a) Find the standard matrix for the linear transformation from R 3 to R 3 that rotates

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Question:

a) Find the standard matrix for the linear transformation from R 3 to R 3 that rotates vectors by π/4 radians about the x–axis, counterclockwise looking towards the y–z plane from the point (1, 0, 0). (Note: under this rotation, points on the positive y–axis will rotate to points with positive y– and z–coordinates.)

b) Find the standard matrix for the linear transformation from R 3 to R 3 that rotates vectors by π/4 radians about the y–axis, counterclockwise looking towards the x–z plane from the point (0, 1, 0). (Note: under this rotation, points on the positive x–axis will rotate to vectors with positive x–coordinates and negative z–coordinates.)

c) Use matrix multiplication to find the standard matrix for the linear transformation from R 3 to R 3 that results from applying the rotation in part a) followed by the rotation in part b). (Be careful about the order in which you multiply your matrices!)

d) It’s a fact that composing two rotations, as in part c), always gives another rotation. Find the axis for the rotation you R computed in part c) by using the fact that rotating around an axis leaves vectors on that axis fixed: in other words, vectors pointing along the axis of rotation are simply eigenvectors with eigenvalue λ = 1, so you can find the axis of rotation by computing the λ = 1 eigenspace for the matrix in part c).

e) Now that you know the axis for the rotation R in part c), what about the angle of rotation? To calculate the angle, note that if you rotate about an axis in R 3 , then the plane perpendicular to that axis just axis gets rotated. Find a vector v perpendicular to the axis you computed in d), and calculate the effect of the rotation R on v. Then compute the angle θ between v and R(v) using the formula cos(θ) = v·R(v) |v||R(v)| .

f) What happens if you replace π/4 in parts a) and b) by another angle τ? Will the axis of rotation for the new composite rotation remain the same? Find the axis and angle of rotation for the case τ = π/2.