Teil 1

1

A = {{-b, b, b}, {a, c, c}, {c, a, c}}; A // MatrixForm

B = {{b, b, c}, {a, -a, a}, {c, c, 
            b}};
B // MatrixForm

cC = {{b, b, a}, {a, -b, a}, {-c, c, 
            b}};
cC // MatrixForm

b1 = {1, -1, 1}

z1 = -1/2 I

z2 = 1/4 + 1/2 I

z3 = 3 - 2  I

a

Det[A]

Det[A] // Factor

b

Det[B]

Det[B] // Factor

c

Det[A] /. {a -> 1, b -> 0, c -> -1}

Det[B] /. {a -> 1, b -> 0, c -> -1}

Det[A . B] /. {a -> 1, b -> 0, c -> -1}

Det[B . A] /. {a -> 1, b -> 0, c -> -1}

d

A . B

(A . B /. {a -> 1, b -> 0, c -> -1}) // 
      MatrixForm

(B . A /. {a -> 1, b -> 0, c -> -1}) // 
      MatrixForm

Faktoren nicht vertauschbar

e

Det[cC] /. {a -> 1, b -> 0, c -> -1}

Det[A . cC] /. {a -> 1, b -> 0, c -> -1}

Det[B . cC] /. {a -> 1, b -> 0, c -> -1}

f

(Det[B . cC] /. {b -> 0, c -> -1}) == 
      16

Solve[(Det[B . cC] /. {b -> 0, c -> -1}) == 
        16, {a}]

N[%]

g

(Inverse[cC] /. {a -> 1, b -> 0, c -> -1}) // 
      MatrixForm

(Inverse[cC] /. {a -> 1, b -> 0, c -> -1}) . 
      b1

h

G = {{-1, 1}, {2, 1}};  G // MatrixForm

Inverse[G] // MatrixForm

N[%] // MatrixForm

i

Inverse[G] . (IdentityMatrix[2] - X^T + G) . G == 
        G . G;

IdentityMatrix[2] - X^T + G == G . G

-X^T == G . G - IdentityMatrix[2] - G

X = Transpose[-(G . G - IdentityMatrix[2] - G)];  
    X // MatrixForm

j

Conjugate[z1] 1/z2

N[%]

k

solv = Solve[z^6 == z1, {z}]

z[k_] := z /. solv[[k]]

tab0 = Table[z[k], {k, 1, 6}]

tab0N = tab0 // N

tab0N[[1]] + tab0N[[2]]

tab0N[[3]] + tab0N[[4]]

tab0N[[5]] + tab0N[[6]]

ListPlot[Table[{Re[tab0N[[k]]], Im[tab0N[[k]]]}, {k, 1, 6}], 
        PlotStyle -> PointSize[0.02], 
        AspectRatio -> Automatic];

l

(z2^2 + z2)/z3

N[%]

2 Matlab - Octave - Output

a

(Eingefügt mit Copy-Paste aus einem Window)

b

c

d

e

f

Teil 2

3

a

P1 = {2, -1, 0};
P2 = {3, 4, 0};
P3 = {1, 6, 0};
Q = {1, 5, 7};

solv = 
      Solve[{Norm[{x, y, 0} - P1] == Norm[{x, y, 0} - P2], 
            Norm[{x, y, 0} - P1] == Norm[{x, y, 0} - P3]}, {x, y}] // 
        Flatten

N[%]

M = {x, y, 0} /. solv

N[%]

b

v1 = (P2 - P1)

g[t_] := P1 + t v1;  g[t]

v1s = {-v1[[2]], v1[[1]], 0}

gM[t_] := M + t v1s;  gM[t]

solv1 = Solve[gM[t1] == g[t2], {t1, t2}] // Flatten

F = g[t2] /. solv1

Ms = M + 2 (F - M)

N[%]

c

winkel = 
      ArcCos[(M - P3) . (Ms - P3)/(Norm[M - P3]  Norm[
                Ms - P3])]

N[%]

%/Degree

d

V = Abs[Det[{(P1 - Q), (P2 - Q), (P3 - Q)}]/6]

e

abst[A_, v_] := 
        Norm[Cross[A, v]/Norm[v]];
abst[M - P1, v1]

abst[M - P2, v1]

abst[Ms - P2, v1]

