OP1 = 8 2, 0, 6 < ; a = 8 3, 1, 4 < ; b = 8 1, 0, 0 < ; c = Cross @ a, b D 8 0, 4, -1 <

Lösungen

1

Remove @ "Global`*" D

Ÿ a

OP1 = 8 2, 0, 6 < ; a = 8 3, 1, 4 < ; b = 8 1, 0, 0 < ; c = Cross @ a, b D 8 0, 4, -1 <

d = Cross @ a, c D

8 -17, 3, 12 <

ea = a • Norm @ a D

9 3

€€€€€€€€€€€€€ •!!!!!!! 26 , 1

€€€€€€€€€€€€€ •!!!!!!! 26

, 2 $%%%%%%%%%% 2

€€€€€€€

13 = ea •• N

8 0.588348, 0.196116, 0.784465 <

ec = c • Norm @ c D

9 0, 4

€€€€€€€€€€€€€ •!!!!!!! 17 , - 1

€€€€€€€€€€€€€ •!!!!!!! 17 =

ec •• N

8 0., 0.970143, -0.242536 <

ed = d • Norm @ d D 9-$%%%%%%%%%% 17

€€€€€€€

26 , 3

€€€€€€€€€€€€€€€ •!!!!!!!!!! 442 , 6 $%%%%%%%%%%%% 2

€€€€€€€€€€

221 = ed •• N

8-0.808608, 0.142695, 0.570782 <

Ÿ b

M = Transpose @8 ea, ec, ed <D ; M •• MatrixForm i

k jjjjj jjjjj jjjjj j

€€€€€€€€€€ •!!!!!!!! 26 3 0 - "######## €€€€€€ ¹⁷ ₂₆

€€€€€€€€€€ •!!!!!!!! 26 1

€€€€€€€€€€ •!!!!!!!! 4 17

€€€€€€€€€€€€ •!!!!!!!!!!! 442 3

2 "######## €€€€€€ ₁₃ ² - €€€€€€€€€€ •!!!!!!!! ¹ 17 6 "########## €€€€€€€€ ₂₂₁ ² y

{ zzzzz zzzzz zzzzz z

M •• N •• MatrixForm i

k jjjjj jj

0.588348 0. -0.808608 0.196116 0.970143 0.142695 0.784465 -0.242536 0.570782

y { zzzzz zz

Ÿ c

Minv = Inverse @ M D

99 3

€€€€€€€€€€€€€ •!!!!!!! 26 , 1

€€€€€€€€€€€€€ •!!!!!!! 26

, 2 $%%%%%%%%%% 2

€€€€€€€

13 = , 9 0, 4

€€€€€€€€€€€€€ •!!!!!!! 17 , - 1

€€€€€€€€€€€€€ •!!!!!!! 17 = , 9-$%%%%%%%%%% 17

€€€€€€€

26 , 3

€€€€€€€€€€€€€€€ •!!!!!!!!!! 442

, 6 $%%%%%%%%%%%% 2

€€€€€€€€€€

221 ==

Minv •• MatrixForm i

k jjjjj jjjjj jjjjj j

€€€€€€€€€€ •!!!!!!!! 26 3

€€€€€€€€€€ •!!!!!!!! 1 26 2 "######## €€€€€€ ₁₃ ² 0 €€€€€€€€€€ •!!!!!!!! ⁴ 17 - €€€€€€€€€€ •!!!!!!!! 17 ¹

- "######## €€€€€€ ¹⁷ ₂₆ €€€€€€€€€€€€ •!!!!!!!!!!! 442 ³ 6 "########## €€€€€€€€ ₂₂₁ ² y

{ zzzzz zzzzz zzzzz z

Minv •• N •• MatrixForm i

k jjjjj jj

0.588348 0.196116 0.784465 0. 0.970143 -0.242536

-0.808608 0.142695 0.570782

y { zzzzz zz

OP1s = Minv.OP1 9 15 $%%%%%%%%%% 2

€€€€€€€

13 , - 6

€€€€€€€€€€€€€ •!!!!!!! 17

, 36 $%%%%%%%%%%%% 2

€€€€€€€€€€

221 - $%%%%%%%%%% 34

€€€€€€€

13 = OP1s •• N

8 5.88348, -1.45521, 1.80748 <

Ÿ d

Dreh @ j_ D := 88 1, 0, 0 < , 8 0, Cos @ j D , -Sin @ j D< , 8 0, Sin @ j D , Cos @ j D<<

Dreh @ 2 Pi • 3 D •• MatrixForm i

k jjjjj jjjjjj

1 0 0

0 - €€€€ ¹ ₂ - €€€€€€€€ ^•!!!!! ₂ ³ 0 €€€€€€€€ ^•!!!!! ₂ ³ - €€€€ ¹ ₂

y

{ zzzzz zzzzzz

Dreh @ 2 Pi • 3 D •• N •• MatrixForm i

k jjjjj jj

1. 0. 0.

