• Status: Solved
  • Priority: Medium
  • Security: Public
  • Views: 795
  • Last Modified:

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.
0
AlephNought
Asked:
AlephNought
1 Solution
 
InteractiveMindCommented:
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

Featured Post

What does it mean to be "Always On"?

Is your cloud always on? With an Always On cloud you won't have to worry about downtime for maintenance or software application code updates, ensuring that your bottom line isn't affected.

Tackle projects and never again get stuck behind a technical roadblock.
Join Now