November 18th, 2017, 07:41 PM  #1 
Newbie Joined: Nov 2017 From: Georgia Posts: 2 Thanks: 0  Orthonormal collection
Why any orthonormal collection of vectors is linearly independent?

November 18th, 2017, 08:48 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 535 Thanks: 306 Math Focus: Dynamical systems, analytic function theory, numerics 
Any collection of mutually orthogonal vectors is linearly independent. It is easy to prove. Suppose $v_1,\dots,v_n$ are orthogonal. Let $w = \sum \lambda_j v_j$ be an arbitrary linear combination of these vectors. Now compute the inner product, $(w,v_k)$ for any $1 \leq k \leq n$. It should be easy to conclude that if $w = 0$ and $v_k \neq 0$, then $\lambda_k = 0$. Give it a try.


Tags 
collection, independant, linear algebra, orthonormal 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Orthonormal Basis in R^3  simons545  Linear Algebra  0  May 6th, 2014 03:09 AM 
orthonormal  wannabe1  Linear Algebra  2  April 20th, 2010 04:47 AM 
Orthonormal basis  md288  Linear Algebra  1  November 13th, 2009 12:32 AM 
Orthonormal basis  omri1_ma  Linear Algebra  2  September 30th, 2008 01:09 AM 
orthonormal basis  shobhitngm  Real Analysis  0  November 16th, 2006 10:27 AM 