# Algebra proof!

1. Nov 6, 2005

### 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).