N[%]

abst[P3 - P1, v1]

abst[P3 - P2, v1]

N[%]

4

a

p[x_] := ((-1 + x - 5 (x^2) - 9 (x^3) - 8 (x^4) - 4 (x^5))/((x^2) (1 + x)));  p[x] // Apart

b

q[
          x_] := ((4 - x - (x^2) - 4 (x^3) - 4 (x^4))/((-1 + x) (1 + x)));
q[x] // Apart

c

(p[x] - q[x]) // Apart

Kein ganzer Anteil

5

a

solv1 = Solve[z^8 == 1, {z}]

solv2 = Solve[z^8 == -1, {z}]

z[k_] := z /. solv1[[k]]

tab1 = Table[z[k], {k, 1, 8}]

tab1a = Table[(E^( I 2 Pi/8))^k, {k, 1, 8}]

tab1aN = tab1a // N

ListPlot[
        Table[{Re[tab1aN[[k]]], Im[tab1aN[[k]]]}, {k, 1, 8}], 
        PlotStyle -> PointSize[0.02], 
        AspectRatio -> Automatic];

tab1uN = -tab1aN

ListPlot[
        Table[{Re[tab1uN[[k]]], Im[tab1uN[[k]]]}, {k, 1, 8}], 
        PlotStyle -> PointSize[0.02], 
        AspectRatio -> Automatic];

Dieselbe Gleichung

b

sum[j_] := Sum[tab1a[[k]], {k, 1, j}]

tab1b = Table[sum[j], {j, 1, 8}]

tab1bN = tab1b // N // Chop

ListPlot[
        Table[{Re[tab1bN[[k]]], Im[tab1bN[[k]]]}, {k, 1, 8}], 
        PlotStyle -> PointSize[0.02], 
        AspectRatio -> Automatic];

a = 
      1; ((Sum[{Re[E^(I ( j 2 Pi/8 + Pi))], 
                    Im[E^(I ( j 2 Pi/8 + Pi))]}, {j, 1, k}] /. 
                k -> 
                  a) + 
(Sum[{Re[
                      E^(I ( j 2 Pi/8 + Pi))], 
                    Im[E^(I ( j 2 Pi/8 + Pi))]}, {j, 1, k}] /. 
                k -> a + 4))/2 // N

a = 
      2; ((Sum[{Re[E^(I ( j 2 Pi/8 + Pi))], 
                    Im[E^(I ( j 2 Pi/8 + Pi))]}, {j, 1, k}] /. 
                k -> 
                  a) + 
(Sum[{Re[
                      E^(I ( j 2 Pi/8 + Pi))], 
                    Im[E^(I ( j 2 Pi/8 + Pi))]}, {j, 1, k}] /. 
                k -> a + 4))/2 // N

a = 
      3; ((Sum[{Re[E^(I ( j 2 Pi/8 + Pi))], 
                    Im[E^(I ( j 2 Pi/8 + Pi))]}, {j, 1, k}] /. 
                k -> 
                  a) + 
(Sum[{Re[
                      E^(I ( j 2 Pi/8 + Pi))], 
                    Im[E^(I ( j 2 Pi/8 + Pi))]}, {j, 1, k}] /. 
                k -> a + 4))/2 // N

a = 
      4; ((Sum[{Re[E^(I ( j 2 Pi/8 + Pi))], 
                    Im[E^(I ( j 2 Pi/8 + Pi))]}, {j, 1, k}] /. 
                k -> 
                  a) + 
(Sum[{Re[
                      E^(I ( j 2 Pi/8 + Pi))], 
                    Im[E^(I ( j 2 Pi/8 + Pi))]}, {j, 1, k}] /. 
                k -> a + 4))/2 // N

Vermutung: Kreis.(Einfach beweisbar mit Hilfe der endlichen geometrischen Reihen)

