Is Definite Integration from 0 to 1 a Linear Transformation from Pn to R?

[Essnov]

I'm hoping I can get some help with the following question:

Does definite integration (from x = 0 to x = 1) of functions in P_n correspond to some linear transformation from Rⁿ⁺¹ to R?

OK, well my original answer was yes, but the textbook says "no, except for P₀" which I do not understand.

So I have p(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀

If I integrate from 0 to 1, I get: P(1) = a_n/(n+1) + a_n-1/n + ... + a₁/2 + a₀

Right?

So I have T(a_n, a_n-1, ..., a₁, a₀) = (a_n/(n+1) + a_n-1/n + ... + a₁/2 + a₀)

and T: Rⁿ⁺¹ -> R

And if I want to show that it is linear, then I show that the transformation has these properties:
T(kx) = kT(x) and
T(x+y) = T(x) + T(y)
and I think both cases are obvious.

And T is multiplication by [1/(n+1) | 1/n . . . 1/2 | 1]

What am I missing?
Thanks in advance for any help.

 

[hunt_mat]

Linear transformations usually mean between vector spaces of the same dimension and therefore can be loosely thought of as matrices. what you have is a linear functional.

Mat