(a.2) Let V be the vector space of polynomials over K of degree n-1 andf: V -> V the map defined by derivative: f(u(r))u(x) u(x)e V. Show that f" 0 and describe all invariant subspaces of V. Is V decomposable? (Hint: Consider two cases: K contains Q (so all nonzcro integers are nonzero in K), or K contains Z, where p > 1 is a prime (so p = 0 in K and i0 for every integer i with 0 < i < p)).

Question
(a.2) Let V be the vector space of polynomials over K of degree
n-1 andf: V -> V
the map defined by derivative:
f(u(r))u(x) u(x)e V.
Show that f"
0 and describe all invariant subspaces of V. Is V decomposable?
(Hint: Consider two cases: K contains Q (so all nonzcro integers are nonzero
in K), or K contains Z, where p > 1 is a prime (so p = 0 in K and i0 for
every integer i with 0 < i < p)).
Expand
Transcribed Image Text

(a.2) Let V be the vector space of polynomials over K of degree n-1 andf: V -> V the map defined by derivative: f(u(r))u(x) u(x)e V. Show that f" 0 and describe all invariant subspaces of V. Is V decomposable? (Hint: Consider two cases: K contains Q (so all nonzcro integers are nonzero in K), or K contains Z, where p > 1 is a prime (so p = 0 in K and i0 for every integer i with 0 < i < p)).

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