Lineare Algebra

1

Remove["Global`*"]

0

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

B = {{1, 2, 1}, {2, 1, 1}, {1, 1, 2}};  B // MatrixForm

b1 = {1, 1, 1};
b2 = {1, -1, 1};
b3 = {-1, 1, 0};

z1 = -I;
z2 = 1/2 - 2  I ;
z3 = 2 - 4  I;

a

Det[A]

b

Det[B]

c

Det[B]/Det[A]

d

A . B // MatrixForm

e

B . A // MatrixForm

f

Inverse[A] // MatrixForm

u = 2

Inverse[B] // MatrixForm

g

Inverse[A] . b1

h

Inverse[A] . b2

i

Inverse[A] . b3

j

Solve[
      A . Inverse[B] . Transpose[Inverse[A]] . {x1, x2, x3} == b1, {x1, 
        x2, x3}]

Inverse[A] . b1

B . Inverse[A] . b1

Transpose[A] . B . Inverse[A] . b1

k

Conjugate[z3]/z2

N[%]

l

(z2)^2 - z2 z3

N[%]

m

Solve[z^4 == z1, {z}]

solv = Solve[z^4 == z1, {z}] // Flatten // N

solv[[1]]

w1 = z /. solv[[1]];    p[1] = 
      Point[{Re[w1], Im[w1]}];

    w2 = z /. solv[[2]];    p[2] = 
      Point[{Re[w2], Im[w2]}];

    w3 = z /. solv[[3]];    p[3] = 
      Point[{Re[w3], Im[w3]}];

    w4 = z /. solv[[4]] ;    p[4] = 
        Point[{Re[w4], Im[w4]}] ;

Show[
        Graphics[{PointSize[0.03],  p[1], p[2], p[3], p[4], 
            Line[{{-1, 0}, {1, 0}}], Line[{{0, -1}, {0, 1}}]}], 
        AspectRatio -> Automatic];

n

Plot[2 Cos[x], {x, 0, Pi}];

0

12   22   32   42   52

p

3   2   1

v = Transpose[{{3, 2, 1}, -{3, 2, 1}, {3, 2, 1}}]; 
    v // MatrixForm

q

(v . Transpose[v]) // MatrixForm

Kontrolle:

(Transpose[v] . v) // MatrixForm

2

Remove["Global`*"]

a

P1 = {1, 2, 1}; 
P2 = {-1, 3, 1};
P3 = {0, -1, 2};
Q = {-1, 0, 8};

a

Norm[Cross[P1, Q]]/2

Norm[Cross[P1, Q]]/2 // N

b

n = Cross[P2 - P1, P3 - P1]

n/Norm[n]  //  Simplify

N[%]

c

Solve[n . P1 + d == 0, {d}]

HNFθ[x_, y_, z_] := (n . {x, y, z} - 12)/
          Norm[n] ; HNFθ[{x_, y_, z_}] :=  
        HNFθ[x, y, z] ;HNFθ[x, y, 
        z] ;

θVec[λ_, μ_] := 
      P1 + λ (P2 - P1) + μ (P3 - 
              P1);  θVec[λ, μ]

gLot[t_] := Q +  t n;   gLot[t]

gLot[t] == θVec[λ, μ]

solv = 
      Solve[gLot[t] == θVec[λ, μ], {t, λ, μ}] // 
        Flatten

gLot[t] /. solv[[1]]

N[%]

d

HNFθ[Q]

N[%]

e

dre[φ_] := {{1, 0, 0}, {0, 
            Cos[φ], -Sin[φ]}, {0, Sin[φ], 
            Cos[φ]}};
dre[φ] // MatrixForm

U1 = dre[30 Degree] . Q

N[%]

U2 = dre[-30 Degree] . Q

N[%]

HNFθ[U1]

N[%]

HNFθ[U2]

N[%]

U1 liegt näher an der Ebene und U2 weiter entfernt von der Ebene als Q.

