Lösungen

1

Remove["Global`*"]

r0 = {1, 2, -1};
a = {-2, 1, 1};
q = {3, 10, 14};
rq = q - r0

gq = Norm[Cross[a, rq]]/Norm[a]

N[%]

2

Remove["Global`*"]

r0 = {1, 2, -1};
a = {2, 1, -1};
b = {1, -1, 2};
q = {3, 10, 14};
rq = q - r0

a

gq = -Det[{a, b, rq}]/Norm[Cross[a, b]]

N[%]

b

v[λ_, μ_] := r0 + λ a + μ b;  
    v[λ, μ ]

w[t_] := q + t Cross[a, b];  w[t]

solv1 = 
      Flatten[Solve[ 
          v[λ, μ ] == w[t], {λ, μ , t}]]

t0 = t /. solv1

w[t0]

N[%]

3

Remove["Global`*"]

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

a

ew = Eigenvalues[M]

N[%]

Abs[ew[[1]]] E^(I Arg[ew[[1]]])

{N[Abs[ew[[1]]]],  I N[Arg[ew[[1]]]]}

Abs[ew[[2]]] E^(I Arg[ew[[2]]])

{N[Abs[ew[[2]]]],  E^(I N[Arg[ew[[2]]]])}

ev = Eigenvectors[M] // ExpandAll

N[%]

b

{ev[[1]]/Norm[ev[[1]]], 
      ev[[2]]/Norm[ev[[2]]]}

N[%]

Eigensystem[M]

c

X = Transpose[ev]

N[%] // MatrixForm

Inverse[X]

N[%] // MatrixForm

dM = {{ew[[1]], 0}, {0, ew[[2]]}}

dM // N // MatrixForm

X . dM . Inverse[X] // N // Chop // MatrixForm

d

Eigenvalues[dM] == Eigenvalues[M]

e

Eigenvectors[dM]

Eigenvectors[dM] == Eigenvectors[M]

f  Komplex

g1[t_] := t ev[[1]];   g1[t] // N

g2[t_] := t ev[[2]];   g2[t] // N

g

(M .  g1[t] // Simplify // ExpandAll)/(1 + 
                I Sqrt[2]) // Simplify // N // Chop

(M .  g2[t] // Simplify // ExpandAll)/(1 - 
                I Sqrt[2]) // Simplify // N // Chop

h

(dM .  g1[t] // N // Simplify // 
            ExpandAll)/(1.` - 
            1.4142135623730951` I) // Chop

(dM .  g2[t] // N // Simplify // 
            ExpandAll)/(1.` - 
            1.4142135623730951` I) // Chop

Keinde Deckung

i

M . M // MatrixForm

ew[[2]]

dM = {{ew[[1]], 0}, {0, 
            ew[[2]]}};
X . (dM^2) . Inverse[X] // N // Chop // 
      MatrixForm

X . (dM^2) . Inverse[X] // Simplify // MatrixForm

j

dM = {{ew[[1]], 0}, {0, 
            ew[[2]]}};
X . (dM^50) . Inverse[X] // N // Chop // 
      MatrixForm

X . (dM^50) . Inverse[X] // Simplify // MatrixForm

4

Remove["Global`*"]

normVec[menge_] := 
      Table[ menge[[k]]/Norm[menge[[k]]], {k, 1, 
          Length[menge]}]

a

a1 = {1, 1, 0} ; a2 = {0, 1, 
          1} ;  a3 = {1, 
          0, -1} ;

A = Transpose[{a1, a2, a3}];  A // MatrixForm

b

Eigenvalues[A]

N[%]

c

Eigenvectors[A]

menge = N[%]

normVec[menge]

d

I3 = IdentityMatrix[3]

A . I3 == A

Volumen = Det[A]

(Zu erwarten, da EW 0 existiert)

e

Det[A]

Det[A]^-1

f

Det[A] == Volumen

g

p = 
      Product[Eigenvalues[A][[k]], {k, 1, 
          Length[Eigenvalues[A]]}]

h

p == Det[A]

i

s = Sum[I3[[k]], {k, 1, Length[I3]}]

A . s

j

sA = Sum[A[[k]], {k, 1, Length[I3]}]

A . s == sA

5

Remove["Global`*"]

a = {-1, 3};  aS = {-3, -1};

v[t_] := t a;  p = {10, -1};

a

{a, aS}

vS = Transpose[{a, aS}];  vS // MatrixForm

dS = {{1, 0}, {0, -1}};  dS // MatrixForm

S = vS . dS . Inverse[vS];  S // MatrixForm

N[%] // MatrixForm

b

p1 = S . p

N[%]

c

α = 10 Degree;  α // N

α1 = -10 Degree

d = {{Cos[α], -Sin[α]}, {Sin[α], 
          Cos[α]}};  d // MatrixForm // N

d1 = {{Cos[-α], -Sin[-α]}, {Sin[-α], Cos[-α]}}

p2 = d . p1

N[%]

d

p3 = S . p2

N[%]

e

p4 = d1 . p

N[%]

g

(p3 // N)  == (p4 // N)

6

Remove["Global`*"]

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

v[λ_,  μ_] := λ a  +  μ b;  
    p0 = {10, 5, -3};  pPr = {10, 10, 20};

a

vS = Transpose[{a, b,  Cross[a, b]}];  vS // MatrixForm

dS = {{1, 0, 0}, {0, 1, 0}, {0, 0, -1}};  
    dS // MatrixForm

S = vS . dS . Inverse[vS];  S // MatrixForm

N[%] // MatrixForm

b

p1 = S . p0

N[%]

c

vPr = Transpose[{a, b,  Cross[a, b]}];  vPr // MatrixForm

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

Pr = vPr . dPr . Inverse[vPr];  Pr // MatrixForm

N[%] // MatrixForm

d

Q = Pr . pPr

7

Remove["Global`*"]

M . (Id - X) . Inverse[M] + M - A . M == 
      A . Transpose[M] - 2 M

Id - M . X . Inverse[M] + M - A . M == 
      A . Transpose[M] - 2 M

Id - A . Transpose[M] + 3 M - A . M == M . X . Inverse[M]

X == Inverse[M] . (Id - A . Transpose[M] + 3 M - A . M) . 
        M

X == Id - Inverse[M] . A . Transpose[M] . M + 3 M - 
        Inverse[M] . A . M . M