Linear transformation

1. Jul 20, 2010

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 Pn 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) = anxn + an-1xn-1 + ... + a1x + a0

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

Right?

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

2. Jul 20, 2010

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