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Basic Solutions and Linear Combinations

  1. Oct 10, 2007 #1
    Just to clarify these concepts: if a homogeneous system of linear equations with four variables z1, z2, z3, and z4 yields a matrix in reduced row echelon form that defines (as an arbitrary example) the linear equations

    z1 = z3 + 0.5z4 = t + 0.5s
    z2 = 2z3 - z4 = 2t - s
    z3 = t
    z4 = s

    then the linear combination should be:

    [tex]
    \left[ \begin{array}{ c } z1 & z2 & z3 & z4 \end{array} \right] = t \left[ \begin{array}{ c } 1 & 2 & 1 & 0 \end{array} \right] + s \left[ \begin{array}{ c } 0.5 & -1 & 0 & 1 \end{array} \right]
    [/tex]

    and the basic solutions are:

    [tex]
    \left[ \begin{array}{ c } 1 & 2 & 1 & 0 \end{array} \right] , \left[ \begin{array}{ c } 0.5 & -1 & 0 & 1 \end{array} \right]
    [/tex]

    My main problem is understanding exactly what the basic solutions are. I'm not sure whether they're just the direction vectors multiplied by each of the parameters or whether they include the parameters as well. Please clarify this for me.

    Thanks.
     
  2. jcsd
  3. Oct 25, 2007 #2
    pls i need some one to help me correct my read subroutine in fortran 77.i must have made some errors and i need to be corrected.Thanks
    walterman

    c**********************************************************
    subroutine read_data(a,b,n,nsys)
    c**********************************************************
    implicit double precision (a-h,o-z)
    dimension a(n,n),b(n)
    charactername*20, form*50
    c
    write(*,*)
    write(*,*) 'Enter configuration file name: '
    read (*,'(a)')
    open(unit=10,file='fname',status='New',iostat=non)
    100 continue
    read(unit=*,fmt=*) nsys
    if(nsys.eq.5) then
    form='(5f10.5)'
    else
    form='(6f10.5)'
    end if
    read(unit=*,fmt=*,err=999) ((a(i,j), j=1,nsys), i=1,nsys)
    read(unit=*,fmt=*,err=999) (b(i), i=1,nsys)
    999 write(*,*)
    write(*,'(a,i3,a,f16.8)') ' System size : ',nsys,' x',nsys
    write(*,*)
    write(*,*) 'xmulticient matrix : '
    write(20,fmt=form) ((a(i,j), j=1,nsys), i=1,nsys)
    write(*,fmt=form) ((a(i,j), j=1,nsys), i=1,nsys)
    write(*,*)
    write(*,*) 'constant vector: '
    write(21,fmt=form) (b(i), i=1,nsys)
    write(*,fmt=form) (b(i), i=1,nsys)
    write(*,*)

    return
    end
     
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