E id k vectorial space and f∈L(E) 1)prove that if f is nilpotent with indice p≥1 ,I −f is bijective and (I−f)^(−1) =Σ_(i=0) ^(p−1) f^i 2)let E=R_n [x] and f∈L(E) / f(p) =p−p^′ prove that f is inversible and find f^(−1) .


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Question Number 37228 by abdo.msup.com last updated on 11/Jun/18

$E i d k v e c t o r i a l s p a c e a n d f \in L (E)$ $1) p r o v e t h a t i f f i s n i l p o t e n t w i t h i n d i c e$ $p ⩾ 1, I - f i s b i j e c t i v e a n d$ ${(I - f)}^{- 1} = \sum_{i = 0}^{p - 1} f^{i}$ $2) l e t E = R_{n} [x] a n d f \in L (E) /$ $f (p) = p - p^{^{'}} p r o v e t h a t f i s i n v e r s i b l e$ $a n d f i n d f^{- 1} .$
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