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.[br]You can redefine the matrix [math]M = \begin{pmatrix}a & c\\b & d\end{pmatrix}[/math].

Try out different matrices. Try for example:[br][br]a=d=1 and b=c=0,[br]a=b=c=d=0,[br]a=2, b=c=0 and d=3,[br]a=0, b=1, c=-1 and d=0,[br]a=b=c=d=1/2.[br][br]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.[br][br]Can you make a matrix that[br][br]reflects all vectors through the x-axis? The y-axis?[br]rotates every vector through an angle of 45 degrees?[br][br]Play on!