By do you mean the space of deg(3) or less polynomials with real coefficiants?
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 ??