From actions on matrices to understanding their essence ...
I really respect people who have the courage to say that they do not understand something. Himself so. What I do not understand is that I have to study, comprehend, and understand. The article " Mathematics on the fingers ", and especially the matrix record of formulas, made me share my small, but it seems to be an important experience in working with matrices.
Some 20 years ago I had a chance to study higher mathematics in high school, and we started with matrices (perhaps, like all students of that time). For some reason, it is believed that the matrix - the easiest topic in the course of higher mathematics. It is possible - because all actions with matrices are reduced to the knowledge of methods for calculating the determinant and several formulas built - again, on the determinant. It would seem, everything is simple. But ... Try to answer the elementary question - what is the determinant, what does the number you get when calculating it? (hint: a variant like “determinant is a number that is determined by certain rules” is not the correct answer, because it refers to the method of receipt, and not to the essence of the determinant). Give up? - then read on ...
Just want to say that I am not a mathematician neither by education nor by position. Unless I'm interested in the essence of things, and sometimes I try to “get to the bottom” of them. It was the same with the determinant: it was necessary to deal with multiple regression, and in this section of econometrics almost everything is done through ... the matrix, be they wrong. So I myself had to do a little research, because none of the familiar mathematicians gave a clear answer to the question, which initially sounded like “what is the determinant”. All argued that the determinant is a number that is specifically counted, and if it is zero, then ... In general, as in any textbook on linear algebra. Thank you, passed.
')
If one person came up with an idea, then the other person should be able to understand it (although, for this, sometimes you have to arm yourself with additional knowledge). Appeal to the "great and powerful" search engine showed that "the area of ​​the parallelogram is equal to the module of the determinant of the matrix formed by the vectors - the sides of the parallelogram ". In simple terms, if a matrix is ​​a way of writing a system of equations, then each equation separately describes a vector. Having constructed from the point of origin of the coordinates the vectors given in the matrix, we thus define a figure in the space. If our space is one-dimensional, then the figure is a segment; if it is two-dimensional, then the figure is a parallelogram, and so on.
It turns out that for one-dimensional space the determinant is the length of the segment, for the plane - the area of ​​the figure, for the three-dimensional figure - its volume ... then there are n-dimensional spaces that we cannot imagine. If the volume of the shape (that is, the determinant for the 3 * 3 matrix) is zero, then this means that the shape itself is not three-dimensional (it can be two-dimensional, one-dimensional, or even a point in general). The rank of the matrix is ​​the true (maximum) dimension of the space for which the determinant is nonzero.
So, almost everything is clear with the determinant: it defines the “volume” of the figure formed by the vectors described by the system of equations (although it is not clear why its value does not depend on whether we are dealing with the original matrix or transposed — perhaps transposing is a kind of affine conversion?). Now we need to deal with the actions on the matrices ...
If the matrix is ​​a system of equations (otherwise why do we need a table of some numbers that have no relation to reality?), Then we can do different things with it. For example, we can add two rows of the same matrix, or multiply a row by a number (that is, each row coefficient is multiplied by the same number). If we have two matrices with the same dimensions, then we can put them together (the main thing is that we don’t add a bulldog with a rhinoceros - but did mathematicians, developing the theory of matrices, think about this scenario?). Intuitively clear, especially as in linear algebra illustrations of such operations are systems of equations.
However, what is the meaning of matrix multiplication? How can I multiply one system of equations by another? What sense will have what I will receive in this case? Why is the transfer rule not applicable for matrix multiplication (that is, the product of matrices B * A is not that not equal to product A * B, but is not always feasible)? Why, if we multiply a matrix by a column vector, we get a column vector, and if we multiply a row vector by a matrix, then we get a row vector?
Well, here it is not that Wikipedia, - even modern textbooks on linear algebra here are powerless to give any intelligible explanation. Since the study of something according to the principle “you first believe - and then you will understand” - is not for me, I dig deep into the centuries (more precisely, I read textbooks of the first half of the 20th century) and find an interesting phrase ...
In the books this is not directly stated, but it turns out that vectors parallel to a certain plane do not have to lie on this plane. That is, they can be in three-dimensional space anywhere, but if they are parallel to this particular plane, then they form two-dimensional space ... From the analogies that come to my mind - a photo: the three-dimensional world is represented on a plane, while the vector is parallel to the matrix (or film) the camera will correspond to the same vector in the picture (subject to the 1: 1 scale). The display of the three-dimensional world on the plane "removes" one dimension (the "depth" of the picture). If I correctly understood complex mathematical concepts, the multiplication of two matrices is precisely a similar reflection of one space in another. Therefore, if the reflection of the space A in the space B is possible, then the admissibility of the reflection of the space B in the space A is not guaranteed.
