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Math Help - Help on Inner product spaces

  1. #1
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    Question Help on Inner product spaces

    I am reading the text book and this was shown and it says left to the reader ..... it would be more then great if some one can show me why this is true

    The statement is exactly as follows

    Help on Inner product spaces-lin-alg-2.jpg
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  2. #2
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    Re: Help on Inner product spaces

    i will assume you mean a REAL inner product space (it is not true for a COMPLEX inner product space, there's a different formula).

    the norm induced by an inner product <_,_> is given by:

    ||w|| = √(<w,w>)

    if a basis {v1,v2} is orthonormal with respect to a given inner product, then:

    <v1,v1> = <v2,v2> = 1

    <v1,v2> = <v2,v1> = 0.

    hence, if w = av1+bv2:

    ||w|| = √(<w,w>) = √(< av1+bv2, av1+bv2>)

    = √[<av1,av1+bv1> + <bv2,av1+bv2>]

    = √[<av1,av1> + <av1,bv2> + <bv2,av1> + <bv2,bv2>]

    = √[a<v1,av1> + a<v1,bv2> + b<v2,av1> + b<v2,bv2>]

    = √[a2<v1,v1> + ab<v1,v2> + ab<v2,v1> + b2<v2,v2>]

    = √[a2(1) + ab(0) + ab(0) + b2(1)]

    = √(a2 + b2).
    Thanks from ohYeah
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  3. #3
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    Re: Help on Inner product spaces

    If saying thank you a 100 times would show you how much i appreciate this I would do it.
    really thank you very much this is excellent
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