What is $f(T)$?

Question

Let a linear transformation $T:\mathbb{R}^3\to \mathbb{R}^3$ defined as $T(v_1, v_2, v_3) = (v_1, v_3 - 2v_2, -v_3)$. Calculate $f(T)$ where $f(X) = -X^2 + 2 \in \mathbb{R}[X]$

I'm not so sure how to evaluate $f(T)$. I'll be glad for an explanation.

Maybe $T(v)$ can be viewed as the polynomial $v_1 + (v_3 - 2v_2)x -v_3x^2$?

Answer 1

We have

$$f(T)=-T^2+2id$$ so

$$f(T)v=-T^2v+2v,\quad\text{where}\; v=(v_1,v_2,v_3)$$