If you are not familiar with the concept of linear algebra and want to study natural science, it seems almost illiterate now. However, according to the current international standards, linear algebra is expressed by axioms and is the second generation mathematical model, which brings difficulties to teaching.
Note that the coordinate system represented by the M matrix consists of a set of bases, and the set of bases also consists of vectors. There is also a question about which coordinate system the vectors are measured in. In other words, the general method of representing a matrix should also indicate its reference coordinate system. The so-called m is actually IM, that is to say, the measurement of the group of bases in m is obtained in the I coordinate system. From this point of view, M×N is not a matrix multiplication, but another coordinate system N measured in the M coordinate system, where M itself is measured in the I coordinate system.
Let's go back to the question of transformation. I just said, "the transformation of an object in a fixed coordinate system is equivalent to the transformation of the coordinate system in which the fixed object is located." We found a "fixed object", which is a vector. But what about the transformation of coordinate system? Why didn't I see it?
Please see:
Ma = Ib
I'm going to change m to I now. How come? By the way, multiplying by M- 1 is the inverse matrix of m, in other words, don't you have the coordinate system m? Now I multiply it by an M- 1 to become I, so that if A in the original M coordinate system is in I, B will be obtained.
I suggest you pick up a pen and paper and draw a picture at this moment to have an understanding of this matter. For example, if you draw a coordinate system, the unit of measurement on the X axis is 2, and the unit of measurement on the Y axis is 3. In such a coordinate system, the point with coordinates (1, 1) is actually the point (2,3) in Cartesian coordinate system. And let it betray oneself, is to put the original coordinate system:
2 0
0 3
The measurement in the X direction is reduced to the original 1/2, and the measurement in the Y direction is reduced to the original 1/3, so that the coordinate system becomes the unit coordinate system i. Keeping the point unchanged, the vector will now become (2,3).
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 make the original coordinate system:
2 0
0 3
Matrix:
1/2 0
0 1/3
Turn left. This matrix is the inverse of the original matrix.
Here we come to an important conclusion:
"The method of applying a transformation to a coordinate system is to multiply the matrix representing the coordinate system with the matrix representing the change."
The multiplication of matrices once again becomes the application of motion. However, it is no longer a vector, but another coordinate system.
If you think you are still clear, please think about the conclusion just mentioned. On the one hand, the matrix MxN represents the transformation result of the coordinate system N under the movement M, on the other hand, M is used as the prefix of N and the environmental description of N, that is to say, there is another coordinate system N under the measurement of the M coordinate system. If this coordinate system n is measured in the I coordinate system, the result is the coordinate system MxN.
Here, I have actually answered a most puzzling question for ordinary people to learn linear algebra, that is, why the multiplication of matrices should be stipulated in this way. Simply put, it is because:
I can't say more. A matrix is a coordinate system and a transformation. It is not clear whether it is a coordinate system or a transformation. Movement and entity are unified here, and the boundary between matter and consciousness disappears. Everything is unspeakable and cannot be defined. Tao can be Tao, extraordinary Tao, famous name, extraordinary name. Matrix is something that can't be said or said. At this point, we have to admit that the definition of matrix in our great linear algebra textbook is extremely correct:
"A matrix is a mathematical object consisting of m rows and n columns."