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May 4th, 2018, 10:58 AM   #1
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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?
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May 4th, 2018, 04:05 PM   #2
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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 Gram-Schmidt orthogonalization process. If you don't know it, here's Wikipedia's article: https://en.wikipedia.org/wiki/Gram%E...chmidt_process
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Last edited by johng40; May 4th, 2018 at 04:28 PM.
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May 6th, 2018, 01:06 AM   #3
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I see so I should just apply the Gram-Schmidt process but use the inner product I'm given right?|
Thank you for your answer

Last edited by Vaki; May 6th, 2018 at 02:02 AM.
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