From actions on matrices to understanding their essence ...

I respect people who have the courage to say that they don’t understand something. Himself so. What I don’t understand, I must study, comprehend, understand. The article “ Mathematics on the Fingers ”, and especially the matrix notation of formulas, made me share my small but, it seems, important experience with matrices.

About 20 years ago I had a chance to study higher mathematics at a university, and we started with matrices (perhaps, like all students of that time). For some reason, it is believed that matrices are the easiest topic in the course of higher mathematics. Perhaps - because all actions with matrices come down to knowing how to calculate the determinant and several formulas built - again, on the determinant. It would seem that everything is simple. But ... Try to answer the basic question - what is a determinant, whatmeans the number that you get when calculating it? (hint: a variant of the type “determinant is a number that follows certain rules” is not the correct answer, because it talks about the method of obtaining, and not about the essence of the determinant). Give up? - then we read further ...

I want to say right away that I am not a mathematician by education or by position. Unless I'm interested in the essence of things, and sometimes I try to "get to the bottom of them." The same thing happened with the determinant: it was necessary to deal with multiple regression, and in this section of econometrics almost everything is done through ... matrices, be they wrong. So I had to do a little research myself, because not one of the familiar mathematicians gave a clear answer to the question posed, which initially sounded like "what is a determinant." Everyone claimed that the determinant is a number that is specially calculated, and if it is equal to zero, then ... In general, as in any textbook on linear algebra. Thank you, passed.

If one person came up with an idea, then another person should be able to understand it (though for this sometimes you have to arm yourself with additional knowledge). An appeal to the "great and mighty" search engine showed that "the area of ​​the parallelogram is equal to the modulus of the determinant of the matrix formed by vectors - the sides of the parallelogram ." In simple terms, if a matrix is ​​a way of writing a system of equations, then each equation individually describes a vector. Having constructed the vectors defined in the matrix from the origin, we will thus define some shape in space. If our space is one-dimensional, then the figure is a segment; if 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 a three-dimensional figure - its volume ... then there are n-dimensional spaces that we cannot imagine. If the volume of the figure (i.e., the determinant for the 3 * 3 matrix) is equal to zero, then this means that the figure itself is not three-dimensional (it can be two-dimensional, one-dimensional, or even represent a point). The rank of the matrix is ​​the true (maximum) dimension of the space for which the determinant is not equal to zero.

So, with the determinant almost everything is clear: it determines the “volume” of the figure formed by the vectors described by the system of equations (although it is unclear why its value does not depend on whether we are dealing with the original matrix or with the transposed one - perhaps transposition is a kind of affine conversions?). Now you 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 nothing to do with 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 add them (the main thing is that we don’t add a bulldog with a rhino - but do the mathematicians, when developing the theory of matrices, think about this scenario?). Intuitively, especially since in linear algebra, illustrations of such operations are systems of equations.

However, what is the point of matrix multiplication? How can I multiply one system of equations by another? What is the meaning of what I get in this case? Why is the displacement rule not applicable for matrix multiplication (that is, the product of the matrices B * A is not only not equal to the product A * B, but it is not always possible)? Why, if we multiply the matrix by the column vector, we get the column vector, and if we multiply the row vector by the matrix, we get the row vector?

Well, here it’s not like Wikipedia, even modern textbooks on linear algebra are powerless to give any clear explanation. Since the study of something on the principle of “you first believe - and you will understand later” - is not for me, I dig back centuries (more precisely, I read textbooks from the first half of the 20th century) and find an interesting phrase ...
If the set of ordinary vectors, i.e. directed geometrical 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 a straight line form a one-dimensional vector space.

The books do not say this directly, but it turns out that vectors parallel to some plane do not have to lie on this plane. That is, they can be located in three-dimensional space anywhere, but if they are parallel to this particular plane, then they form a two-dimensional space ... From the analogies that come to my mind - a photograph: the three-dimensional world is represented on a plane, with this vector parallel to the matrix (or film) the camera will correspond to the same vector in the picture (subject to a 1: 1 scale). Displaying a three-dimensional world on a plane "removes" one dimension (the "depth" of the picture). If I correctly understood complex mathematical concepts, the multiplication of two matrices just represents a similar reflection of one space in another. Therefore, if the reflection of the space A in the space B is possible,

Any article ends at the moment when the author bothers her IPR and be. Since I did not set myself the goal of embracing the immensity, I solely wanted to understand the essence of the described operations on matrices and how the matrices are related to the systems of equations that I solve, I did not go into further wilds of linear algebra, but returned to econometrics and multiple regression, but made it already more consciously. Understanding what and why I am doing and why only this way and not otherwise. What I got in this material can be entitled 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 don’t read textbooks, right? To be honest, when I was at university, I really lacked understandingthe issues raised here, so I hope that by setting out this difficult material in simple words as possible, I’m doing a good deed and helping someone understand the very essence of matrix algebra by transferring operations on matrices from the section “shredding with a tambourine” to the section “practical tools used consciously. "

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