of that
, let's look at some of the most basic features of such a space that we are familiar with. Think carefully and we'll
Will know, this three-dimensional space: 1. Consists of a number of (in fact, infinitely multiple) position points; 2. This
The relative relationship between the points; 3. Length, angle, 4 can be defined in space. This space can
To accommodate the movement, here we call the movement from one point to another point of movement (transformation), rather than
The "continuous" movement in the sense 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 counting
is a space-specific property, all the discussion of mathematical problems, there must be a set, most of the set
To define some structures (relationships) is not to say that there are spaces. And the 3rd is too special, its
His space does not need to be possessed, nor is it a crucial 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. Fact
, no matter what the space, it must accommodate and support the rule-conforming Motion (transformation) in which it occurs.
You will find that in some space there is often a relative transformation, such as topological space with topological
Transformation, there are linear transformations in linear space, affine transformations in affine spaces, but these transformations are only
is the permissible form of motion in the corresponding space.
So, as long as you know, "space" is a collection of objects that accommodate motion, whereas transformations specify the corresponding space
of movement.
Let's take a look at the linear space. The definition of a linear space is available in any book, but since we
If linear space is a space, then there are two basic problems that must be solved first:
1. Space is a collection of objects, the linear space is also a space, so is also a collection of objects. So the line
What is a collection of objects of sexual space? 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 straight
The answer is given by the cut-off. Any of the objects in a linear space, by selecting the base and coordinate methods, are
Can be expressed in the form of vectors. The usual vector space I won't say, give two less mundane examples:
L1. The whole of a polynomial with a maximum of not more than n times constitutes a linear space, that is, the line
Each object in the sex space is a polynomial. If we take x0, x1, ..., xn as the base, then any
One such polynomial can be expressed as a set of n+1-dimensional vectors, in which each component AI is actually
is the coefficient of the X (i-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 first, mention a
The next.
L2. The whole of the N-order continuous micro-function on the closed interval [a, b] constitutes a linear space. Other words
Each object in this linear space is a continuous function. For any one of the continuous functions, depending on
Weierstrass theorem, we can find the polynomial function with the highest item not greater than N, and make it with the
The difference of a 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, the vector is very powerful, as long as you find the right base, with a vector can represent the linear space in any
What an object. 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 the number itself carries, it can be carried in the corresponding position of each number
with information. 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.
Problem.
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, how does the linear transformation table
Show it? Very interesting, in linear space, when you select a group of bases, you can not only use a vector to
Describe any object in the space, and you can use a matrix to describe any motion in the space
(transform). The way to make an object correspond to motion is to use the matrix that represents that movement, multiply
To represent 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 object's motion, using
The matrix and the multiplication of vectors exert motion.
Yes, the essence of the matrix is the description of motion. If someone later asks you what the matrix is, then you can
To tell him loudly that the essence of the matrix is the description of the movement. (Chensh, say you!) ）
But how interesting is it that the vectors themselves can also be viewed as n x 1 matrices? It's really amazing,
Objects and motions in a space can be represented 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
Have a direct relation to this coincidence.
In the previous article, "The matrix is a description of the movement", 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 final turn. Because the concept of movement in mathematics and things
Rilly is associated with calculus. When we learn calculus, someone always tells
To you, elementary mathematics is the study of constants of mathematics, is the study of static mathematics, advanced mathematics is a variable of mathematics,
is to study the mathematics of motion. Everyone is word of mouth, and almost everyone knows it. But I do know that.
People who say what they mean, don't seem to have much. In short, in our human experience, movement is
A continuous process, from point A to point B, even the fastest light can take a time to point
Through the path between AB, which leads to the concept of continuity. And continuous this thing, if uncertain
The concept of the limits of righteousness cannot be explained at all. The ancient Greeks were very strong in mathematics, but they lacked the concept of limits,
So I can't explain sports, the famous paradox of Zeno (flying Arrows, Scud Achilles, but
Turtles and other four paradoxes). Because this article is not about calculus, so I don't have much.
Said it. Interested readers can take a look at the "relive calculus" written by the professor of civil friends. I just read this.
At the beginning of this book, we understand that "advanced mathematics is the study of the Mathematics of Motion".
But in my article on understanding the Matrix, the concept of "movement" is not a continuum in calculus.
Movement, but a change that happens instantaneously. Such as this moment at point a, after a "movement", suddenly
"Jump" to point B, where no point between point A and point B is required. Such a
"Movement", or "jump", is a violation of our daily experience. But understanding a little bit of quantum physics often
The quantum (for example, electrons) jumps in different energy-level orbits and is instantaneous
Such a transition behavior occurs between the So, there is no such movement in nature,
But we can't see it on the macro level. But anyway, the word "sport" is used here, or
It is easy to create ambiguity and, more precisely, "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--transform, to describe the thing. Such a saying that the big
Home should understand that the so-called transformation, in fact, is the space from one point (Element/object) to another
The jump of a point (Element/object). Topological transformations, for example, are in the topological space from one point to another
A jump of a point. In other words, affine transformations are in the affine space from one point to another.
