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Math Help - Gram-Schmidt Proof

  1. #1
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    Gram-Schmidt Proof

    I'm stuck on this one part for the proof of the Gram-Schmidt which states if (v_1,\dots,v_m_) is a linearly independent list (my text uses the term list, but I believe the term set is used more often) of vectors in V, then there exist an orthonormal list (e_1,\dots,e_m) of vectors in V such that

    span(v_1,\dots,v_j)=span(e_1,\dots,e_j)

    for j=1,...,m

    The proofs follows:

    Suppose (v_1,\dots,v_j) is a linearly independent list of vectors in V. To construct the e's, start by setting e_1=\frac{v_1}{||v_1||}. This then satisfies the above equation.

    So far in the proof I understand how it works for e_1.

    We will choose e_2,\dots,e_m inductively, as follows. Suppose j>1 and an orthonormal list (e_1,\dots,e_{j-1}) has been chosen so that

    span(v_1,\dots,v_{j-1})=span(e_1,\dots,e_{j-1}).

    Here is where I am confused. I don't understand how an orthonormal list can be chosen such that the above is true. Isn't that similar to what we're trying to prove? And why can't the index j-1 just be j?

    I did not post the complete proof; if this information is not sufficient enough I can post the rest.

    Thank you.
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  2. #2
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    Quote Originally Posted by Anthonny View Post
    I'm stuck on this one part for the proof of the Gram-Schmidt which states if (v_1,\dots,v_m_) is a linearly independent list (my text uses the term list, but I believe the term set is used more often) of vectors in V, then there exist an orthonormal list (e_1,\dots,e_m) of vectors in V such that

    span(v_1,\dots,v_j)=span(e_1,\dots,e_j)
    for j=1,...,m

    The proofs follows:

    Suppose (v_1,\dots,v_j) is a linearly independent list of vectors in V. To construct the e's, start by setting e_1=\frac{v_1}{||v_1||}. This then satisfies the above equation.

    So far in the proof I understand how it works for e_1.

    We will choose e_2,\dots,e_m inductively, as follows. Suppose j>1 and an orthonormal list (e_1,\dots,e_{j-1}) has been chosen so that

    span(v_1,\dots,v_{j-1})=span(e_1,\dots,e_{j-1}).

    Here is where I am confused. I don't understand how an orthonormal list can be chosen such that the above is true. Isn't that similar to what we're trying to prove? And why can't the index j-1 just be j?

    I did not post the complete proof; if this information is not sufficient enough I can post the rest.

    Thank you.
    Well, how's e_j constructed? First, you define w_j:=\frac{v_j}{||v_j||} , and then you take

    e_j:=\sum\limits^j_{k=1}\langle w_j,e_k\rangle e_k . The inductive problem is to show that e_j\neq 0 and that it is

    lin. indep. from \{e_1,\ldots ,e_{j-1}\}...and this is your job.

    Tonio
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