?
Solved

Orthogonal Operator Proof

Posted on 2009-04-07
1
Medium Priority
?
791 Views
Last Modified: 2012-06-21
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
Comment
Question by:AlephNought
[X]
Welcome to Experts Exchange

Add your voice to the tech community where 5M+ people just like you are talking about what matters.

  • Help others & share knowledge
  • Earn cash & points
  • Learn & ask questions
1 Comment
 
LVL 25

Accepted Solution

by:
InteractiveMind earned 1000 total points
ID: 24095716
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

Free Tool: ZipGrep

ZipGrep is a utility that can list and search zip (.war, .ear, .jar, etc) archives for text patterns, without the need to extract the archive's contents.

One of a set of tools we're offering as a way to say thank you for being a part of the community.

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Complex Numbers are funny things.  Many people have a basic understanding of them, some a more advanced.  The confusion usually arises when that pesky i (or j for Electrical Engineers) appears and understanding the meaning of a square root of a nega…
This article seeks to propel the full implementation of geothermal power plants in Mexico as a renewable energy source.
Although Jacob Bernoulli (1654-1705) has been credited as the creator of "Binomial Distribution Table", Gottfried Leibniz (1646-1716) did his dissertation on the subject in 1666; Leibniz you may recall is the co-inventor of "Calculus" and beat Isaac…
I've attached the XLSM Excel spreadsheet I used in the video and also text files containing the macros used below. https://filedb.experts-exchange.com/incoming/2017/03_w12/1151775/Permutations.txt https://filedb.experts-exchange.com/incoming/201…
Suggested Courses

777 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question