Monic Generator (Minimal Polynomial)

[gain01]

1. Homework Statement [/b]
Let V be the space of all polynomials of degree less than or equal to 2 over the reals. Define the transformation, H, as a mapping from V to R[x] by $$(Hp)(x)=\int^x_{-1}p(t)dt\\$$. a) Find the monic generator, d, which generates the ideal, M, containing the range of H.
b) If $$f\in{M\cap Range(H)$$ then $$f=dg$$ where d is given above. Show that $$g\in{V}$$.

Homework Equations

The Attempt at a Solution

I know the monic generator is the minimal polynomial which annihilates H. The way I would find the minimal polynomial is
1. find a basis for V
2. find the matrix representation for H with respect the basis
3. find the characteristic eqn of the matrix
4. the minimal polynomial divides the characteristic polynomial.

But when I try step 2, I keep getting a 4 x 3 matrix which doesn't have a determinant. So i don't know how I would find the characteristic eqn.

 

[HallsofIvy]

H maps V into a 3 dimensional subspace of R[x]. Use that three dimensional subspace as the "range" space rather than R[x] and you will get a 3 by 3 matrix.