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W is a subspace in R^n. T is a linear operator on R^n defined by T(v)= w-z, where v= w+z, w in W, and z in W-perp. (v, w, z are vectors)

Prove T is an orthogonal operator.

Prove T is an orthogonal operator.

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So let T(u)=w-z, and T(v)=w'-z'

Then

<T(u), T(v)>

= <w-z, w'-z'>

= <w,w'> - <w,z'> - <z,w'> + <z,z'>

= <w,w'> + <w,z'> + <z,w'> + <z,z'> (because <z,w'>=<w,z'>=0)

= <w+z, w'+z'>

= <u, v>

Q.E.D.