Lösungen

Remove["Global`*"]

Aufgabe 1a

g1= (1 x + 1 y + 1 z == 1 );
g2= (0 x + 1 y + 1 z == 1 );
g3= (0 x + 0 y + 1 z == 1 );
Solve[{g1,g2,g3},{x,y,z}]

Aufgabe 1b

g1= (1 x + 1 y + 1 z == 1 );
g2= (1 x + 2 y + 3 z == 1 );
g3= (1 x + 4 y + 9 z == 1 );
Solve[{g1,g2,g3},{x,y,z}]

Aufgabe 2

g1 =  (λ x  -  5 y  +  6 z  +  7 w  ==  
            8 );
g2 =  (4 x  -  8 y  +  9 z  +  1 w  ==  
            1 );
g3 =  (1 x  +  1 y  -  1  z  +  1 w  ==  
            0 );

a

Solve[{g1, g2, g3}, {x, y, z}] /. λ -> 3    //  
      Flatten

w  ist Parameter!

b

Solve[{g1, g2, g3}, {x, y, z}]  //  Flatten

Solve[{g1, g2, g3}, {x, y, z}]  /. λ -> 7  //  
      Flatten

λ = 7  !

c

Clear[λ]; λ

Ord=4, Dim = 2 -> Rang = 2     ->  λ so wählen, dass g1= μ1 g2 +μ2 g3 :

Solve[{λ, -5, 6, 7, 
          8} ==  μ1 {4, -8, 9, 1, 1} +  μ2 {1, 1, -1, 1, 
              0}, {λ, μ1, μ2}]

Keine Lösung

Aufgabe 3

g11 = {1, 1, 1, 1, 1};
g21 = {1, 2, 3, 4, 2};
g31 = {1, 4, 9, 16, 3};
g41 = {1, 8, 27, 64, 4};
{g11, g21, g31, g41} //  MatrixForm

{g12 = g11, g22 = g21 - g11, g32 = g31 - g11, g42 = g41 - g11} //  
      MatrixForm

{g13 = g12 - g22, g23 = g22, g33 = g32 - 3 g22, 
        g43 = g42 - 7 g22} //  MatrixForm

{g14 = g13 +  1/2  g33, g24 = g23 - g33, g34 = 1/2 g33, 
        g44 = g43 - 12  g34} //  MatrixForm

g45 = 1/6 g44;
{g15 = g14  -  g45, g25 = g24 + 3 g45, g35 =  g34 - 3  g45, 
        g45} //  MatrixForm

Aufgabe 4

Clear[x, y, z, w, g1, g2, g3, g4]

g1 = (1 x  +  1 y  +  1 z  +  1 w  ==  
            1);
g2 = (1 x  +  2 y  +  3 z  +  4 w  ==  
            2);
g3 = (2  x  +  3 y  +  4 z  +  5 w  ==  
            3);
g4 = (0  x  +  1 y  +  2 z  +  3 w  ==  
            1);

Solve[{g1, g2, g3, g4}, {x, y, z, w}] // Flatten

Ord=4, Dim=2, Rang r =Rang r0 =2

Aufgabe 5

g1 = (1 x  +  1 y  +  1 z  +  1 w  ==  
            1);
g2 = (1 x  +  2 y  +  3 z  +  4 w  ==  
            2);
g3 = (2  x  +  3 y  +  4 z  +  5 w  == 
            1);
g4 = (0  x  +  1 y  +  2 z  +  3 w  ==  
            3);
Solve[{g1, g2, g3, g4}, {x, y, z, w}] // Flatten

Ord=4, Rang r>2, Rang  r0 =2, keine Lösung