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Math Help - find a vector in an orthnormal basis

  1. #1
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    find a vector in an orthnormal basis

    suppose I have 3 vectors, a b and c that form an orthonormal basis. How would I find a vector v = (1,2,3) in the a b c system? I don't really understand what its asking. Do I have to find a linear combination from a b c that forms v?
    so basically for some scalars x y and z, I have to find

    xa + by + cz = v?

    if I'm right, how would I go about doing that?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: find a vector in an orthnormal basis

    Quote Originally Posted by Kuma View Post
    suppose I have 3 vectors, a b and c that form an orthonormal basis. How would I find a vector v = (1,2,3) in the a b c system? I don't really understand what its asking. Do I have to find a linear combination from a b c that forms v?
    so basically for some scalars x y and z, I have to find

    xa + by + cz = v?

    if I'm right, how would I go about doing that?
    That's the beauty of the orthonormal bases. We know that v=xa+by+cz for some x,y,z\in\mathbb{C} but then \langle v,a\rangle=x\langle a,a\rangle+y\langle b,a\rangle+z\langle c,a\rangle=x\cdot1 +y\cdot0+z\cdot 0 and so \langle v,a\rangle=x. Doing the same analysis we arrive at the fact that v=\langle v,a\rangle a+\langle v,b\rangle b+\langle v,c\rangle c.
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    Re: find a vector in an orthnormal basis

    wait I'm a little lost. How did x(a,a) turn into x? (a,a) is just ((a1,a2,a3),(a1,a2,a3)) in R3. So its just a set of vectors. How did it turn into 1? Unless I'm misunderstanding. What do the pointy brackets mean?
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    MHF Contributor Drexel28's Avatar
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    Re: find a vector in an orthnormal basis

    Quote Originally Posted by Kuma View Post
    wait I'm a little lost. How did x(a,a) turn into x? (a,a) is just ((a1,a2,a3),(a1,a2,a3)) in R3. So its just a set of vectors. How did it turn into 1? Unless I'm misunderstanding. What do the pointy brackets mean?
    The angle brackets mean inner product, i.e. \langle a,a\rangle=a\cdot a=1. Make sense now?
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    Re: find a vector in an orthnormal basis

    ahh ok. Thanks. Although I think I got the answer through row reduction before your post. Basically I made a matix with the vectors abc and set them equal to v. I got a unique scalar solution for x y and z Those scalars multiplied by a b and c gives me v.
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    MHF Contributor Drexel28's Avatar
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    Re: find a vector in an orthnormal basis

    Quote Originally Posted by Kuma View Post
    ahh ok. Thanks. Although I think I got the answer through row reduction before your post. Basically I made a matix with the vectors abc and set them equal to v. I got a unique scalar solution for x y and z Those scalars multiplied by a b and c gives me v.
    That works too haha
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