0. -0.5 -0.866025

0. 0.866025 -0.5

y { zzzzz zz

Dreh @ 4 Pi • 3 D •• MatrixForm i

k jjjjj jjjjjj

1 0 0

0 - €€€€ ¹ ₂ €€€€€€€€ ^•!!!!! ₂ ³ 0 - €€€€€€€€ ^•!!!!! ₂ ³ - €€€€ ¹ ₂

y

{ zzzzz zzzzzz

Dreh @ 4 Pi • 3 D •• N •• MatrixForm i

k jjjjj jj

1. 0. 0.

0. -0.5 0.866025

0. -0.866025 -0.5

y { zzzzz zz

OP2s = Dreh @ 2 Pi • 3 D .OP1s 9 15 $%%%%%%%%%% 2

€€€€€€€

13 , 3

€€€€€€€€€€€€€ •!!!!!!! 17 - 1

€€€€ 2

•!!!! 3 i

k jjjjjj 36 $%%%%%%%%%%%% 2

€€€€€€€€€€

221 - $%%%%%%%%%% 34

€€€€€€€

13 y

{ zzzzzz , -3 $%%%%%%%%%% 3

€€€€€€€

17 + 1

€€€€ 2 i

k jjjjjj -36 $%%%%%%%%%%%% 2

€€€€€€€€€€

221 + $%%%%%%%%%% 34

€€€€€€€

13 y { zzzzzz=

OP2s •• N

8 5.88348, -0.837713, -2.16399 <

OP3s = Dreh @ 4 Pi • 3 D .OP1s 9 15 $%%%%%%%%%% 2

€€€€€€€

13 , 3

€€€€€€€€€€€€€ •!!!!!!! 17 + 1

€€€€ 2

•!!!! 3 i

k jjjjjj 36 $%%%%%%%%%%%% 2

€€€€€€€€€€

221 - $%%%%%%%%%% 34

€€€€€€€

13 y

{ zzzzzz , 3 $%%%%%%%%%% 3

€€€€€€€

17 + 1

€€€€ 2 i

k jjjjjj -36 $%%%%%%%%%%%% 2

€€€€€€€€€€

221 + $%%%%%%%%%% 34

€€€€€€€

13 y { zzzzzz=

OP3s •• N

8 5.88348, 2.29293, 0.356514 <

OP3as = Dreh @ 4 Pi • 3 D .OP2s 9 15 $%%%%%%%%%% 2

€€€€€€€

13 ,

€€€€ 1 2

i k jjjjjj - 3

€€€€€€€€€€€€€ •!!!!!!! 17 + 1

€€€€ 2

•!!!! 3 i

k jjjjjj 36 $%%%%%%%%%%%% 2

€€€€€€€€€€

221 - $%%%%%%%%%% 34

€€€€€€€

13 y { zzzzzz y

{ zzzzzz + 1

€€€€ 2

•!!!! 3 i

k jjjjjj -3 $%%%%%%%%%% 3

€€€€€€€

17 + 1

€€€€ 2 i

k jjjjjj -36 $%%%%%%%%%%%% 2

€€€€€€€€€€

221 + $%%%%%%%%%% 34

€€€€€€€

13 y { zzzzzz y

{ zzzzzz ,

€€€€ 1 2

i

k jjjjjj 3 $%%%%%%%%%% 3

€€€€€€€

17 + 1

€€€€ 2 i

k jjjjjj 36 $%%%%%%%%%%%% 2

€€€€€€€€€€

221 - $%%%%%%%%%% 34

€€€€€€€

13 y { zzzzzz y

{ zzzzzz - 1

€€€€ 2

•!!!! 3 i k jjjjjj 3

€€€€€€€€€€€€€ •!!!!!!! 17 - 1

€€€€ 2

•!!!! 3 i

k jjjjjj 36 $%%%%%%%%%%%% 2

€€€€€€€€€€

221 - $%%%%%%%%%% 34

€€€€€€€

13 y { zzzzzz y

{ zzzzzz=

OP3as •• N

8 5.88348, -1.45521, 1.80748 <

Ÿ e

OP2 = M.OP2s; OP2 •• N

8 5.21136, 0.0323537, 3.58339 <

OP3 = M.OP3s; OP3 •• N 8 3.17326, 3.42918, 4.26276 <

OP3a = M.OP3as; OP3a •• N 8 2., 0., 6. <

Ÿ f

V = Det @8 OP1, OP2, OP3 <D • 6 525 "######## €€€€€€ ₂₆ ³

€€€€€€€€€€€€€€€€€€€€€€€€

13 V •• N 13.718

Va = Det @8 OP1, OP2, OP3a <D • 6 0

Va •• N 0.

2

Remove @ "Global`*" D

Ÿ a

a = 8 3, 1, 2 < ; b = 8-1, -1, 0 < ; c = 8 1, -2, -2 < ; M = Transpose @8 a, b, c <D ; M •• MatrixForm i

