3. Matrix multiplication and affine transformations. In week 3 you saw that the matrix cos sin xo MA= = sin 0 cos transformed the first two components of a vector by rotating 0 0 it through an angle and adding the vector a = (2o, 3o). Another way to represent this cos sin transformation is an ordered pair A = (R(0), ), where R(0) = ( 4) is sin 0 30 1 is the 2 x 2 cos matrix in the upper-left corner of the 3 x 3 matrix MA- This is an affine transformation which acts on a 2-component vector 7 and returns w = R(0) + . (a) (15 points) If we apply a second affine transformation B = (R(o), b) to w, what is the resulting 2-component vector in terms of R(0), R(6), a, b, and ? (b) (15 points) If we apply the transformations of part (a) in the opposite order (B applied to , then A applied to the result), what is the resulting two-component vector? You should find that it is not the same an the answer from part (a). Explain this difference geometrically (using a sketch if you find it helpful). (c) (20 points) Letting b= (, 1), write B as a 3 x 3 matrix Mg in the same form as MA- Compute the matrix products MBMA and MAMB, and explain how applying these matrices to V = (v, 12, 1) agrees with your computations from (a) and (b).