Proof based on projections

1. Jul 9, 2009

evilpostingmong

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$$\in$$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$$\in$$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.

Last edited: Jul 9, 2009
2. Jul 10, 2009

evilpostingmong

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