Matrix Transformation Composition

Take the matrix with columns $1, 1$ and $-2, 0$ (Matrix $M_1$) and another matrix with columns $0, 1$ and $2, 0$ (Matrix $M_2$). Determine the matrix that represents the total effect of applying $M_1$ then $M_2$ as a single transformation. Solve this without visual aids, using only the numerical entries in each matrix.

In this problem, you are tasked with finding a single matrix that represents two successive linear transformations in two dimensions. The main concept here is matrix multiplication, which allows us to combine two linear transformations into one. When you apply matrix M1 then M2, the resulting transformation can be found by multiplying these matrices, with the first transformation matrix multiplying the second from the right.

This order is crucial because matrix multiplication is not commutative, meaning the order in which you multiply matrices matters greatly - reversing the order could lead to a completely different result or an invalid operation.

This problem touches on the fundamental idea of how linear transformations, represented by matrices, can be combined. When you combine transformations, you need to think of each transformation as a function that affects the space. By multiplying the matrices, we are chaining these functions together, which is key in understanding processes like rotating, scaling, translating vectors, or any combination of these.

For students, the task hones skills in performing matrix multiplications accurately and understanding the implications of such operations. It's an excellent exercise to delve into the practical applications of matrix operations in linear algebra and how these concise, abstract representations are used in various fields like computer graphics, physics, and engineering to model and transform spaces.