Visualizing Multiplication of Two Matrices

This example demonstrates matrix multiplication geometrically by applying two linear transformations to a polygon (or a set of points). It shows that performing the first matrix transformation and then the second is equivalent to applying the matrix product once. The key concepts are matrix multiplication (including its non-commutativity), the relationship between matrices and linear geometric transformations, and how point coordinates change under matrix action—often producing effects such as rotation and scaling. Construction highlights: the setup uses Random to generate parameters or random configurations, LaTeX to label matrices/expressions, and Polygon to build the base and transformed shapes (e.g., poly1, poly2, poly4). A collection of labeled points (a, b, c, …) and text objects (text1, text2, A–R) represent vertices before and after transformations, while indexed point coordinates (such as c_2, d_1, e_1, f_1, g_1, h_1, i, j) encode how matrix entries combine to produce the final vertex locations corresponding to the product matrix. Suitable for high school through early university linear algebra, as a classroom demonstration or self-study tool for understanding how matrix products correspond to composition of transformations.