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Math Help - Check my answer for least square solution of Ax=b

  1. #1
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    Question Check my answer for least square solution of Ax=b

    I was just trying to solve a couple of questions from the past exam tests and I've got the following answer for this question. Can someone check if my answer is right. I am not so sure. Thanks!
    Let,
    A = [ 1 1 0
    1 1 0
    1 0 1
    1 0 1]

    and b = [1,3,8,2]^T. Find the least squares solutions of the system Ax=b.

    ANSWER:
    The general least squares solution to this sort of problem is:

    x = ((A^T)A)^(-1)*((A^T)*b)

    In this case this will not yield a solution, since you have two pairs of identical rows, the A matrix represents only 2 independent realtions. As such, ((A^T)A) is singular so no solution exists.
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  2. #2
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    Quote Originally Posted by Skelly View Post
    I was just trying to solve a couple of questions from the past exam tests and I've got the following answer for this question. Can someone check if my answer is right. I am not so sure. Thanks!
    Let,
    A = [ 1 1 0
    1 1 0
    1 0 1
    1 0 1]

    and b = [1,3,8,2]^T. Find the least squares solutions of the system Ax=b.

    ANSWER:
    The general least squares solution to this sort of problem is:

    x = ((A^T)A)^(-1)*((A^T)*b)

    In this case this will not yield a solution, since you have two pairs of identical rows, the A matrix represents only 2 independent realtions. As such, ((A^T)A) is singular so no solution exists.
    The question says find the least squares solutions (plural). Because A^T A is singular, there are an infinite number of solutions. They are the solutions to the normal equations A^T A x = A^T b. When A^T A is nonsingular, there is a unique solution to those equations, which you give. Here you must form the normal equations and find the general form of the solutions, which is x1 + x2 = 2 and x1 + x3 = 5.
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