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Orthogonal Operator Proof

Posted on 2009-04-07
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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.
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Question by:AlephNought
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One definition of orthogonality of an operator is that the inner product is preserved. Viz. <u,v> = <T(u), T(v)>

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.
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