# T-cyclic Operator - linear algebra

## Homework Statement

Let T be a linear operator on a vector space V, and suppose that V is a T-cyclic subspace of itself. Prove that if U is a linear operator on V, then UT=TU if and only if U=g(T) for some polynomial g(t).

## Homework Equations

They suggest supposing that V is generated by v.
and choose g(t) such that g(T)(v) = U(v).

## The Attempt at a Solution

I could get some coherent stuff for one way..

=> assume U=g(T), prove UT=TU.

first, {v, T(v), T^2(v), ... , T^n-1(v)} is a basis of V (thus V is n-dimensional).
U=g(T)
let y=U(x) choosem g(t) such that y = g(T)(v).

then UT(x)=U(T(x)) = U(T(a_0v + a_1T(v)+...+a_n-1T^n-1(v)) = U(a_0T(v) + a_1T^2(v) + ... +a_n-1T^n(v))
= a_0UT(v) + a_1UT^2(v) + ... +a_n-1UT^n(v)

then TU(x)= T(U(x))

(where x is a linear comb of basis vectors, x=a_0v + a_1T(v)+...+a_n-1T^n-1(v))

thats as far as i got.. =(

<=

the other direction is more difficult for me.. i'm not really sure where to get the proper expression for g(t) from at all.

AKG