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evilpostingmong
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Homework Statement
Suppose T ∈ L(V) and U is a subspace of V. Prove that U is
invariant under T if and only if PUTPU = TPU.
Homework Equations
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.
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