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**1. Homework Statement**

Suppose T is in L(V) and U is a subspace of V. Prove that U is invariant under T if and only if Uperp is invariant under T

^{*}.

**2. Homework Equations**

V = U [tex]\oplus[/tex] Uperp

if v [tex]\in[/tex] V, u [tex]\in[/tex] U, w [tex]\in[/tex] Uperp, then v = u + w.

<Tv, w> = <v, T

^{*}w>

**3. The Attempt at a Solution**

If U is invariant under T, this means that if u [tex]\in[/tex] U, Tu [tex]\in[/tex] U. Basically the same thing for Uperp. Not really sure where to go from here. Any ideas? Thanks!