Transitions. Incidentally, this affine space is a brother with vector space. Be a friend of computer graphics
All know, although the description of a three-dimensional object requires only three-dimensional vector, but all the computer graphics transformation moment
The array is 4 x 4. Say the reason, many books are written "for the convenience of use", which in my opinion Jane
Straight is an attempt to muddle through. The real reason is that the graphic transformations applied in computer graphics,
is actually carried out in affine space rather than in vector space. Think about it, in vector space in a direction
The same vector is still the same when the volume is moved parallel, and the two parallel segments of the real world are certainly not
is considered the same thing, so the living space of computer graphics is actually affine space. and affine change
The matrix representation of the change is 4 x 4 at all. Again, interested readers can go to see the computer diagram
Geometry--A detailed description of the geometric tools algorithm.
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 just a few more words to say.
It is generally said in the textbook that a linear transformation t in a linear space V, when a group of bases is selected
After that, it can be represented as a matrix. 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 makes the line
Any two different objects X and Y in the middle of the sex space V, as well as any real numbers a and B, are:
T (ax + by) = at (x) + BT (y),
Then it is called t as a linear transformation.
Definitions are written in this way, but there is no intuitive understanding of the definition of light. What is a linear transformation?
What kind of transformation? As we have just said, the transformation is from one point in space to another, and the linear change
is to jump from one point of a linear space V to another point of another linear space W.
Movement. This sentence contains a layer of meaning, that a point can not only be transformed into the same linear space
and 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, the transformation must be a linear transformation, it will be able to
To describe with a non-singular matrix. And you use a non-singular matrix to describe a transformation that must be
A linear transformation. Some people may ask, why do we emphasize the non-singular matrix here? So-called non-singular,
It is only meaningful to the square, so what about the non-phalanx? This is going to be a lengthy one, and finally
The linear transformation as a kind of mapping, and discusses its mapping properties, as well as the linear transformation of the kernel and image, etc.
To be completely clear. I think this is not the point, if you do have time, then write a little later.
Here we only explore one of the most common and useful transformations that are linear within the same linear space
Transform. In other words, the matrix, which is not described below, is a square and a non-singular square
Array. To learn a learning, the most important thing is to grasp the backbone of the content, and quickly establish a general overview of this knowledge
Think, do not have to start with all the details and special circumstances, self-disorder.
Then, what is a base? This question is going to be a big one at the back, as long as the base is
The coordinate system in the linear space is available. Note that the coordinate system, not the coordinate value, is a "pair
Unity of contradictions ". In this way, "Select a group of bases" means that a coordinate system is selected in the linear space.
That's what it means.
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 set of bases, any linear transformation can be described with a definite matrix. ”
The key to understanding this sentence is to distinguish between "linear transformation" and "a description of linear transformation". One
One is the object, the other is the expression of that object. It's like we're familiar with object-oriented programming,
An object can have multiple references, each of which can be called a different name, but all refer to the same object.
If the image is not yet, then a very vulgar analogy.
Like a pig, you're going to take a picture of it, so long as you've chosen a camera position
Can take a picture of the pig. This picture can be seen as a description of the pig, but only a
A one-sided description, because taking a camera position to take a picture of the pig, can get a different picture,
is another one-sided description of this pig. All the pictures in this picture are the same pig.
But it is not the pig itself.
Similarly, for a linear transformation, as long as you select a set of bases, you can find a matrix to trace
The linear transformation is described. To change a group of bases, you get a different matrix. All of these matrices are the same
A linear transformation, but not a linear transformation itself.
But then, here's the problem. If you give me two pictures of a pig, how do I know the two pictures
Is the same pig? Similarly, you give me two matrices, how do I know that both matrices are described in the same
What about a linear transformation? If it's a different matrix description of the same linear transformation, that's the clan brother.
, did not know the meeting, not become a joke.
Fortunately, we can find a property of the same linear transformation of the Matrix Brothers, that is:
If matrices A and B are two different descriptions of the same linear transformation (the difference is
A different base, i.e. a different coordinate system is selected), a non-singular matrix p must be found,
So that the relationship between a and B is satisfied:
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
The similarity matrix is a different description matrix of the same linear transformation. According to this definition, the same pig
Photos of different angles can also be similar photos. Vulgar a little, but can make people understand.
In the above equation, the matrix p is actually the basis of the base and B matrix on which a matrix is based.
