Characteristic polynomial and polynomial vector space

  • Thread starter Caims
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Homework Statement


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

Homework Equations



The Attempt at a Solution


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

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

Answers and Replies

  • #2
Dick
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Homework Statement


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

Homework Equations



The Attempt at a Solution


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

What should I do next? What's up with [tex]t^3[/tex] ? (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.
 

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