Finding an orthonormal basis


by DWill
Tags: basis, orthonormal
DWill
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#1
Nov13-08, 09:41 PM
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1. The problem statement, all variables and given/known data
Find an orthonormal basis for the subspace of R^3 consisting of all vectors (a, b, c) such that a + b + c = 0.


2. Relevant equations



3. The attempt at a solution
I know how to find an orthonormal basis just for R^3 by taking the standard basis vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1) and applying the Gram-Schmidt process to make them orthonormal. The condition that a + b + c must equal 0 is throwing me off, however. Can anyone give me any suggestions for how to approach problems like these? thanks!
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Dick
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#2
Nov13-08, 10:02 PM
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If you know Gram-Schmidt, that's the hard part. What's so hard about about finding two vectors (a,b,c) such that a+b+c=0 to do Gram-Schmidt with? (1,-1,0) sounds like a good start. Give me another one.
DWill
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#3
Nov13-08, 10:07 PM
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I think I need 3 vectors, right? such that a + b + c = 0. Can I start with any vector? Or is there a reason why (1, -1, 0) is a good choice? If it can be any 3 vectors that add to 0 it should be simpler, but then I can't be sure they are a basis?

Dick
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#4
Nov13-08, 10:10 PM
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Finding an orthonormal basis


Why do you need 3 vectors? That's a subspace of R^3. It looks to me like it's two dimensional.
DWill
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#5
Nov13-08, 10:16 PM
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Ohh.. sorry I was being confused, I thought a, b, and c referred to vectors and not the components.

So I can choose (1, -1, 0) like you said and also (1, 0, -1)? I would still have to check if the 2 vectors I choose are actually a basis (span R^3 and are linearly independent) right?
Dick
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#6
Nov13-08, 10:18 PM
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Sure, exactly. That's a good choice. There aren't too many dumb choices. (1,-1,0) and (-1,1,0) would have been a dumb choice because they're linearly dependent. But you didn't make the dumb choice. Now do Gram-Schmidt.
Mark44
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#7
Nov13-08, 11:57 PM
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I know how to find an orthonormal basis just for R^3 by taking the standard basis vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1) and applying the Gram-Schmidt process to make them orthonormal.
If you apply Gram-Schmidt to these vectors, you'll just get the same ones. They are already orthogonal and have length 1.
Mark44
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#8
Nov14-08, 12:13 AM
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Quote Quote by DWill View Post
Ohh.. sorry I was being confused, I thought a, b, and c referred to vectors and not the components.

So I can choose (1, -1, 0) like you said and also (1, 0, -1)? I would still have to check if the 2 vectors I choose are actually a basis (span R^3 and are linearly independent) right?
I'm not sure it's clear to you where Dick got the vectors he did. Start with your equation:
a + b + c = 0

If you solve for a, you get
a = -b - c, where b and c are arbitrary
b =  b
c =      c
That last two equations are obviously true.
To make things even more explicit,
a = -1b - 1c
b =  1b + 0c
c =  0b  +1 c
or,
[tex]
\left[
\begin{array}{ c }
a \\
b \\
c
\end{array} \right] = b\left[
\begin{array}{ c }
-1 \\
1 \\
0
\end{array} \right] + c\left[
\begin{array}{ c }
-1 \\
0 \\
1
\end{array} \right][/tex]
Voila, there's your basis.
HallsofIvy
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#9
Nov14-08, 04:17 AM
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Quote Quote by DWill View Post
Ohh.. sorry I was being confused, I thought a, b, and c referred to vectors and not the components.

So I can choose (1, -1, 0) like you said and also (1, 0, -1)? I would still have to check if the 2 vectors I choose are actually a basis (span R^3 and are linearly independent) right?
You need to review basic definitions. Those two vectors CAN'T be a basis for R^3 and can't span R^3 because that requires THREE vectors. You must have misread the original problem. As Dick said, the set of all (a, b, c) such that a+ b+ c= 0 is a two dimensional subspace of R^3. NO set of such vectors can be a basis for R^3. (1, -1, 0) and (1, 0, -1) form a basis for that two dimensional subspace.
jinksys
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#10
Jul27-10, 12:09 AM
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I'm currently doing the same problem so I figured I'd revive this old thread.

Once I have my two vectors,
u1=<-1,1,0> and u2=<-1,0,1>

I then go through the Gram-Schmidt process to find the normalized basis?
Mark44
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#11
Jul27-10, 01:08 AM
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If you already have a basis, you don't need Gram-Schmidt to find a normalized basis.

Do you know what the term "normalized" means?
jinksys
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#12
Jul27-10, 01:15 AM
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Quote Quote by Mark44 View Post
If you already have a basis, you don't need Gram-Schmidt to find a normalized basis.

Do you know what the term "normalized" means?
Sorry, I meant a orthonormal basis.
Mark44
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#13
Jul27-10, 02:15 PM
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OK, in that case you need to use G-S.


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