k jjjjj jj

3 -1 1 1 -1 -2 2 0 -2

y { zzzzz zz

Dl = 88 1, 0, 0 < , 8 0, 1, 0 < , 8 0, 0, 0 << ; Dl •• MatrixForm i

k jjjjj jj

1 0 0 0 1 0 0 0 0

y

{ zzzzz

zz

A = M.Dl.Inverse @ M D ; A •• MatrixForm i

k jjjjj jjjjj

€€€€ 4 ₅ €€€€ ¹ ₅ €€€€ ¹ ₅

€€€€ 2 ₅ €€€€ ³ ₅ - €€€€ ² ₅

€€€€ 2 ₅ - €€€€ ² ₅ €€€€ ³ ₅ y

{ zzzzz zzzzz A •• N •• MatrixForm i

k jjjjj jj

0.8 0.2 0.2 0.4 0.6 -0.4 0.4 -0.4 0.6

y { zzzzz zz

Ÿ b

OA = 8 1, 0, 0 < ; OB = 8 0, 1, 0 < ; OC = 8 0, 0, 1 < ; A.OA

9 4

€€€€ 5 , 2

€€€€ 5 , 2

€€€€ 5 =

N @ % D

8 0.8, 0.4, 0.4 <

A.OB

9 1

€€€€ 5 , 3

€€€€ 5 , - 2

€€€€ 5 =

N @ % D

8 0.2, 0.6, -0.4 <

A.OC

9 1

€€€€ 5 , - 2

€€€€ 5 , 3

€€€€ 5 =

N @ % D

8 0.2, -0.4, 0.6 <

Ÿ c

Norm @ Cross @ OB - OA, OC - OA DD • 2

•!!!! 3

€€€€€€€€€€

2 N @ % D 0.866025

Norm @ Cross @ A.OB - A.OA, A.OC - A.OA DD • 2 3 •!!!! 3

€€€€€€€€€€€€€€

10

N @ % D 0.519615

3

Remove @ "Global`*" D

Ÿ a

88 8, 1, -2 < , 8 4, 5, -4 < , 8 1, -1, 5 << •• MatrixForm i

k jjjjj jj

8 1 -2 4 5 -4 1 -1 5

y { zzzzz zz

M3 = 1 • 3 88 8, 1, -2 < , 8 4, 5, -4 < , 8 1, -1, 5 << ; M3 •• MatrixForm i

k jjjjj jjjjj

€€€€ 8 ₃ €€€€ ¹ ₃ - €€€€ ² ₃

€€€€ 4 ₃ €€€€ ⁵ ₃ - €€€€ ⁴ ₃

€€€€ 1 ₃ - €€€€ ¹ ₃ €€€€ ⁵ ₃ y

{ zzzzz zzzzz EW = Eigenvalues @ M3 D 8 3, 2, 1 <

EV = Eigenvectors @ M3 D

88 1, 1, 0 < , 8 1, 0, 1 < , 8 0, 2, 1 <<

EV @@ 1 DD • Norm @ EV @@ 1 DDD

9 1

€€€€€€€€€€ •!!!! 2 , 1

€€€€€€€€€€ •!!!! 2 , 0 =

N @ % D

8 0.707107, 0.707107, 0. <

EV @@ 2 DD • Norm @ EV @@ 2 DDD 9 €€€€€€€€€€ •!!!! 1 2

, 0, €€€€€€€€€€ •!!!! 1 2 =