A transformation relationship between the two groups of bases. On this conclusion, it is possible to use a very intuitive method to testify
(Not the form of the general textbook), if there is time, I later in the blog
Add this proof.
The discovery is too important. Originally a family of similar matrices are the same linear transformation of the description Ah! No wonder 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
Transformation, such as what is similar to the standard type, diagonalization and other content, all require the transformation of the moment to get
The array is similar to the previous matrix, why is this required? Because this is the only way to guarantee
The two matrices before and after the transformation describe the same linear transformation. Of course, the different moments of the same linear transformation
Array description, from the actual operation of the nature of the ring is not good. Some descriptor matrices are more than other matrices.
The nature is much better. This is easy to understand, the same pig's picture also has a beautiful ugly point. So the similarity of matrices
Transforms can turn an ugly matrix into a more beautiful matrix, and ensure that both matrices are
The same linear transformation is described.
In this way, the matrix as a linear transformation of the side of the description, basically clear. But, things don't have that
Simple, or, linear algebra has a more wonderful nature than this, that is, matrices can not only be used as
A description of a linear transformation, and can be described as a set of bases. And as a matrix of transformation, not only can the
A point in a linear space gives the transformation to another point, and it can also put a coordinate in the linear space
The system (base) table is swapped to another coordinate system (base) to go. Moreover, the transformation point and the transformation coordinate system, have the different curvature
The effect of working with colleagues. The most interesting mysteries of linear algebra are contained therein. Understanding of the content, the linear
Many theorems and rules in algebra become clearer and more intuitive.
Let's begin by summarizing some of the main conclusions of the previous two parts:
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 to behave as different matrices, but they are the same nature
, so the intrinsic value is the same.
Let's focus our eyesight on a bit to change the way we look at matrices in the past. We know
Tao, the basic object in a linear space is a vector, and the vector is that:
[A1, A2, A3, ..., an]
What about the matrix? This is what the matrix says:
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, n-dimensional
The square of a linear space is made up 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
All are accidents, all have to deal with the nasty situation, big can put on one side. Here's a more mouth to learn
Things to catch the mainstream, do not dwell on the offshoot stub. Unfortunately, most of our textbook textbooks are about the main
The line buried in the details, make everyone still do not understand how to get dizzy first. such as mathematical analysis,
Clearly, the most important idea is that an object can be expressed as an infinite number of reasonably selected objects of linear
And, this concept is throughout, but also the essence of mathematical analysis. But the textbook doesn't talk about it from the beginning.
Sentence, anyway is let you do Demidovich, master a lot of solution tricky skills, remember all kinds of special situation
Conditions, two kinds of discontinuity, weird micro and integrable condition (who remembers Cauchy condition, Durrich Bar
Pieces ...? ), after the final exam, all forget the light. I'd say it's better to stress this one thing over and over again.
Engrave it deeply in the mind, other things forget to forget, really encountered the problem, and then check the Math Handbook,
Why Pound foolish?
Anyway If a group of vectors are linearly independent of each other, then they can become degrees
A set of bases in this linear space, which in fact becomes a coordinate system in which each vector is
Lie on an axis and become the basic unit of measure on that axis (length 1).
Now it's a critical step. It appears that the matrix is made up of a set of vectors, and if the moment
Array of non-singular words (I said, considering only this), then the set of vectors that make up this matrix also
is linearly unrelated, it can also be a coordinate system for measuring linear space. Conclusion: Matrix description
a coordinate system.
"Wait!" "You yell up," you liar! Didn't you say that the matrix is motion
It? 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,
It's the coordinate system again, because--
"Motion is equivalent to a coordinate system transformation."
I'm sorry, but it's not accurate, I just want to impress you. The exact statement is:
"The transformation of an object is equivalent to the transformation of a coordinate system."
Or:
The transformation of the next object in a fixed coordinate system is equivalent to the coordinate system transformation where the fixed object is located. ”
To be blunt:
"Movement is relative. ”
Let's think about achieving the same transformation results, such as turning points (1, 1) to a point (2, 3),
There are two ways you can do it. First, the coordinate system is fixed, move, move (1, 1) point to (2, 3) to go. Second
Point, change the coordinate system, let the x-axis measure (unit vector) become the original 1/2, let the y-axis measure
(unit vector) becomes the original 1/3, this point or that point, but the point coordinates will become (2, 3)
The In different ways, the result is the same.
In the first way, that's what I said in the understanding Matrix 1/2, The matrix as
Is the motion description, the matrix and vector multiplication is the process of making the vector (point) movement. In this way,
Ma = b
The meaning is:
"Vector a passes the transformation described by the matrix M and becomes the vector b." ”
In the second way, The matrix M describes a coordinate system, which is called M.
