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Pearce_09
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Prove: If V is a finite dimensional vector space and T is in L(V), then there exists a finite list of scalars ao,a1,a2,...,an, not all 0 such that
aoX + a1T x + a2T^2 x... + anT^n x = theata
for all x in V
my hint for the question is:
the powers of T are defined as T^0 = I, T^1 = 1, T^2 = TT, T^3 = T^2T
consider the sequence I, T, T^2, T3,... in the finite-dimensional vector space L(V).
please help, have have no clue what to do. Any help would be greatly appriciated.
aoX + a1T x + a2T^2 x... + anT^n x = theata
for all x in V
my hint for the question is:
the powers of T are defined as T^0 = I, T^1 = 1, T^2 = TT, T^3 = T^2T
consider the sequence I, T, T^2, T3,... in the finite-dimensional vector space L(V).
please help, have have no clue what to do. Any help would be greatly appriciated.