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.
You can redifine the matrix .

Try out different matrices. Try for example:
a=d=1 og b=c=0,
a=b=c=d=0,
a=2, b=c=0 og d=3,
a=0, b=1, c=-1 og d=0,
a=b=c=d=1/2.
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?
What happens to x = (0,0)?
Can you determine any eigenvectors of M?
Can you make a matrix that
reflects all vectors through the x-axis? The y-axis?
rotates every vector through an angle of 30 degrees?
adds 2 to the first coordinate of every vector? (Why is this impossible?)
Play on!