Thread: Invariant subspaces

I really need some help with this problem:

For each of the following linear operators T on the vector space V, determine whether the given subspace W is a T-invariant subspace of :

V=P(R), T(f(x)) = xf(x), and W=P2(R)

I was thinking that in order for this to be T-invariant, that means that it needs to be contained in W because T(W) needs to be contained in W.

Am I on the right track?

Clarification: is P(R) the set of all polynomials on the reals? And is P2(R) the set of all quadratics? If so, you tell me: is multiplication by x going to leave a quadratic a quadratic?

I should also point out that "T-invariant" might mean different things depending on definition. Does it mean that T is a bijection from a space onto itself? Or is it allowed to map a subspace to a subset of the subspace (i.e., can it fail to be onto?)

Sorry about my notation; yes P(R) is the set of all real polynomials and P2(R) are all real polynomials of the third degree.

: T(W)=x(1+x+x^2) = x+x^2+x^3 <-- that is different than P2(R) because its not in the same dimension, the result is P4(R)

So does this mean that it is not T-invariant?

So does this mean that it is not T-invariant?

I would say so. Multiplication by x is never going to leave PN(r) invariant; it'll always raise the highest degree polynomial's degree by one, thus pushing you out of the subspace.

Oh that makes sense! Thanks so much~

You're very welcome. Have a good one!