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 or green) 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?
Multiplying Matrices
Suppose you could represent one transformation with Matrix A and another transformation with Matrix B. What should Matrix B*A represent?
Conjugation
As we saw in the previous lesson (on the Cartesian Plane and analytic geometry) it is sometimes helpful to represent transformations through conjugations (like a-1ba, or even a-1b-1cba) involving simpler representations. So, we can apply the same idea to matrix representations. The applet below will perform the conjugation for us, if we can represent the intended transformation with simpler transformations, using matrices A, B, and C.