Is U Invariant under T if PUTPU = TPU?

[evilpostingmong]

Homework Statement

Suppose T ∈ L(V) and U is a subspace of V. Prove that U is
invariant under T if and only if P_UTP_U = TP_U.

Homework Equations

The Attempt at a Solution

Consider u$$\in$$U. Now let U be invariant under T. Now let P_U project
v to U so that P_U(v)=u. Therefore TP_U(v)=T(u). Now
since T(u) is$$\in$$U, P_U should project T(u) back to U and
by the definition of P^2=P, P_U must be an identity operator since T(u) is in U,
the space P_U projects T(u) to, so P_UTP_U(v)=P_UT(u)=T(u)
which is equivalent to TP_Uv since TP_Uv =T(u) thus P_UTP_U=TP_U.

 

[evilpostingmong]

would like to know whether or not its right or wrong, thank you!