Thread: 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 = x²

b) T = d/dx: R_<=3[x] --> R_<=3[x], v = x³

c) T is rotation of R² by 180 degrees, v = [3,4]

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

e) T: R³ --> R³ 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 ??

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

Yes you are correct jakncoke.

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

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?

the basis i got for b) was { }

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)}?

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