Lösungen/ Solutions

Aufgabe 1 /Problème 1

Remove["Global`*"]

g1 = (a1 == 3 e1  -  e2 +  e3);
g2 = (a2 == - e1  +  e2 -  
              e3);
g3 = (a3 ==  e1  + 2 e2 + 4 e3);

Solve[{g1, g2, g3}, {e1, e2, e3}] //  ExpandAll

Solve[{g1, g2, g3}, {e1, e2, e3}] //  N   //  
      ExpandAll

Aufgabe 2 /Problème 2

Remove["Global`*"]

a = {0, 0}; b = {7, 0}; c = {4, 5};
OA = a; OB = b; OC = c; BC = OC - OB; AC = OC - OA; OA1 = OB + 1/3 BC; 
    OB1 = 3/4 AC;

aA1[t1_] := OA + t1 (OA1 - OA);
bB1[t2_] := OB + t2 (OB1 - OB);

s1 = Solve[aA1[t1] == bB1[t2], {t1, t2}] // Flatten

s2 = {aA1[t1], bB1[t2]}/.s1

sS = s2[[1]]

cC1[t3_] := OC + t3 (sS - OC);  cC1[t3]

s3 = 
      Solve[ cC1[t3][[2]] == 0, {t3}]  // Flatten

OC1 = cC1[t3] /. s3

7:6 -> 6/1; 1:(1/3) -> 1/2;  1:(3/4) -> 1/3;  ===>

(6/1) * (1/2) * (1/
          3)

Aufgabe 3 /Problème 3

Remove["Global`*"]

z1[a_, b_] = Re[a] + Re[b] I;  z2[a_, b_] = Conjugate[z1[a, b]];  
    z3[a_, b_] = z1[a, b] + z2[a, b];
r = Sqrt[z1[a, b] z2[a, b]] // ExpandAll

g1 = (r == 1) /. {Re[a] -> x, 
          Re[b] -> y}

h = (z1[a, b] - z2[a, b])/I

s = 
      Sqrt[(z1[a, b] - z3[a, b])*Conjugate[(z1[a, b] - z3[a, b])]] // 
        ExpandAll

g2 = (h == s) /. {Re[a] -> x, 
          Re[b] -> y}

Solve[{g1, g2}, {x, y}]

N[%]

Aufgabe 4 /Problème 4

a

Remove["Global`*"]

w = 3 + 4 I;  u[a_, b_] := Re[a] + Re[b] I;

q[z_] := Re[z]^2 + Im[z]^2

q[w + u[a, b]]

q[w + u[a, b]] + q[w - u[a, b]] == 2 (q[w] + q[u[a, b]]) // 
      ExpandAll

===> Alle u sind Lösungen / Tous les u sont des solutions

b

|w+u|, |w-u| Diagonalen eines Rechtecks / diagonales d'un rectangle
|w|, |u| Seiten des Rechtecks / arrêts du rectangle

Aufgabe 5 /Problème 5

a,b

Remove["Global`*"]

z1 = 3  -  2  I;
s1 = 
      NSolve[Evaluate[
              2 I (z1^2 - z1)] == (z - 1)^5, {z}] //  
        Flatten

2 I (z1^2 - z1)

(2 I (z1^2 - z1))^(1/5)  + 1

(2 I (z1^2 - z1))^(1/5)  + 1 // N

p[z_] := Point[{Re[z], Im[z]}];
SetAttributes[p, Listable]

tabz = Table[z /. s1[[n]], {n, 1, 5}]

pz = p[tabz]

Prepend[pz, PointSize[0.03]]

zus = Plot[0, {x, -0.53, 3}];

Show[zus, 
        Graphics[
g = 
            Prepend[pz, PointSize[0.03]]]
, 
        AspectRatio -> 01];

c

am = Sum[tabz[[n]], {n, 1, Length[tabz]}]

am5 = p[am/5]

Append[g, am5]

Show[zus, 
        Graphics[
Append[g, am5]]
, 
        AspectRatio -> 01];

Aufgabe 6 /Problème 6

Remove["Global`*"]

p[x_] := 
        a x^3 + b x^2  +  c x  +  d;
q[x_] :=  
      a (x - (-1)) (x - (-2)) (x - 3) // 
        Expand

CoefficientList[p[x], x]

CoefficientList[q[x], x]

s6 = 
      Solve[CoefficientList[p[x], x] == CoefficientList[q[x], x], {b, c, 
              d}] /.  a -> 1  // Flatten

p[x_] :=  x^3 + b x^2  +  c x  +  d  /. s6

Plot[p[x], {x, -3, 4}];