How to find an explicit formula for linear transformation?

Question

Let $T\colon \Bbb R_3[x]\rightarrow \Bbb R^3 $ be a linear transformation. Suppose that : $T(1)=(1,1,1), T(x+1)=(0,1,1), T(x^2+x+1)=(0,0,1)$

find an explicit formula for $T(p(x))$ for any $p(x)=ax^2+bx+c\in\Bbb R_3[x].$

I tried to put the polynomial expression of $T$ in a matrix and saw that I get the standard basis, but I don't know if it helps me or how to find an explicit formula . Any help is greatly appreciated

Answer 1

Observe that $p(x) = c+bx+ax^2$ can be written as $$p(x) = (c-b)+(b-a)(1+x)+a(1+x+x^2).$$ Hence $$T(p(x)) = (c-b)T(1)+(b-a)T(1+x)+aT(1+x+x^2).$$

Answer 2

HINT

Let observe that

• $T(1)=(1,1,1)$
• $T(x)=T(x+1)-T(1)=(-1,0,0)$
• $T(x^2)=T(x^2+x+1)-T(x+1)=(0,-1,0)$

and

• $T(ax^2+bx+c)=aT(x^2)+bT(x)+cT(1)$