[2 days left] What’s wrong with your cloud strategy? Learn why multicloud solutions matter with Nimble Storage.Register Now

x
Solved

Orthogonal Operator Proof

Posted on 2009-04-07
Medium Priority
794 Views
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
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
1 Comment

LVL 25

Accepted Solution

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

Question has a verified solution.

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

Have you ever thought of installing a power system that generates solar electricity to power your house? Some may say yes, while others may tell me no. But have you noticed that people around you are now considering installing such systems in their …
This is a research brief on the potential colonization of humans on Mars.
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
Course of the Month14 days, left to enroll