N @ % D

8 0.707107, 0., 0.707107 <

EV @@ 3 DD • Norm @ EV @@ 3 DDD

9 0, 2

€€€€€€€€€€ •!!!! 5 , 1

€€€€€€€€€€ •!!!! 5 =

N @ % D

8 0., 0.894427, 0.447214 <

88l, 0, 0 < , 8 0, l, 0 < , 8 0, 0, l<< •• MatrixForm i

k jjjjj jj

l 0 0

0 l 0 0 0 l

y { zzzzz zz

Id @l_ D := 88l, 0, 0 < , 8 0, l, 0 < , 8 0, 0, l<<

Ÿ b

p @l_ D := Det @ M3 - Id @lDD ; p @lD Š 0 6 - 11 l + 6 l ² - l ³ Š 0

p @lD u •• Expand 6 u - 11 u l + 6 u l ² - u l ³

H p @ l D u •• Expand L • . 8H l ^ 3 L ® M3.M3.M3, H l ^ 2 L ® M3.M3, l ® M3, u -> Id @ 1 D< •• MatrixForm i

k jjjjj jj

0 0 0 0 0 0 0 0 0

y { zzzzz zz

exp = H u H p @lD - 6 L • H-6 lL •• Expand L • . 8 u l ® l, u l ^ 2 ® l^ 2 <

€€€€€€€€€€€ 11 u

6 - l + €€€€€€€ l ² 6 exp

€€€€€€€€€€€ 11 u

6 - l + €€€€€€€ l ² 6

H exp • . 8 u -> Id @ 1 D , H l^ 2 L ® M3.M3, l ® M3 <L •• MatrixForm i

k jjjjj jjjjj

€€€€€€ ₁₈ 7 - €€€€€€ ₁₈ ¹ €€€€ ¹ ₉ - €€€€ ⁴ ₉ €€€€ ⁷ ₉ €€€€ ⁴ ₉ - €€€€ ¹ ₆ €€€€ ¹ ₆ €€€€ ² ₃

y

{ zzzzz zzzzz

res = H exp • . 8H l^ 2 L ® M3.M3, l ® M3, u -> Id @ 1 D<L ; res •• MatrixForm i

k jjjjj jjjjj

€€€€€€ ₁₈ 7 - €€€€€€ ₁₈ ¹ €€€€ ¹ ₉ - €€€€ ⁴ ₉ €€€€ ⁷ ₉ €€€€ ⁴ ₉ - €€€€ ¹ ₆ €€€€ ¹ ₆ €€€€ ² ₃

y

{ zzzzz zzzzz

Inverse @ M3 D •• MatrixForm i

k jjjjj jjjjj

€€€€€€ ₁₈ 7 - €€€€€€ ₁₈ ¹ €€€€ ¹ ₉ - €€€€ ⁴ ₉ €€€€ ⁷ ₉ €€€€ ⁴ ₉ - €€€€ ¹ ₆ €€€€ ¹ ₆ €€€€ ² ₃

y

{

zzzzz

zzzzz

res == Inverse @ M3 D

True

Inverse @ M3 D == 1 • 6 M3.M3 - M3 + 11 • 6 Id @ 1 D True

4

Remove @ "Global`*" D

Ÿ a

M4 = 88 0, 1, 0, 0 < , 8 0, 0, 1, 0 < , 8 0, 0, 0, 1 < , 8 0, 0, 0, 0 << ; M4 •• MatrixForm i

