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May 4th, 2018, 11:58 AM  #1 
Member Joined: Apr 2014 From: Greece Posts: 58 Thanks: 0  Orthonormal basis B:{e1,e2,e3} with respect to an inner product space
We have the inner product $\displaystyle <(x_1,x_2,x_3),(y_1,y_2,y_3)>=3x_1y_1+x_1y_3+y_1x_ 3+x_2y_2+2x_3y_3$ I'm asked to find the orthonormal basis of $\displaystyle R^3$ that is given from the normal basis $\displaystyle B=(e_1,e_2,e_3)$, $\displaystyle e_1=(1,0,0), e_2=(0,1,0), e_3=(0,0,1)$ with respect to the above inner product I guess I should apply the above inner product to the basis and then find the new one but I'm not sure how to do it.. Any help? 
May 4th, 2018, 05:05 PM  #2 
Member Joined: Jan 2016 From: Athens, OH Posts: 92 Thanks: 47 
First, it is fairly easy to verify that the given scalar product is actually positive definite. So this is a straight forward application of the GramSchmidt orthogonalization process. If you don't know it, here's Wikipedia's article: https://en.wikipedia.org/wiki/Gram%E...chmidt_process Last edited by johng40; May 4th, 2018 at 05:28 PM. 
May 6th, 2018, 02:06 AM  #3 
Member Joined: Apr 2014 From: Greece Posts: 58 Thanks: 0 
I see so I should just apply the GramSchmidt process but use the inner product I'm given right? Thank you for your answer Last edited by Vaki; May 6th, 2018 at 03:02 AM. 

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basis, be1, orthonormal, product, respect, space 
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