Proving Finite Scalar Series for V and T in L(V)

[Pearce_09]

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.

 

[HallsofIvy]

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.

"theata"? Do you mean the 0 vector? If that is the case then saying
aoX + a1T x + a2T^2 x... + anT^n x = 0 with not all a0,a1,... zero is the same as saying that {x, Tx, T^2x, T^3x, ..., T^n x} are linearly dependent.

Suppose all of {x, Tx, T^x, ..., T^n x} were distinct for n larger than the dimension of T. What does that say about the linear independence of the set? On the other hand, suppose T^k x= T^j x for some k and j. What does THAT say about the linear independence of the set?