For each of the following find a basis for the corresponding cyclic subspace:

a) T = d/dx : R_{<=3}[x] --> R_{<=3}[x], v = x^{2}

b) T = d/dx: R_{<=3}[x] --> R_{<=3}[x], v = x^{3}

c) T is rotation of R^{2}by 180 degrees, v = [3,4]

d) T is rotation by 30 degrees, v = [3,4]

e) T: R^{3}--> R^{3}given by T(x,y,z) = (x+2z, 2x-y, z), and v = [1,0,0]

My thoughts on what bases are:

a) Would a basis be (x^2, 2x, 2)?

b) Would a basis be (x^3, 3x^2, 6x)?

c) Would a basis be {(3,4),(-3,-4)}?

d) Would a basis be {(3,4),(-3,-4)}? *Not too sure on this one*

e) I have no idea ??