Orthogonal Operator Proof

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.
AlephNoughtAsked:
Who is Participating?
 
InteractiveMindConnect With a Mentor Commented:
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.
0
Question has a verified solution.

Are you are experiencing a similar issue? Get a personalized answer when you ask a related question.

Have a better answer? Share it in a comment.

All Courses

From novice to tech pro — start learning today.