Expert: David Hemmer - 10/6/2008

Question
two 2x2 matrices A= (0.5,-sqrt3/2)(sqrt3/2, 0.5) B= (1,2)(0,1)   a) calculate matrix powers A^2 and A^3. Describe the linear transformations corresponding to each of there powers.   For this question I have found A^# and A^3. But dont understand how to describe the linear transformations. Does it have something to do with the fact that AA^2=AA..??   Another question:   consider two transformations of a plane with centre 0. T: R^2->R^2, a rotation about 0, anticlockwise through pi/3 radians. R: R^2->R^2, a shear of the plane in the x1 direction, shear factor 2.   (note: R^2->R^2 means all real numbers squared)   b) Find the image of a unit square under both the transformations ToR and RoT, corressponding to matrices AB and BA.   totally stuck on this question!! thanks heaps for you help :)

Answer
The linear map given by your matrix A is just a rotation around the origin ccwise of 60 degrees. The matrix B is the shear transformation described in part b of your question.

Just look at what happens to the unit square, in one case it gets sheared into a parallelogram then rotated. In the other case it gets rotated then sheared.