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Math Help - Orthogonal set

  1. #1
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    Orthogonal set

    Can anyone help me with this problem please?
    Let {v_1,v_2,...v_k} be an orthogonal set in V, and let a_1,a_2,...,a_k be scalars. Prove that (the norm of the summation of a_i*v_i)^2 is equal to the summation of l a_i l^2* (norm of v_i)^2
    i runs from 1 to k.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by namelessguy View Post
    Can anyone help me with this problem please?
    Let {v_1,v_2,...v_k} be an orthogonal set in V, and let a_1,a_2,...,a_k be scalars. Prove that (the norm of the summation of a_i*v_i)^2 is equal to the summation of l a_i l^2* (norm of v_i)^2
    i runs from 1 to k.
    {\left| \left|{ \sum_{i=1}^k {a_i v_i}} \right| \right|}^2 = \left<{{ \sum_{i=1}^k {a_i v_i}},{ \sum_{i=1}^k {a_i v_i}}}\right> = \sum_{i=1}^k\left<{a_i v_i,a_i v_i}\right>

    = \sum_{i=1}^k {a_i}^2\left<{v_i,v_i}\right> = \sum_{i=1}^k {a_i}^2\| v_i \| ^2

    im just trying my luck here.. please, someone verify my proof.. thx..
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  3. #3
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    Quote Originally Posted by kalagota View Post
    im just trying my luck here.. please, someone verify my proof.. thx..
    It looks right.
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    Thanks a lot for your help, kalagota. Where can I learn to type mathematical symbols and does it take a lot of time to learn this?
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  5. #5
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    Quote Originally Posted by kalagota View Post
    {\left| \left|{ \sum_{i=1}^k {a_i v_i}} \right| \right|}^2 = \left<{{ \sum_{i=1}^k {a_i v_i}},{ \sum_{i=1}^k {a_i v_i}}}\right> = \sum_{i=1}^k\left<{a_i v_i,a_i v_i}\right>
    This should be \Bigl\|\sum_{i=1}^k a_i v_i \Bigr\|^2 = \Bigl\langle\sum_{i=1}^k a_i v_i, \sum_{j=1}^k a_j v_j\Bigr\rangle = \sum_{i,j=1}^k \langle a_i v_i,a_j v_j\rangle. The two sums are independent, so should have different dummy variables. However, \langle v_i,v_j\rangle = 0 when i≠j. So the only terms that survive are those for which j=i, and the rest of kalagota's solution is correct:

    Quote Originally Posted by kalagota View Post
    = \sum_{i=1}^k {a_i}^2\left<{v_i,v_i}\right> = \sum_{i=1}^k {a_i}^2\| v_i \| ^2
    The only other comment is that if the scalar field is the complex numbers then when a_i comes out of the right side of the inner product it becomes \bar{a}_i (the complex conjugate). But that's okay, because we then get a_i\bar{a}_i=|a_i|^2, as required.
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  6. #6
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Opalg View Post
    This should be \Bigl\|\sum_{i=1}^k a_i v_i \Bigr\|^2 = \Bigl\langle\sum_{i=1}^k a_i v_i, \sum_{j=1}^k a_j v_j\Bigr\rangle = \sum_{i,j=1}^k \langle a_i v_i,a_j v_j\rangle. The two sums are independent, so should have different dummy variables. However, \langle v_i,v_j\rangle = 0 when i≠j. So the only terms that survive are those for which j=i, and the rest of kalagota's solution is correct:
    ....

    oh yeah.. i was thinking about this last night (i min here, it's about 8:30pm here).. thanks for teh comments!!
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  7. #7
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by namelessguy View Post
    Thanks a lot for your help, kalagota. Where can I learn to type mathematical symbols and does it take a lot of time to learn this?
    hehe.. not really since you'll just need to learn the basics.. besides, almost every codes are similar..
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