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    Linear Algrebra - Vectors in R^n, Orthogonal Spaces and Lines of Best Fit

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    1) Let u and v be vectors in R^n.
    a) Prove that llull = llvll if and only if u + v and u - v are orthogonal.

    b)Let (proj of u onto v) be the vector projection of u onto v. For u, v does not equal to 0, prove that (projection of u onto v - u) is orthogonal.

    2) Find a basis for the space orthogonal to [1,1,0]^T in R^3.

    3) The following data shows U.S. Health Expenditures (in billions of dollars):
    x=1985, 1990, 1991, 1992, 1993, 1994
    y=376.4, 614.7, 676.2, 739.8, 786.5, 831.7

    a) Determine the line of best fit to the given data.
    b) Predict the amount of Health expenditures in the year 2004.

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    Solution Preview

    (a) If ||u||=||v||, then (u,u)=(v,v). Thus (u+v,u-v)=(u,u)+(v,u)-(u,v)-(v,v)=0 since (v,u)=(u,v). Thus u+v and u-v is orthorgonal.
    If u+v and u-v are orthogonal, then (u+v,u-v)=0 and we can induce (u,u)=(v,v). Thus ||u||=||v||.
    (b) Suppose {x_1,...,x_m} ...

    Solution Summary

    Vectors in R^n, Orthogonal Spaces and Lines of Best Fit are investigated.