6

A = {{1, 1, 0, 0, 0}, {1, 1, 1, 0, 0}, {0, 1, 1, 1, 0}, {0, 0, 1, 1, 
          1}, {0, 0, 0, 1, 1}}

A // MatrixForm

B = {{1, -1, 0, 0, 0}, {1, 1, -1, 0, 0}, {0, 1, 1, -1, 
          0}, {0, 0, 1, 1, -1}, {0, 0, 0, 1, 1}}

B // MatrixForm

a

A . A // MatrixForm

A . A . A // MatrixForm

Matrix wird nach aussen aufgefüllt, Nullen werden eliminiert. Symmetrisch da A symmetrisch.

b

B . B // MatrixForm

B . B . B // MatrixForm

Matrix wird nach aussen aufgefüllt, Nullen werden eliminiert. Symmetrisch da B symmetrisch.

c

A . B // MatrixForm

B . A // MatrixForm

An der x=y-Achse gespiegelt, identisch bis auf Endelemente der Hauptdiagonalen.

A . B == B . A

d

Inverse[B]

Inverse[B] // MatrixForm

N[%] // MatrixForm

A . B . A + A == X . B . B + B;
X == (A . B . A + A - B) . Inverse[B] . Inverse[B]

(A . B . A + A - B) . Inverse[B] . Inverse[B] // 
      MatrixForm

e

x1 = {0, 0, 0, 0, 0};
x2 = {1, -1, 0, 1, -1}

A . x1

A . x2

A . x1 == A . x2

Det[A]

Zwei verschiedene Urbilder haben dieselben Bilder. A ist nicht regulär.

7

Remove["Global`*"]

Für die Ebene braucht es 3 Punkte. Beispiel:

r1 = 
      5;   k1[x_, y_, z_] := {x, y, z} . {x, y, z} - 
        r1^2;
r2 = 6;  M2 = {5, 5, 5};
k2[x_, y_, z_] := ({x, y, z} - M2) . ({x, y, z} - M2) - 
          r2^2;

Solve[{k1[x, y, z] == 0, k2[x, y, z] == 0}, {x, y, 
        z}]

solv1 = 
      Solve[({k1[x, y, z] == 0, k2[x, y, z] == 0} /. 
              z -> 1), {x, y, z}] // N

p1 = {x, y} /. solv1[[1]];  p1 = Join[p1, {1}]

solv2 = 
      Solve[({k1[x, y, z] == 0, k2[x, y, z] == 0, 
                k1[x, y, z] == k2[x, y, z]} /. z -> 2), {x, y, 
            z}] // N

p2 = {x, y} /. solv2[[1]];  p2 = Join[p2, {2}]

solv3 = 
      Solve[({k1[x, y, z] == 0, k2[x, y, z] == 0, 
                k1[x, y, z] == k2[x, y, z]} /. z -> 3), {x, y, 
            z}] // N

p3 = {x, y} /. solv3[[1]];  p3 = Join[p3, {3}]

Φ[λ_, μ_] := 
        p1 + λ (p2 - p1) + μ (p3 - 
                p1);
Φ[λ, μ]

solve = 
      Solve[Φ[λ, μ] == {t, 0, 
              0}, {t, λ, μ}] // Flatten

x = t /. solve

8

Remove["Global`*"]

P1 = {1, 0, 0};  P2 = {0, 2, 0}; 
    P3 = {0, 0, 3};
Q1 = {4, 0, 0};  Q2 = {1, 5, 1};
Φ[λ_, μ_] := 
        P1 + λ (P2 - P1) + μ (P3 - 
                P1);
gSQ2Φ[t_] := 
        Q2 + t Cross[ (P2 - P1),  (P3 - 
                  P1)];
solvQ2F = 
      Solve[Φ[λ, μ] == 
            gSQ2Φ[t], {t, λ, μ}] // 
        Flatten

FusspunktQ2 = gSQ2Φ[t] /. solvQ2F

N[%]

Q2Gespiegelt = Q2 + 2 (FusspunktQ2 - Q2)

N[%]

gQ2GespQ1[t_] := Q1 = 
          t (Q2Gespiegelt - Q1);
solvgQ2GespQ1 = 
      Solve[Φ[λ, μ] == 
            gQ2GespQ1[t], {t, λ, μ}] // Flatten

L = gQ2GespQ1[t] /. solvgQ2GespQ1

N[%]