This applet allows you to experiment with 2x2-matrices and linear transformations of the plane. You can move the vector x (the blue arrow) and see how the vector y = Mx (the black arrow) moves. The red lattice illustrates how the entire plane is effected by the transformation represented by M. You can redefine the matrix .
Try out different matrices. Try for example: a=d=1 and b=c=0, a=b=c=d=0, a=2, b=c=0 and d=3, a=0, b=1, c=-1 and d=0, a=b=c=d=1/2. For each matrix consider the following: What happens to x = (1,0) and x = (0,1)? Notice the connection between these and the matrix M. Can you make a matrix that reflects all vectors through the x-axis? The y-axis? rotates every vector through an angle of 45 degrees? Play on!