Lösungen

1

Remove["Global`*"]

a

m1={{3,2,1},{1,2,3},{4,0,4}};
Det[m1]

b

m2={{1,2,3},{1,2,4},{0,1,4}};
Det[m2]

c

Det[m1.m2]

d

m1.m2//MatrixForm

e

m2.m1//MatrixForm

f

b1={1,0,-1}

Solve[m1.{x1,x2,x3}==b1,{x1,x2,x3}]

g

b2={8,0,-8}

m1.{x1,x2,x3}==b1

Solve[m1.{x1,x2,x3}==b2,{x1,x2,x3}]

h

bT=Transpose[{b2}];
m2.bT==Transpose[Transpose[bT].Transpose[m2]]

Solve[(m2.m1).{x1,x2,x3}==m2.b2,{x1,x2,x3}]

i

m3={{3,2},{1,2}};
Inverse[m3]//MatrixForm

N[%]

j

z1=3+2I;
z2=1-I;
z1 z2

1/(z1 z2)

N[%]

k

1/(Conjugate[z1] Conjugate[z2])

N[%]

l

(* 4*(1:5)+10  umgesetzt in Mathematica *)

4*Range[5]+10

>> 4*(1:5)+10
       14       18       22       26       30

m

Mod[70, 12]

>> rem(70,12)
ans = 10

n

g = I + 3;  Im[g] Conjugate[g]

>> g=j+3; imag(g)*conj(g) 
ans = 3 - 1i

o

g = I + 3;  Im[g] Conjugate[g]

a = {1, 2, 3, 4};  b = Transpose[{a, 2  a}];  
    b // MatrixForm

>> a=[1 2 3 4];b=[a',2*a']
b =

        1        2
        2        4
        3        6
        4        8

p

b . Transpose[b] // MatrixForm

>> a=[1 2 3 4];b=[a',2*a'];b*b'
ans =

        5       10       15       20
       10       20       30       40
       15       30       45       60
       20       40       60       80

2

Remove["Global`*"]

a

P0={1,1,-1}; P1={2,0,2}; P2={3,2,1};

Inhalt= Norm[Cross[P1-P0,P2-P0]]/2

N[%]

b

v={2,5,1};
hnf[x_,y_,z_]:=(v.{x,y,z}+d)/Norm[v];
hnf[{x_,y_,z_}]:=hnf[x,y,z];
hnf[x,y,z]

N[%]//Expand

c, d

hnf[1,1,-1]==0

solv1=Solve[hnf[1,1,-1]==0,{d}]//Flatten

hnf[x_,y_,z_]:=(v.{x,y,z}+d)/Norm[v]/.solv1;
hnf[{x_,y_,z_}]:=hnf[x,y,z];
hnf[x,y,z]

p1={2,0,2};
hnf[p1]

p2={3,2,1};hnf[p2]

N[%]

(* Nur P1 liegt in der Ebene *)

e

g[t_]:=t p2

hnf[g[t]]

solv2=Solve[hnf[g[t]]==0,{t}]//Flatten

PS=g[t]/.solv2

N[%]

tanA =  PS[[3]]/Sqrt[PS[[1]]^2+PS[[2]]^3]

N[%]

f   Der Abstand ist minimal, wenn die erste Spur der Ebene senkrecht zu g ist.

solv3=Solve[hnf[x,y,0]==0,{y}]//Flatten

N[%]

y[t_]:=(y/.solv3)/.x->t; y[a]

N[%]

y[1]-y[0]

yVec= 5 {1,y[1]-y[0]}

yVecSenkr= {-yVec[[2]],yVec[[1]]}

lyVecSenkr=Norm[yVecSenkr]

N[%]

gyVecSenkr[t_]:= t {yVecSenkr[[1]]/Norm[yVecSenkr],yVecSenkr[[2]]/Norm[yVecSenkr],tanA}; gyVecSenkr[t]

N[%]

(hnf[gyVecSenkr[t]]//Simplify)==0

N[%]

solv1 = Solve[(hnf[gyVecSenkr[t]]//Simplify)==0,{t}] // Simplify // Flatten

N[%]

gyVecSenkr1[t_]:=gyVecSenkr[t]/.solv1;  gyVecSenkr1[t]//Simplify

N[%]

Solve[hnf[gyVecSenkr1[t]]==0,{d}]

N[%]

3

Remove["Global`*"]

