1. The problem statement, all variables and given/known data Suppose T ∈ L(V) and U is a subspace of V. Prove that U is invariant under T if and only if PUTPU = TPU. 2. Relevant equations 3. The attempt at a solution Consider u[tex]\in[/tex]U. Now let U be invariant under T. Now let PU project v to U so that PU(v)=u. Therefore TPU(v)=T(u). Now since T(u) is[tex]\in[/tex]U, PU should project T(u) back to U and by the definition of P^2=P, PU must be an identity operator since T(u) is in U, the space PU projects T(u) to, so PUTPU(v)=PUT(u)=T(u) which is equivalent to TPUv since TPUv =T(u) thus PUTPU=TPU.