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Math Help - Orthogonal Basis Using Gram-Schmidt

  1. #1
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    Orthogonal Basis Using Gram-Schmidt

    Consider the inner product space  \mathbb{C}^3 with the usual inner product. Starting with the basis

    { u_1, u_2, u_3} = { (1,1,1),(4i,-i,0),(2,0,-1)},

    find an orthogonal basis using the Gram-Schmidt process.
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  2. #2
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    Quote Originally Posted by funnyinga View Post
    Consider the inner product space  \mathbb{C}^3 with the usual inner product. Starting with the basis

    { u_1, u_2, u_3} = { (1,1,1),(4i,-i,0),(2,0,-1)},

    find an orthogonal basis using the Gram-Schmidt process.
    Hello funnyinga,

    Can you show us where you are getting stuck? If you are not understanding the whole process, then I suggest working out a solved example...
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  3. #3
    MHF Contributor arbolis's Avatar
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    I don't remember where I downloaded this example of the Gram-Schmidt so instead of posting the link, I post the PDF.
    Attached Files Attached Files
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  4. #4
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    Im really stuck here. The complex numbers are confusing me. Here is what I have done (although its probably all wrong).

    Let  v_1 = u_1 = (1,1,1), u_2 = (4i,-i,0), u_3 = (2,0,-1)

    Then  v_2 = u_2 - \frac{<u_2,v_1>}{||v_1||^2} v_1

    So  v_2 = (4i, -i, 0) - \frac{(4i, -i, 0) * (1,1,1)}{3} (1,1,1) \ = ??


    and  v_3 = (2,0,1) - \frac{(2,0,-1) * (v_2)}{3} (1,1,1) - \frac{(2,0,-1)*(v_2)}{||v_2||^2} v_2  \ = ???

    Ive tried to get the answer but havent had any results.
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  5. #5
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    [QUOTE=funnyinga;324406]Im really stuck here. The complex numbers are confusing me. Here is what I have done (although its probably all wrong).

    Let  v_1 = u_1 = (1,1,1), u_2 = (4i,-i,0), u_3 = (2,0,-1)

    Then  v_2 = u_2 - \frac{<u_2,v_1>}{||v_1||^2} v_1

    So  v_2 = (4i, -i, 0) - \frac{(4i, -i, 0) * (1,1,1)}{3} (1,1,1) \ = ??
    Just treat "i" as if it were a variable.
    That is (4i,-i,0)- \frac{4i- i}{3}(1,1,1)= (4i, -i, 0)- \frac{3i}{3}(1,1,1) = (4i, -i, 0)- (i, i, i)= (3i, 0, -i).


    and  v_3 = (2,0,1) - \frac{(2,0,-1) * (v_2)}{3} (1,1,1) - \frac{(2,0,-1)*(v_2)}{||v_2||^2} v_2  \ = ???

    Ive tried to get the answer but havent had any results.
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