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: 276 Thanks: 141 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.


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collection, independant, linear algebra, orthonormal 
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