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Math Help - Gram-Schmidt process vectors

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    Gram-Schmidt process vectors

    Prove that if  \{ w_{1}, w_{2}, ... , w_{n} \} is an orthogonal set of nonzero vectors, then the vectors  v_{1}, v_{2}, . . . , v_{n} derived from the Gram-Schmidt process satisfy  v_{i} = w_{i} \ \ \ \forall i

    my proof so far:

    I intend to use induction.

    Now, v_{1} = w_{1} is travial.

    Suppose that n = k is true, then I have  v_{k} = w{k-1} - \sum ^{k-2}_{j=1} \frac {<w_{k-1},v_{j}>}{ || v_{j} || ^2 } v_{j} = w_{k}

    Now, how would I use that information as well as the fact that the vecters are orthogonal to get k+1 is true?

    thanks
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Prove that if  \{ w_{1}, w_{2}, ... , w_{n} \} is an orthogonal set of nonzero vectors, then the vectors  v_{1}, v_{2}, . . . , v_{n} derived from the Gram-Schmidt process satisfy  v_{i} = w_{i} \ \ \ \forall i

    my proof so far:

    I intend to use induction.

    Now, v_{1} = w_{1} is travial.

    Suppose that n = k is true, then I have  v_{k} = w{k-1} - \sum ^{k-2}_{j=1} \frac {<w_{k-1},v_{j}>}{ || v_{j} || ^2 } v_{j} = w_{k}

    Now, how would I use that information as well as the fact that the vecters are orthogonal to get k+1 is true?
    This is easy if you use strong induction (in other words, assume that the inductive hypothesis holds for all n≤k, not just for n=k). The formula for v_{k+1} is v_{k+1} = w_{k+1} - \sum ^{k}_{j=1} \frac {<w_{k+1},v_{j}>}{ || v_{j} || ^2 } v_{j} . But if v_j = w_j for 1≤j≤k then each term in that sum will vanish, because the w's are orthogonal to each other.
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