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Math Help - determine weather the given formula defines an inner product

  1. #1
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    Question determine weather the given formula defines an inner product

    Hi this was an exercise left for the reader, so i did part (a) and (c) but i really wanted to know (b) and (d) and i really tried but can't seem to figure it out

    the exercise is as follows
    determine weather the given formula defines an inner product-bryan-1.jpg

    here are the axioms i believe its referring to
    determine weather the given formula defines an inner product-bry.jpg

    and here is an example
    determine weather the given formula defines an inner product-bry2.jpg

    I am really curious to see the solution, since i spent so much time on it.
    Thank you in advance
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  2. #2
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    Re: determine weather the given formula defines an inner product

    Quote Originally Posted by ohYeah View Post
    I am really curious to see the solution, since i spent so much time on it.
    Thank you in advance
    Why don't you tell us how far you've gotten? Then we can focus on where you need the help.

    -Dan
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  3. #3
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    Re: determine weather the given formula defines an inner product

    ok this is what I did for (a) and (c)
    determine weather the given formula defines an inner product-photo-10-.jpg

    and this is what i was thinking for (b)
    just take f ∈ C[0,1] to be:

    f(x) = 0, x∈ [0,1/2]

    f(x) = x − 1/2, x∈ [1/2,1]

    Then <f,f> = 0, but f ≠ 0.

    <f,f> ≥ 0 is not enough for an inner product; it must satisfy the stronger condition:

    <f,f> = 0 ⇔ f = 0

    but i believe the approach i am taking for ( b ) is not correct and i don't know what to do for (d)
    Attached Thumbnails Attached Thumbnails determine weather the given formula defines an inner product-photo-10-.jpg  
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  4. #4
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    Re: determine weather the given formula defines an inner product

    is anyone able to do this
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  5. #5
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    Re: determine weather the given formula defines an inner product

    what you did for (b) is just fine. it's not positive-definite (although it is positive), so its not an inner product.

    let's just try to verify the axioms for (d):

    <u,v> = (Au).(Av) = (Av).(Au) (since the standard dot product is symmetric)

    = <v,u> well, that was easy.

    <u,v+w> = (Au).(A(v+w)) = (Au).(Av+Aw) (since matrix multiplication is linear)

    = (Au).(Av) + (Au).(Aw) (since the standard dot product is linear in the second variable)

    = <u,v> + <u,w>.

    i'm pretty sure you can handle the 3rd axiom yourself.

    so it all boils down to the 4th axiom:

    <u,u> > 0 when u ≠ 0 (it's pretty clear that <0,0> = (A0).(A0) = 0.0 = 0).

    suppose u ≠ 0.

    we need to know that Au ≠ 0 that is, that: null(A) = {0}. recall that n = dim(null(A)) + rank(A). under what conditions on rank(A) can we be sure that dim(null(A)) = 0?
    Thanks from ohYeah
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  6. #6
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    Re: determine weather the given formula defines an inner product

    so the rank(A) would have to be zero
    so A would have to consist of zero vectors
    so it does pass axiom 4
    so its an inner product
    is my logic near correct
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  7. #7
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    Re: determine weather the given formula defines an inner product

    no. it's very exactly totally wrong.

    let me ask you, does 0 + 0 = n (if n > 0)?
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  8. #8
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    Re: determine weather the given formula defines an inner product

    oh wow yeah that makes absolutely no sense
    so then the rank(A) = n
    and since n = dim(null(A)) + rank(A) so if rank(A) = n then dim(null(A)) would have to be 0
    Last edited by ohYeah; December 4th 2012 at 11:23 PM.
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  9. #9
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    Re: determine weather the given formula defines an inner product

    yes, that's the right way to deduce it. sometimes this is phrased as: a matrix of full rank gives a 1-1 function.
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