Any article ends at the moment when the author is bored with her writing. Since I did not set myself the goal of embracing the immense, I only wanted to understand the essence of the described operations on matrices and how exactly matrices are related to the systems of equations I solve, I did not go further into the wilds of linear algebra, but returned to econometrics and multiple regression, but did it more consciously. Understanding what and why I am doing and why it is the only way and not otherwise. What I did in this material can be titled as “a chapter on the essence of the basic operations of linear algebra, which for some reason was forgotten to be printed in textbooks”. But we do not read textbooks, right? To be honest, when I was at university, I really lacked an understanding of the issues raised here, so I hope that, having stated this difficult material in simple words, I am doing a good deed and helping someone to get into the essence of matrix algebra, transferring matrix operations from the section “Rite with a Tambourine” to the section “Practical Tools Used Consciously.”
Some 20 years ago I had a chance to study higher mathematics in high school, and we started with matrices (perhaps, like all students of that time). For some reason, it is believed that the matrix - the easiest topic in the course of higher mathematics. It is possible - because all actions with matrices are reduced to the knowledge of methods for calculating the determinant and several formulas built - again, on the determinant. It would seem, everything is simple. But ... Try to answer the elementary question - what is the determinant, what does the number you get when calculating it? (hint: a variant like “determinant is a number that is determined by certain rules” is not the correct answer, because it refers to the method of receipt, and not to the essence of the determinant). Give up? - then read on ...
Just want to say that I am not a mathematician neither by education nor by position. Unless I'm interested in the essence of things, and sometimes I try to “get to the bottom” of them. It was the same with the determinant: it was necessary to deal with multiple regression, and in this section of econometrics almost everything is done through ... the matrix, be they wrong. So I myself had to do a little research, because none of the familiar mathematicians gave a clear answer to the question, which initially sounded like “what is the determinant”. All argued that the determinant is a number that is specifically counted, and if it is zero, then ... In general, as in any textbook on linear algebra. Thank you, passed.
')
If one person came up with an idea, then the other person should be able to understand it (although, for this, sometimes you have to arm yourself with additional knowledge). Appeal to the "great and powerful" search engine showed that "the area of ​​the parallelogram is equal to the module of the determinant of the matrix formed by the vectors - the sides of the parallelogram ". In simple terms, if a matrix is ​​a way of writing a system of equations, then each equation separately describes a vector. Having constructed from the point of origin of the coordinates the vectors given in the matrix, we thus define a figure in the space. If our space is one-dimensional, then the figure is a segment; if it is two-dimensional, then the figure is a parallelogram, and so on.
It turns out that for one-dimensional space the determinant is the length of the segment, for the plane - the area of ​​the figure, for the three-dimensional figure - its volume ... then there are n-dimensional spaces that we cannot imagine. If the volume of the shape (that is, the determinant for the 3 * 3 matrix) is zero, then this means that the shape itself is not three-dimensional (it can be two-dimensional, one-dimensional, or even a point in general). The rank of the matrix is ​​the true (maximum) dimension of the space for which the determinant is nonzero.
So, almost everything is clear with the determinant: it defines the “volume” of the figure formed by the vectors described by the system of equations (although it is not clear why its value does not depend on whether we are dealing with the original matrix or transposed — perhaps transposing is a kind of affine conversion?). Now we need to deal with the actions on the matrices ...
If the matrix is ​​a system of equations (otherwise why do we need a table of some numbers that have no relation to reality?), Then we can do different things with it. For example, we can add two rows of the same matrix, or multiply a row by a number (that is, each row coefficient is multiplied by the same number). If we have two matrices with the same dimensions, then we can put them together (the main thing is that we don’t add a bulldog with a rhinoceros - but did mathematicians, developing the theory of matrices, think about this scenario?). Intuitively clear, especially as in linear algebra illustrations of such operations are systems of equations.
However, what is the meaning of matrix multiplication? How can I multiply one system of equations by another? What sense will have what I will receive in this case? Why is the transfer rule not applicable for matrix multiplication (that is, the product of matrices B * A is not that not equal to product A * B, but is not always feasible)? Why, if we multiply a matrix by a column vector, we get a column vector, and if we multiply a row vector by a matrix, then we get a row vector?
Well, here it is not that Wikipedia, - even modern textbooks on linear algebra here are powerless to give any intelligible explanation. Since the study of something according to the principle “you first believe - and then you will understand” - is not for me, I dig deep into the centuries (more precisely, I read textbooks of the first half of the 20th century) and find an interesting phrase ...
If the set of ordinary vectors, i.e. directional geometric segments is a three-dimensional space, then the part of this space consisting of vectors parallel to a certain plane is a two-dimensional space, and all vectors parallel to some straight line form a one-dimensional vector space.
In the books this is not directly stated, but it turns out that vectors parallel to a certain plane do not have to lie on this plane. That is, they can be in three-dimensional space anywhere, but if they are parallel to this particular plane, then they form two-dimensional space ... From the analogies that come to my mind - a photo: the three-dimensional world is represented on a plane, while the vector is parallel to the matrix (or film) the camera will correspond to the same vector in the picture (subject to the 1: 1 scale). The display of the three-dimensional world on the plane "removes" one dimension (the "depth" of the picture). If I correctly understood complex mathematical concepts, the multiplication of two matrices is precisely a similar reflection of one space in another. Therefore, if the reflection of the space A in the space B is possible, then the admissibility of the reflection of the space B in the space A is not guaranteed.
Source: https://habr.com/ru/post/277421/
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