Basic Solutions and Linear Combinations

[lapo3399]

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:

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

and the basic solutions are:

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

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.

 

[walterman]

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

 

[tacman]

The basic solutions are the specific values for the parameters (t and s) that satisfy the system of equations. In this case, the basic solutions are the vectors [1, 2, 1, 0] and [0.5, -1, 0, 1] when t and s are set to any real number. These solutions form the basis for all other solutions to the system of equations.

To further clarify, the basic solutions represent the direction vectors (or the coefficients of the variables) that, when multiplied by the parameters, give the specific solutions to the system of equations. So in this case, [1, 2, 1, 0] represents the direction vector for t and [0.5, -1, 0, 1] represents the direction vector for s.

In summary, the basic solutions are the specific values for the parameters that make up the linear combination of the direction vectors, which in turn give the solutions to the system of equations.