Lösungen

1

Remove["Global`*"]

a

OP1 = {2, 0, 6};

    a = {3, 1, 4};

    b = {1, 0, 0};

    c = Cross[a, b]

d=Cross[a, c]

ea = a/Norm[a]

ea // N

ec = c/Norm[c]

ec // N

ed = d/Norm[d]

ed // N

b

M = Transpose[{ea, ec, ed}];  M // MatrixForm

M // N // MatrixForm

c

Minv = Inverse[M]

Minv // MatrixForm

Minv // N // MatrixForm

OP1s = Minv . OP1

OP1s // N

d

Dreh[φ_] := {{1, 0, 0}, {0, 
          Cos[φ], -Sin[φ]}, {0, Sin[φ], 
          Cos[φ]}}

Dreh[2  Pi/3] // MatrixForm

Dreh[2  Pi/3] // N // MatrixForm

Dreh[4  Pi/3] // MatrixForm

Dreh[4  Pi/3] // N // MatrixForm

OP2s = Dreh[2  Pi/3] . OP1s

OP2s // N

OP3s = Dreh[4  Pi/3] . OP1s

OP3s // N

OP3as = Dreh[4  Pi/3] . OP2s

OP3as // N

e

OP2 = M . OP2s;  OP2 // N

OP3 = M . OP3s;  OP3 // N

OP3a = M . OP3as;  OP3a // N

f

V = Det[{OP1, OP2, OP3}]/6

V // N

Va = Det[{OP1, OP2, OP3a}]/6

Va // N

2

Remove["Global`*"]

a

a = {3, 1, 2};  b = {-1, -1, 0};  
    c = {1, -2, -2};

M = Transpose[{a, b, c}];  M // MatrixForm

Dl = {{1, 0, 0}, {0, 1, 0}, {0, 0, 0}};  Dl // MatrixForm

A = M . Dl . Inverse[M];  A // MatrixForm

A // N // MatrixForm

b

OA = {1, 0, 0};  OB = {0, 1, 0};  OC = {0, 0, 1};

A . OA

N[%]

A . OB

N[%]

A . OC

N[%]

c

Norm[Cross[OB - OA, OC - OA]]/2

N[%]

Norm[Cross[A . OB - A . OA, A . OC - A . OA]]/2

N[%]

3

Remove["Global`*"]

a

{{8, 1, -2}, {4, 5, -4}, {1, -1, 5}} // 
      MatrixForm

M3 = 1/3 {{8, 1, -2}, {4, 5, -4}, {1, -1, 5}};  
    M3 // MatrixForm

EW = Eigenvalues[M3]

EV = Eigenvectors[M3]

EV[[1]]/Norm[EV[[1]]]

N[%]

EV[[2]]/Norm[EV[[2]]]

N[%]

EV[[3]]/Norm[EV[[3]]]

N[%]

{{λ, 0, 0}, {0, λ, 0}, {0, 0, λ}} // 
      MatrixForm

Id[λ_] := {{λ, 0, 0}, {0, λ, 0}, {0, 
          0, λ}}

b

p[λ_] := Det[M3 - Id[λ]];  
    p[λ] == 0

p[λ]  u // Expand

(p[λ]  u // Expand) /. {(λ^3) -> 
            M3 . M3 . M3, (λ^2) -> 
            M3 . M3, λ -> M3, u -> Id[1]} // 
      MatrixForm

exp = (u (p[λ] - 6)/(-6 λ)  // 
            Expand) /. {u λ  -> λ,  
          u λ^2 -> λ^2}

exp

(exp /. {u -> Id[1], (λ^2) -> 
              M3 . M3, λ -> M3}) // MatrixForm

res = (exp /. {(λ^2) -> M3 . M3, λ -> 
              M3, u -> Id[1]});  res // MatrixForm

Inverse[M3] // MatrixForm

res == Inverse[M3]

Inverse[M3] == 1/6  M3 . M3  - M3 + 11/6 Id[1]

4

Remove["Global`*"]

a

M4 = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {0, 0, 0, 0}};  
    M4 // MatrixForm

(M4 . M4) // MatrixForm

(M4 . M4 . M4) // MatrixForm

(M4 . M4 . M4 . M4) // MatrixForm

b

(M4 . M4) . (M4 . M4) // MatrixForm

(M4) . (M4 . M4 . M4) // MatrixForm

(M4) . (M4 . M4 . M4 . M4) // MatrixForm

5

Remove["Global`*"]

a

\(M5 = {{1, \ 2, \ 3, \ 4, \ 5\ } + {1, \ 1, \ 2, \ 1, \ 2}, 
{3, \ 2, \ 
          1, \ 5, \ 4}, 
{7, \ 2, \(-3\), \ 7, \ 
          2\ }, 
{1, \ 1, \ 2, \ 1, \ 2} - {1, \ 2, \ 3, \ 4, \ 5\ }}; 
    M5 // MatrixForm\)

b1 = {4 + 2, 3, 1, 2 - 4}

b2 = {4 + 2, 3, 0, 2 - 4}

b

x = {x1, x2, x3, x4, x5};

Solve[M5 . x == b2, x]

Solve[M5 . x == b1, x]

c

Fall für b1: Dim Lösungsraum = 2

d

Rang = Ordnung - Dimension = 5 - 2 = 3

6

Remove["Global`*"]

S = {{-2, 2/Sqrt[3]}, {2/Sqrt[3], 2}};  S // MatrixForm

X = {{x1}, {x2}}

Flatten[Transpose[X] . S . X  // Simplify][[1]]

syst = Eigensystem[S] // Simplify

Dl = {{syst[[1]][[1]], 
          0}, {0, syst[[1]][[2]]}};   Dl // 
      MatrixForm

n[v_] := v/Norm[v]

M = 
      Transpose[{n[syst[[2]][[1]]], 
          n[syst[[2]][[2]]] }];   M // MatrixForm

M // N // MatrixForm

N[Inverse[M]] == N[Transpose[M]]

Det[M] // N

Y = Inverse[M] . X  // Simplify

Flatten[Transpose[X] . S . X  // Simplify][[1]] == Flatten[
          Transpose[Y] . Dl . Y  // Simplify][[1]]

Y1 = {{y1}, {y2}}

Flatten[   Transpose[Y1] . Dl . Y1 ] [[1]]

N[%]  ==  52

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