- #1

#### member 545369

## Homework Statement

Let P

_{2}be the vector space of all polynomials of a degree at most 2 with real coefficients. Let T: P

_{2}→ℝ be the functioned defined by:

##T(p(t)) = p(2) - p(1)##

a) Find a non-zero element of the Kernel of T. (I think I figured this one out, but I'm not too sure).

b) Find a specific element of p(t) for which T(p(t)) = π. Describe the image of T as an explicit subset of ℝ. ("Image" and "Range" mean the same.)

## Homework Equations

##p(t) = a_0 + a_1t + a_2t^2##

## The Attempt at a Solution

[/B]

a) What I understood by the "non-zero element of ker(T)" part of the question was that they didn't want me to just write f(x) = 0. So, this is what I wrote down:

##ker(T) = \{f(x) ∈ P^2 : f(2) = f(1)\}##

b) I'll write down my attempt at a solution:

If we're look for a ##p(t)## such that ##T(p(t)) = π##, then we're going to need an equation that follows the requirements of ##p(2) - p(1) = π##. I was thinking solving the following linear system:

##a_0 + 2a_1 + 4a_2t = π + 2##

##a_0 + a_1 + a_2 = 2##

But I feel like I'm overcomplicating things, because the margin given to me in this question doesn't permit the space required to solve this system of linear equations. There must be a clever way of doing this, no? I was thinking about substituting ##a_0 = 0##, since I was asked for a specific solution but I don't know… It seems as if I'm over-complicating things.