This applet allows you to experiment with 2x2-matrices and linear transformations of the plane. You can move the vector x and see how the vector y = Mx moves. The red lattice illustrates how the entire plane is effected by multiplication with M.[br]You can redifine 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 og b=c=0,[br]a=b=c=d=0,[br]a=2, b=c=0 og d=3,[br]a=0, b=1, c=-1 og 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)? Can you make a rule based on your experiments?[br][br]What happens to x = (0,0)?[br][br]Can you determine any eigenvectors of 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 30 degrees?[br]adds 2 to the first coordinate of every vector? (Why is this impossible?)[br][br]Play on!