1. Prove that mapping

$\displaystyle T:\mathbb{R}_{2}[x]\rightarrow \mathbb{R}_{2}[x] , T(ax^{2}+bx+c)= 3ax^{2}+2(b-c)x +a $

is linear operator and find some basis of subspace Im(T), and Ker (T),and find dim Im(T) and dim Ker(T)

It's easy to prove that is linear operator. But, how to find basis for Im, and Ker ?

2. Let A $\displaystyle A:\mathbb{R}^{3}\rightarrow R_{3}[x]$ be linear operator such that

$\displaystyle A(1,0,0)=2x + x^{3}, A(0,1,0)=-2x + x^{2}, A(0,0,1)= x^{2}+x^{3}$. Find linear operator a, some basis of subspace Im(T), and Ker (T),and find dim Im(T) and dim Ker(T).

In this one I dont know how to find A, and later, basis of Im and Ker.