My Math Forum  

Go Back   My Math Forum > College Math Forum > Linear Algebra

Linear Algebra Linear Algebra Math Forum

Thanks Tree1Thanks
  • 1 Post By mathman
LinkBack Thread Tools Display Modes
July 29th, 2014, 11:21 AM   #1
Joined: Mar 2013

Posts: 71
Thanks: 4

prove that $x$ and $y$ are multiples

I've got the following problem:
"Let $x,y \in \mathbb R^n$. If every $z \in \mathbb R^n$ which is orthogonal to $x$ is also orthogonal to $y$, prove that $x$ and $y$ are multiples".

Proof: consider two vectors $x,y$. The projection of $y$ into $x$ is $w=\frac{<x,y>x}{<x,x>}$. Then we have $<y-w,x>=0$. By hypothesis of the problem, $<y-w,y>=0 \Rightarrow |y|^2=<w,y>$. If I could get that $|x|^2|y|^2=<x,y>^2$, it's done. But how?
walter r is offline  
July 29th, 2014, 02:15 PM   #2
Global Moderator
Joined: May 2007

Posts: 6,661
Thanks: 648

Let y = ax + z, where (x,z) = 0, with a = (x,y)/(x,x). However, since (x,z) = 0, then (y,z) = 0. Therefore (y,y) = [(x,y)]^2/(x,x), which is what you are trying to show.
Thanks from walter r
mathman is offline  

  My Math Forum > College Math Forum > Linear Algebra

$x$, $y$, multiples, prove

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Common multiples of two numbers md9 Abstract Algebra 1 May 10th, 2013 02:26 PM
multiples nitin1 Number Theory 11 December 14th, 2012 10:59 AM
why is my function so stable at multiples of 22? mark212 Algebra 5 April 10th, 2012 10:06 PM
converting multiples Tylerman Applied Math 4 January 30th, 2012 02:15 PM
multiples of 2 pi Icevox Number Theory 8 March 25th, 2011 02:11 PM

Copyright © 2019 My Math Forum. All rights reserved.