So:
Ma = b
The meaning is:
"There is a vector, which is a vector of measure results under the measure of the coordinate system m, that
It is measured in the coordinate system I, the result of this vector is B. ”
Here I refers to the unit matrix, that is, the main diagonal is 1, the other is a zero matrix.
And these two approaches are inherently equivalent.
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, it forms a MA
Style, we can think of this as an environmental declaration on vector A. It is equivalent to saying:
"Watch out!" Here is a vector, which is measured in the coordinate system m and the resulting measurement
Can be expressed as a. But it is measured in other coordinate systems, and it gets different results. To clear
Indeed, 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 lone vector B:
B
How many times do you see it? It is not actually B, it is:
Ib
In other words: "In the unit coordinate system, which is what we usually call the Cartesian coordinate system I, there is a
A vector, the result of the measure is B. ”
The meaning of Ma = Ib means:
"The vector A in the M coordinate system, which is the amount of vector b in the I coordinate system,
The truth is a vector! ”
This is where the multiplication calculation is, is the identity.
In this sense we re-understand the vector. Vector this thing exists objectively, but to
Show it, put it in a coordinate system to measure it, and then put the result of the measurement (vector
The projected values on each axis) are listed in a certain order, which is what we see as a vector table.
Display form. You choose a different coordinate system (base), the representation of the vector is different. Vector or that?
Vector, choose a different coordinate system, and its representation is different. So, as a matter of principle, each written
The representation of a vector, it should be stated in which coordinate system the representation is measured. Represents the
Is the MA, that is, a vector that is measured in the coordinate system represented by the M-Matrix.
The result is a. We usually say that a vector is [2 3 5 7]t, implying that this vector is seated in I
The measurement in the standard 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, and that group of bases
is made up of vectors, and there is the problem that the set of vectors is measured in which coordinate system. That is
said that the general method of expressing a matrix should also indicate the datum coordinate system in which it is located. So-called M,
In fact, it is IM, that is to say, the metric of the group of bases in M is derived in the I coordinate system. From this view
Angle, MXN is not a matrix multiplication, but it declares a measure in the M coordinate system
Another coordinate system n, where m itself is measured in the I coordinate system.
Back to the problem of transformation. I just said, "the transformation equivalence of the next object in fixed coordinate system
The coordinate system in which the fixed object is transformed ", the" fixed object "we found, that's 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, which is the inverse of M
Matrix. In other words, don't you have a coordinate system m, now I multiply it by a M-1, I,
In this case, the A in the M coordinate system is a quantity of a 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. Like what
You draw a coordinate system, the unit of measure on the x-axis is the unit of measure on the 2,y axis is 3, in such a sitting
In the standard system, the point at which the coordinates are (in) is actually the points in the Cartesian coordinate system (2, 3). and let
Its way of true colors, that is, 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
The coordinate system becomes the unit coordinate system I. Keep the point constant, and that vector now becomes (2, 3)
The
How can we 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.
Here we come to an important conclusion:
"The method of applying transformations to a coordinate system is to let the matrix representing that coordinate system be represented by that
multiplied by the changing matrix. ”
Once again, the multiplication of matrices becomes the exertion of motion. But the movement is no longer
is a vector, but another coordinate system.
If you think you can make it clear, think again of the conclusion that has just been mentioned, matrix
MxN, on the one hand, indicates the transformation result of the coordinate system n under the motion m, on the other hand, the M as N
As an environment description of N, 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've actually answered the general people in learning linear algebra is one of the most confusing questions
, 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 constituent coordinate system n
The M transform is applied to each vector.
2. From the point of view of the coordinate system, the second coordinate system of n is represented in the M coordinate system, which
It is also attributed to each vector of the n coordinate system base, which is found in the coordinates of the I coordinate system, and then
Sinks into a new matrix.
3. As to why the matrix is multiplied by the vector, it is because a metric in M is
A vector, if you want to restore the true image in I, you have to separate each vector in M with the internal
Product operation. I leave the derivation of this conclusion to the interested friend. It should be said that, in fact, to this step,
It's easy enough.
Synthesize above 1/2/3, the multiplication of matrix must be so stipulation, all substantiated, never is which
A psychotic freak out.
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 not clear, movement and entity here unified, the boundary between material and consciousness has been
Disappeared, and all was unspeakable, unable to define. 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,
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. Matrix
The determinant of M is actually the various vectors that make up m, according to parallelogram principle, into an n-dimensional cubic
Volume of the body. For this, I can only sigh at its subtlety, but can not uncover the mystery. Maybe I
There is not enough mathematical tools to master, 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. In my work situation, the near
It's hard to keep the brain in this field, although I'm still interested in it. But if
Also, there may be some application-level considerations, such as the calculation of computer graphics
Understanding of the law. But I do not promise that these discussions will emerge in the near future.

In-depth understanding of matrices-matrix revolution (full version)