"Linear Algebra and its Applications"-linear transformation

Linear transformations:

As we mentioned earlier, when we discuss the matrix equation ax = b and the vector equation X1a1+x2a2+x3a3+...+xnan = b, we have said that this will echo the rules of matrix multiplication. But here we first introduce a concept of transition-the linear transformation.

To investigate the matrix equation ax = B,a is the n x m matrix, X is the r^n vector, by the rule we defined earlier, B must be the r^m vector. We abstract this process, from the point of view of set theory or function, to see such a process of obvious mapping, we treat vector x as the original image, Vector B as the image, and multiply matrix A as a correspondence relation.

Why build such a mapping model? Because this dynamic matrix multiplication concept will help us to apply this computing tool to the actual system (for example, to establish a mathematical model of a time-varying physical system and an in-depth understanding of the operation of the linear algebra itself).

In order to better understand this concept, we give the following simple examples.

"Linear Algebra and its Applications"-linear transformation