Find a basis for teh corresponding cyclic subspace

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 ??

Re: Find a basis for teh corresponding cyclic subspace

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

Re: Find a basis for teh corresponding cyclic subspace

Yes you are correct jakncoke.

Re: Find a basis for teh corresponding cyclic subspace

b and d seem to be incorrect. recall for d, the transformation matrix of rotation is given by For the last one, transformation matrix is given by

Re: Find a basis for teh corresponding cyclic subspace

So for b) would I have to go a step further and say the basis is {3x^2, 6x, 6}? or am I missing something between a and b that makes a correct and b incorrect?

Re: Find a basis for teh corresponding cyclic subspace

the basis i got for b) was { }

Re: Find a basis for teh corresponding cyclic subspace

For e) I am having trouble coming up with a basis. Do I multiply the given vector by the transformation matrix, or is the basis just {(1,0,2),(2,-1,0),(0,0,1)}?

Re: Find a basis for teh corresponding cyclic subspace

the basis is

so you have we see that and

so since the third vector makes this linearly dependent, throw it away

and thus resulting is the basis