213 - A4.5 - Undetermined Systems

239 days ago by Professor213

John Travis

Mississippi College

These are some transportation problems from Lay's Linear Algebra text.

var('o1,o2,o3,o4,i1,i2,i3,i4,m') assume(o1+o2+o3+o4==i1+i2+i3+i4) a=matrix([[1, 1, 0, 0, 0, 0, o1+o2], [0, -1, 1, 0, 0, 0, m], [0, 0, -1, 1, 0, 0, o3-i1], [0, 0, 0, -1, 1, 0, o4-i2], [0, 0, 0, 0, -1, 1, -1*m], [-1, 0, 0, 0, 0, -1, -1*i3-i4]]); print a print show(a.echelon_form()) 
       
[       1        1        0        0        0        0  o1 + o2]
[       0       -1        1        0        0        0        m]
[       0        0       -1        1        0        0 -i1 + o3]
[       0        0        0       -1        1        0 -i2 + o4]
[       0        0        0        0       -1        1       -m]
[      -1        0        0        0        0       -1 -i3 - i4]

[       1        1        0        0        0        0  o1 + o2]
[       0       -1        1        0        0        0        m]
[       0        0       -1        1        0        0 -i1 + o3]
[       0        0        0       -1        1        0 -i2 + o4]
[       0        0        0        0       -1        1       -m]
[      -1        0        0        0        0       -1 -i3 - i4]

var('o1,o2,o3,o4,i1,i2,i3,i4,m') a=matrix([[1, 1, 0, 0, 0, 0, 850], [0, -1, 1, 0, 0, 0, 400], [0, 0, -1, 1, 0, 0, 850], [0, 0, 0, -1, 1, 0, 1150], [0, 0, 0, 0, -1, 1, -400], [-1, 0, 0, 0, 0, -1, -2850]]); print a print show(a.echelon_form()) 
       
[    1     1     0     0     0     0   850]
[    0    -1     1     0     0     0   400]
[    0     0    -1     1     0     0   850]
[    0     0     0    -1     1     0  1150]
[    0     0     0     0    -1     1  -400]
[   -1     0     0     0     0    -1 -2850]

[    1     1     0     0     0     0   850]
[    0    -1     1     0     0     0   400]
[    0     0    -1     1     0     0   850]
[    0     0     0    -1     1     0  1150]
[    0     0     0     0    -1     1  -400]
[   -1     0     0     0     0    -1 -2850]

var('a,b,c') a=5 b=6 c=3 A=matrix([[2,2,a], [-2,1,b], [1,-1,c]]); print A print show(A.echelon_form()) 
       
[ 2  2  5]
[-2  1  6]
[ 1 -1  3]

[ 2  2  5]
[-2  1  6]
[ 1 -1  3]

m=matrix(QQ,2,3,[1,2,3,4,5,6]); m; m.parent() 
       
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field
m=matrix(QQ,2,2,1); m; m.parent() 
       
[1 0]
[0 1]
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
[1 0]
[0 1]
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
# vandermonde matrix x = [2,4,5,-1] n = len(x) m = matrix(QQ, n, n, lambda i, j: x[i]^j) show(m) 
       

                                
                            

                                
len(x) 
       
4
4