Matrices as Geometric Transformations
Representing Linear Transformations with 3x3 Matrices
Using the applet below, you can enter various 3x3 matrices, M (as shown in rows 1-3; and leaving the bottom row as 0,0,1), and various 3x1 vectors, v (as shown in rows 5-7 of columns A, B, C, and D; and leaving the third term as 1). Row 9 shows each product, Mv. Plotting the first two coordinates for v (in blue) and Mv (in red) shows the geometric transformation, on the right.
Representing Isometries in General
We can represent the most challenging isometries of the plane by breaking them down into a composition of simpler transformations and representing each of those simpler transformations using either analytic geometry or matrices. How could you represent a reflection of point P over the line y=mx+b, as shown in the applet below?