3

Remove["Global`*"]

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

B = Transpose[A];  B // MatrixForm

a

A . A // MatrixForm

b

B . B // MatrixForm

c

A . B // MatrixForm

d

B . A // MatrixForm

e

A . A . B . B // MatrixForm

f

Im Falle, dass man den transponierten Vektor nimmt:

A . {1, 2, 3, 4, 5}

A . {2, 3, 4, 5, 6}

A . {3, 4, 5, 6, 7}

g

A1 = A + 2  B;  Inverse[A1] // MatrixForm

N[ Inverse[A1]] // MatrixForm

h

Im Falle, dass man statt einem Vektor nimmt eine Matrix X nimmt:

X = 
      Inverse[A1] . (A1 + Inverse[A1] - A1 . A1) // MatrixForm

N[%] // MatrixForm

4

Remove["Global`*"]

a

Solve[z^4 == -1 - I, {z}]

solv = Solve[z^4 == -1 - I, {z}] // Flatten // 
        N

solv[[1]]

w1 = z /. solv[[1]];    p[1] = 
      Point[{Re[w1], Im[w1]}];

    w2 = z /. solv[[2]];    p[2] = 
      Point[{Re[w2], Im[w2]}];

    w3 = z /. solv[[3]];    p[3] = 
      Point[{Re[w3], Im[w3]}];

    w4 = z /. solv[[4]] ;    p[4] = 
        Point[{Re[w4], Im[w4]}] ;

w1^4

Norm[-1 - I] // Simplify

Norm[-1 - I]^(1/4)

N[%]

ArcTan[-1/(-1)]

ArcTan[-1/(-1)] + Pi

(ArcTan[-1/(-1)] + Pi)/4

{(ArcTan[-1/(-1)] + Pi)/
        4, (ArcTan[-1/(-1)] + Pi)/4 + 
        Pi/2, (ArcTan[-1/(-1)] + Pi)/4 + 
        Pi, (ArcTan[-1/(-1)] + Pi)/4 + 3  Pi/2}

N[%]

Show[
        Graphics[{PointSize[0.03],  p[1], p[2], p[3], p[4], 
            Line[{{-1, 0}, {1, 0}}], Line[{{0, -1}, {0, 1}}]}], 
        AspectRatio -> Automatic];

b

(x - 1) (x + 2) (x^2 + 3) // Expand

-6 + 3 x + (x^2) + (x^3) + (x^4) // Factor

solv = 
      Solve[-6 + 3 x + (x^2) + (x^3) + (x^4) == 0, {x}] // 
        Flatten

N[%]

w1 = x /. solv[[1]];    p[1] = 
      Point[{Re[w1], Im[w1]}];

    w2 = x /. solv[[2]];    p[2] = 
      Point[{Re[w2], Im[w2]}];

    w3 = x /. solv[[3]];    p[3] = 
      Point[{Re[w3], Im[w3]}];

    w4 = x /. solv[[4]] ;    p[4] = 
        Point[{Re[w4], Im[w4]}] ;

Show[
        Graphics[{PointSize[0.03],  p[1], p[2], p[3], p[4], 
            Line[{{-1, 0}, {1, 0}}], Line[{{0, -1}, {0, 1}}]}], 
        AspectRatio -> Automatic];

Inhalt = 2 3*Sqrt[3]/2

N[%]

q[x_] := 84/(-6 + 3 x + (x^2) + (x^3) + (x^4));  q[x]

Apart[q[x]]

Apart[84/(-6 + 3 x + (x^2) + (x^3) + (x^4))] - 7/(x - 1) + 
      3 (x + 5)/(x^2 + 3)

5

Remove["Global`*"]

a

InhW[a_] := (2  a)^3; InhW[a]

InhOktW[a_] := 1/2 a^2 *4*a*2/3;  InhOktW[a]

rK[a_] := Sqrt[3  a^2];  rK[a]

InhOktK[a_] := InhOktW[rK[a]] ; InhOktK[
      a]

InhOktW[rK[a]]/InhW[a] // Simplify

InhOktW[rK[1]]/InhW[1] // Simplify

N[%]

b

p1[a_] := {a, 0, 0}; 
p2[a_] := {0, a, 0}; 
p3[a_] := {0, 0, a}; 
q1[a_] := (p1[a] + p2[a])/2;
s[a_] := q1[a] + 1/3 (p3[a] - q1[a])

hOkt[a_] := Norm[s[a]];  hOkt[a]

hOkt[1]

Statt diesen Abstand steht im Falle von O2 der Radius von O1, d.h.  rK[a]. Das Streckungsverhältnis für das Volumen ist damit ( rK[1]/ hOkt[1] )^3 :

InhOkt2K[a_] := InhOktW[rK[a]] (rK[1]/hOkt[1])^3;  
    InhOkt2K[1]

InhOkt2K[1]/ InhOktW[1]

N[%]