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

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

a

u1=A.A;u1//MatrixForm

v1=B.B; v1//MatrixForm

b

u2=u1.A;u2//MatrixForm

v2=v1.B; v2//MatrixForm

c2=A.B;c1//MatrixForm

c

A.B//MatrixForm

B.A//MatrixForm

A.A.B.B//MatrixForm

A.A.A.B.B.B//MatrixForm

d

A1=Inverse[A]; A1//MatrixForm

B1=Inverse[B]; B1//MatrixForm

A1.B1//MatrixForm

B1.A1//MatrixForm

e

(B1.A.B.A.B1//MatrixForm)

f

A+B//MatrixForm

Inverse[B1-A1].(A+B)//MatrixForm

4

Remove["Global`*"]

a

i

solv=Solve[z^3==1,{z}]//Flatten

z1=z/.solv[[1]]

zz[1]=z1//N

z2=z/.solv[[2]]

zz[2]=z2//N

z3=z/.solv[[3]]

zz[3]=z3//N

p[k_]:={Re[zz[k]],Im[zz[k]]}
s1=Show[Graphics[{PointSize[0.03],Point[p[1]],Point[p[2]],Point[p[3]],Line[{p[1],p[2],p[3],p[1]}],Line[{{-1,0},{1,0}}],Line[{{0,-1},{0,1}}]}],AspectRatio->Automatic];

ii

w1=z1+k

w2=z2+k

w3=z3+k

(z-w1)(z-w2)(z-w3)//Expand//Simplify//Expand

iii

w1=z1+2

w2=z2+2

w3=z3+2

(z-w1)(z-w2)(z-w3)//Expand//Simplify//Expand

ww[1]=w1//N; ww[2]=w2//N; ww[3]=w3//N;
p[k_]:={Re[ww[k]],Im[ww[k]]}
s2=Show[s1,Graphics[{PointSize[0.03],Point[p[1]],Point[p[2]],Point[p[3]],Line[{p[1],p[2],p[3],p[1]}],Line[{{-1,0},{1,0}}],Line[{{0,-1},{0,1}}]}],AspectRatio->Automatic];

iv

w1=z1+2+4I

w2=z2+2+4I

w3=z3+2+4I

(z-w1)(z-w2)(z-w3)//Expand//Simplify//Expand

www[1]=w1//N; www[2]=w2//N; www[3]=w3//N;
p[k_]:={Re[www[k]],Im[www[k]]}
s3=Show[Graphics[{PointSize[0.03],Point[p[1]],Point[p[2]],Point[p[3]],Line[{p[1],p[2],p[3],p[1]}],Line[{{-1,0},{4,0}}],Line[{{0,-1},{0,5}}]}],s1,s2,AspectRatio->Automatic];

v

Komplex -- reell

b

Remove["Global`*"]

i

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

Apart[q[x]]

ii

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

solv=Solve[(x-1)(x^2+1)==0,{x}]//Flatten

w[1]=x/.solv[[1]]

w[2]=x/.solv[[2]]

w[3]=x/.solv[[3]]

p[k_]:={Re[w[k]],Im[w[k]]}

s4=Show[Graphics[{PointSize[0.03],Point[p[1]],Point[p[2]],Point[p[3]],Line[{p[1],p[2],p[3],p[1]}],Line[{{-1,0},{2,0}}],Line[{{0,-2},{0,2}}]}],AspectRatio->Automatic];

iii

Flaeche[v1_,v2_]:=v1[[1]] v2[[2]] - v1[[2]] v2[[1]]

A1 = Flaeche[p[3]-p[1],p[2]-p[1]]/2

Umfang1 = Norm[p[3]-p[2]]+Norm[p[2]-p[1]]+Norm[p[1]-p[3]]

N[%]

A1/Umfang1

N[%]

iv

pp[k_]:={Re[1/w[k]],Im[1/w[k]]}

A2 = Flaeche[pp[3]-pp[1],pp[2]-pp[1]]/2

N[%]

Umfang2 = Norm[pp[3]-pp[2]]+Norm[pp[2]-pp[1]]+Norm[pp[1]-pp[3]//Simplify]

N[%]

A2/Umfang2

N[%]

v

ww[1]=w[2]+2

ww[2]=w[3]+2

pw[1]={Re[ww[1]],Im[ww[1]]};
pw[2]={Re[ww[2]],Im[ww[2]]};

s5=Show[s4,Graphics[{PointSize[0.03],Line[{pw[1],pw[2]}],Line[{{-1,0},{6,0}}],Line[{{0,-1},{0,1}}]}],AspectRatio->Automatic];

winkel=-(pw[1]-pw[2]).{1,0}/Norm[pw[1]-pw[2]]

ArcCos[winkel]

ArcCos[winkel]/Degree//N

5

Das Oktaeder ist belanglos. Es dient nur der Einkleidung der Aufgabe, kann also überlesen werden.

Die weitere Lösung ist dem Leser überlassen.