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Cramers rule on a set of linear equations, notation help

  1. Jan 17, 2013 #1
    1. The problem statement, all variables and given/known data

    I don't have a particular problem to solve, but I need help with understanding the notation I have found in an academic paper.

    I have a system of linear equations which each take the form:

    pi xi r1 + pi yi r2 - pi u + qi xi r4 + qi yi r5 - qi v + xi x + yi y - w/2 + mi = 0 (for i=1-6)


    Where mi=((Li)^2- (xi)^2- (yi)^2- (pi)^2- (qi)^2 )/2

    u=r1x+r2y+r3z
    v=r4x+r5y+r6z
    w= x^2+ y^2+ z^2

    Where: pi, qi, xi, yi, Li and mi are all known quantities.

    The paper outlines a method to solve for r1, r2, r4, r5, v, u, w, x, y, z


    2. Relevant equations
    The paper enters the equations into a 6 x10 matrix and called this matrix 'M'

    The following equation then holds true:

    M * t = 0

    Where t is a 10 x 1 matrix

    t = [r1; r2; u; r4; r5; v; x; y; z; 1]

    (so all variables in the 'M' matrix are known and we are solving for the variables in the 't' matrix)

    The paper then goes on to say that we can solve the system of equations symbolically using Cramer algorithm, regarding r1,r2,r4,r5,u,v as linear unknowns, and obtain the following expressions of those variables with respect to x,y,z.

    a0r1 + a11x + a12y + a13w + a14 = 0
    a0r2 + a21x + a22y + a23w + a24 = 0
    a0u + a31x + a32y + a33w + a34 = 0
    a0r4 + a41x + a42y + a43w + a44 = 0
    a0r5 + a51x + a52y + a53w + a54 = 0
    a0v + a61x + a62y + a63w + a64 = 0

    Where

    a0 = det(c1,c2,c3,c4,c5,c6)

    And

    aij = det(c1,...,ci-1,cj+6,ci+1,...,c6)


    Now this is where I am getting stuck.

    I am unsure of exactly what the last line of notation above means.


    3. The attempt at a solution

    I understand that a0 is equal to the determinant of the matrix 'M' when it is reduced to M(6x6).

    But what exactly does the next line mean? I dont know how to calculate the coefficients aij.

    If was looking for the coefficient a11 for instance would this constitute the following matrix:

    aij = det(c1,...,ci-1,cj+6,ci+1,...,c6)
    a11 = det(c1,..., c0,c7,c2,...,c6) ??????

    There is no column C0 is there? And I'm unsure of what columns of the matrix go into the two gaps.

    Could somebody please help, I'm sure this must be relatively easy for somebody who works with this notation all the time, but I dont understand.

    Thank you

    MissMk1
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 17, 2013 #2

    haruspex

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    I can answer that part. In this case, it would simply mean that the lead-in sequence is empty:
    a11 = det(c7,c2,...,c6)
     
  4. Jan 18, 2013 #3
    Thank you!

    I think I understand it now

    Thanks for your reply :)
     
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