# Characteristic polynomial and polynomial vector space

1. Apr 13, 2013

### Caims

1. The problem statement, all variables and given/known data
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.
2. Relevant equations

3. 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 ?

2. Apr 13, 2013

### Dick

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.