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Math Help - Inner products

  1. #1
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    Inner products

    We define: <u,v>=u2v2+u3v3
    For vectors u=(u1,u2,u3) and v=(v1,v2,v3) in R3.
    Explain the reasons why this is not an inner product on R3.

    I have completed the 4 axioms as below:
    1. <u,v>= u2v3 + u3v3
    =v2u2 + v3u3
    =<v,u>

    2.<cu,v> = cu1v2+cu2v2
    = c(u2v2+u3v3)
    = c<u,v>
    3.<u,v+w>=u2(v2+w2)+u3(v3,w3)
    = u2v2+u2w2+u3v3+u3w3
    = <u,v>+<u,w>
    4.a) <u,u>=u22 + u32
    greater than or equal to zero as u2^2 greater than or equal to zero and u3^2 is greater than or equal to zero
    b) <u,u>=0 then u2=0 and u3=0.

    Somehow ive wrongly proved all the axioms :S im assuming this does not define an inner product as it does not include u1 and v1 in the inner product, therefore cannot be an inner product in R^3. I would greatly appreciate anyone looking over my work to help me! Thanks
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  2. #2
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    If u=(1,0,0) then what is <u,u>?
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  3. #3
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    Even if u=(1,0,0) <u,u> does not depend on u1 as stated in the inner product equation. <u,u>=u2^2 + u3^3 which is equal to zero in this case. I understand now that 4b) is not satisfied as u does not equal the zero vector, but doesnt axiom 4a still hold?
    (thanks for your reply )
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