Arr Lin. algebra problem (frustrating)

[georgeh]

the question asks;
Consider the following matrices. What is the corresponding transfrmation on polynomials? Indicate the domain P_i and the codomain p_j
The matrix is: [1,1;1,-1]
I looked at the sol. manual and it states
T(ax+b)=(a+b)x+(a-b)
I seriously have no idea how they came up with that. I read the section and i don't see any examples on how to do it.

 

[nocturnal]

Given an arbitrary polynomial $ax+b \epsilon P_1(\mathbb{R})$, what is its coordinate vector with respect to the standard ordered basis for $P_1(\mathbb{R})$?

What is the result when you multiply this coordinate vector (on the left) by your matrix?

What is the element of $P_1(\mathbb{R})$ that corresponds to this result?

 

[nocturnal]

the first line should be $ax+b \ \epsilon \ P_1(\mathbb{R})$. For some reason the symbol $\epsilon$ isn't showing.