# Characteristic polynomial and polynomial vector space

## Homework Statement

Let $$V:= ℝ_{2}[t]$$
$$V \in f: v \mapsto f(v) \in V, \forall v \in V (f(v))(t) := v(2-t)$$
a) Check that $$f \in End(V)$$
b) Calculate the characteristic polynomial of f.

## The Attempt at a Solution

a) Is it sufficient to check that $$(f+g)(t)=f(t)+g(t)$$ ?
b) Standard basis of polynomials is $$1+t+t^2$$, so $$f(1)=(2-t) \\ f(t)=2t - t^2 \\ f(t^2)=2t^2-t^3$$

What should I do next? What's up with $$t^3$$ ? (The space is of polynomials of degree at most 2). How do I calculate this map into a matrix ?

Dick
Homework Helper

## Homework Statement

Let $$V:= ℝ_{2}[t]$$
$$V \in f: v \mapsto f(v) \in V, \forall v \in V (f(v))(t) := v(2-t)$$
a) Check that $$f \in End(V)$$
b) Calculate the characteristic polynomial of f.

## The Attempt at a Solution

a) Is it sufficient to check that $$(f+g)(t)=f(t)+g(t)$$ ?
b) Standard basis of polynomials is $$1+t+t^2$$, so $$f(1)=(2-t) \\ f(t)=2t - t^2 \\ f(t^2)=2t^2-t^3$$

What should I do next? What's up with $$t^3$$ ? (The space is of polynomials of degree at most 2). How do I calculate this map into a matrix ?

Well, as you've observed, the given map doesn't take second degree polynomial into second degree polynomials, it maps them into the space of third degree polynomials. That wouldn't stop you from writing a matrix for it, but it won't be square. Not sure what to do with the rest of the question at all.