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Math Help - Probably an easy question: finding a transformation R^3 -> R^3

  1. #1
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    Probably an easy question: finding a transformation R^3 -> R^3

    Hello,

    My overall goal is to find a linear transformation which takes R^3 -> R^3, taking the inner product space with squared norm x^2 + 2xy + 4y^2 + 8z^2 into the std. inner product.

    Now, I know how to use the gram-schmidt method to orthonormalize a basis, but I am just completely blanking on how to find a basis to orthonormalize! Any ideas would be great. I don't need the problem solved, just the right nudge

    Thanks!
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  2. #2
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    I've done some more work on the problem and have reduced it further. I believe this is a notational problem on my end, as I have two books for the class and the notation is inconsistent.

    I am given that the "squared norm" is as given above. How do I extract from that the inner product?

    I assume for a vector v = (x, y, z), ||v||^2 = x^2 + 2xy + 4y^2 + 8z^2 = <v, v>, but how would I define this for different vectors <v, u>?

    I hope I'm being clear. Thanks for the help!
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  3. #3
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    Quote Originally Posted by arcketer View Post
    I assume for a vector v = (x, y, z), ||v||^2 = x^2 + 2xy + 4y^2 + 8z^2 = <v, v>, but how would I define this for different vectors <v, u>?
    Write the expression for the norm as \|v\|^2 = (x+y)^2+3y^2+8z^2. Then the inner product will be given by \langle v,u\rangle = (x_1+y_1)(x_2+y_2) + 3y_1y_2 + 8z_1z_2, where v=(x_1,y_1,z_1) and u=(x_2,y_2,z_2).
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  4. #4
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    Oh wow, thank you very much. I'm embarassed for not having noticed that haha.

    Question answered!
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