Lösungen / Solutions

1

Solve[
      4 (a - 2  x) + b == 1/3 x - 7  a + 6  c, {x}]

% // ExpandAll

% // N

2

<< Graphics`Arrow`

v[k_, n_, r_] :=  r {Cos[k 2  Pi/n], Sin[k 2  Pi/n]}

a

Show[
        Graphics[{Circle[{0, 0}, 1], 
            Arrow[{0, 0}, v[0, 3, 1]], 
Arrow[{0, 0}, 
              v[1, 3, 1]], 
Arrow[{0, 0}, 
              v[2, 3, 1]], 
Line[{v[0, 3, 1], v[1, 3, 1], 
                v[2, 3, 1], v[0, 3, 1]}]}, 
          AspectRatio -> Automatic]];

v[0, 3, 1] + v[1, 3, 1] + v[2, 3, 1]

b

v[0, 5, 1] + v[1, 5, 1] + v[2, 5, 1] + v[3, 5, 1] + v[4, 5, 1] // 
      Simplify

b

Sum[v[k, 17, 1], {k, 0, 16}] // Simplify

Sum[v[k, 17, 1], {k, 0, 16}] // N // Chop

3

a = {3, -4}; b = {5, 4}; c = {2, 11}

Solve[λ a  +  μ b  ==  
        c, {λ, μ}]

N[%]

4

a1 = {Sqrt[2]/2, Sqrt[2]/2}; 
    a2 = {-Sqrt[2]/2, Sqrt[2]/2};
a3 = -a1;  a4 = -a2;
s = 
      Solve[a1 + a2 + λ a3 + 
              a4  μ == {0, -1}, {λ, μ}] // 
        Flatten

{s1, s2} = {λ, μ} /. s

N[%]

Show[
        Graphics[{Circle[{0, 0}, 1], 
Arrow[{0, 0}, 
              a1], 
Arrow[{0, 0}, 
              a2], 
Arrow[{0, 0}, 
              a3], 
Arrow[{0, 0}, 
              a4], 
Arrow[{0, 
                0}, {0, -1}], 
Arrow[a1, 
              a1 + a2], 
Arrow[a1 + a2, 
              a1 + a2 + s1 a3], 
Arrow[a1 + a2 + s1 a3, 
              a1 + a2 + s1 a3 + 
                s2 a4], 
Line[{{-1, 0}, {1, 0}}], 
            Line[{{0, -1}, {0, 1}}]}, 
          AspectRatio -> Automatic]];

5

a

Man sieht: Reflexiv, symmetrisch, transitiv.

b

Klassen mit Resten 0, 1, 2, 3, 4.

c

?Mod

Mod[2,  5] Mod[x,  5]  + Mod[1,  5] ==Mod[4,  
        5]

Mod[2,  5] Mod[x,  5]  + Mod[1,  5]  - Mod[4,  5]  == 
        Mod[0,  5]  // Simplify

Table[ Mod[2 x, 5] == 3, {x, 0, 4}]

Mod[2*4 + 1,  5] ==Mod[4,  5]