k jjjjj jjjjj j

0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 y

{ zzzzz zzzzz z

H M4.M4 L •• MatrixForm i

k jjjjj jjjjj j

0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 y

{ zzzzz zzzzz z

H M4.M4.M4 L •• MatrixForm i

k jjjjj jjjjj j

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 y

{ zzzzz zzzzz z

H M4.M4.M4.M4 L •• MatrixForm i

k jjjjj jjjjj j

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y

{ zzzzz zzzzz z

Ÿ b

H M4.M4 L . H M4.M4 L •• MatrixForm i

k jjjjj jjjjj j

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y

{ zzzzz zzzzz z

H M4 L . H M4.M4.M4 L •• MatrixForm i

k jjjjj jjjjj j

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y

{

zzzzz

zzzzz

z

H M4 L . H M4.M4.M4.M4 L •• MatrixForm i

k jjjjj jjjjj j

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y

{ zzzzz zzzzz z

5

Remove @ "Global`*" D

Ÿ a

M5 = 88 1, 2, 3, 4, 5 < + 8 1, 1, 2, 1, 2 < , 8 3, 2, 1, 5, 4 < ,

8 7, 2, -3, 7, 2 < ,

8 1, 1, 2, 1, 2 < - 8 1, 2, 3, 4, 5 << ; M5 •• MatrixForm i

k jjjjj jjjjj j

2 3 5 5 7

3 2 1 5 4

7 2 -3 7 2

0 -1 -1 -3 -3 y

{ zzzzz zzzzz z

b1 = 8 4 + 2, 3, 1, 2 - 4 <

8 6, 3, 1, -2 <

b2 = 8 4 + 2, 3, 0, 2 - 4 <

8 6, 3, 0, -2 <

Ÿ b

x = 8 x1, x2, x3, x4, x5 < ; Solve @ M5.x Š b2, x D 8<

Solve @ M5.x Š b1, x D

99 x1 ® - 1

€€€€ 4 + 3 x4

€€€€€€€€€€€

4 + 3 x5

€€€€€€€€€€€

4 , x2 ® 7

€€€€ 4 - 17 x4

€€€€€€€€€€€€€€

4 - 13 x5

€€€€€€€€€€€€€€

4 , x3 ® 1

€€€€ 4 + 5 x4

€€€€€€€€€€€

4 + x5

€€€€€€€

4 ==

Ÿ c

Fall für b1: Dim Lösungsraum = 2

Ÿ d

Rang = Ordnung - Dimension = 5 - 2 = 3

6

Remove @ "Global`*" D

S = 88-2, 2 • Sqrt @ 3 D< , 8 2 • Sqrt @ 3 D , 2 << ; S •• MatrixForm i

k jjjjj jj

-2 €€€€€€€€ •!!!!! ² 3

€€€€€€€€ •!!!!! 2 3 2 y { zzzzz zz

X = 88 x1 < , 8 x2 <<

88 x1 < , 8 x2 <<

Flatten @ Transpose @ X D .S.X •• Simplify D@@ 1 DD

-2 x1 ² + 4 x1 x2

€€€€€€€€€€€€€€€€€€ •!!!! 3 + 2 x2 ²

syst = Eigensystem @ S D •• Simplify

99- 4

€€€€€€€€€€ •!!!! 3 , 4

€€€€€€€€€€ •!!!! 3 = , 99-2 - •!!!! 3 , 1 = , 9 2 - •!!!! 3 , 1 ===

Dl = 88 syst @@ 1 DD@@ 1 DD , 0 < , 8 0, syst @@ 1 DD@@ 2 DD<< ; Dl •• MatrixForm i

k jjjjj jj

- €€€€€€€€ •!!!!! ⁴ 3 0 0 €€€€€€€€ •!!!!! ⁴ 3

y { zzzzz zz

n @ v_ D := v • Norm @ v D

M = Transpose @8 n @ syst @@ 2 DD@@ 1 DDD , n @ syst @@ 2 DD@@ 2 DDD <D ; M •• MatrixForm i

k jjjjj jjjjj

-2-

•!!!!! 3

€€€€€€€€€€€€€€€€€€€€€€€€€€ "################################# 1+I 2+ •!!!!! 3 M

²

2- •!!!!! 3

€€€€€€€€€€€€€€€€€€€€€€€€€€ "################################# 1+I 2- •!!!!! 3 M

²

€€€€€€€€€€€€€€€€€€€€€€€€€€ "################################# 1+I 2+ 1 •!!!!! 3 M

²

€€€€€€€€€€€€€€€€€€€€€€€€€€ "################################# 1+I 2- 1 •!!!!! 3 M

²

y

{ zzzzz zzzzz

M •• N •• MatrixForm

J -0.965926 0.258819

0.258819 0.965926 N

N @ Inverse @ M DD Š N @ Transpose @ M DD True

Det @ M D •• N

-1.

Y = Inverse @ M D .X •• Simplify

99- 1

€€€€ 2

"################# 2 + •!!!! 3 I x1 + I-2 + •!!!! 3 M x2 M= , 9 1

€€€€ 2

"################# 2 - •!!!! 3 I x1 + I 2 + •!!!! 3 M x2 M==

Flatten @ Transpose @ X D .S.X •• Simplify D@@ 1 DD ==

Flatten @ Transpose @ Y D .Dl.Y •• Simplify D@@ 1 DD True

Y1 = 88 y1 < , 8 y2 <<

88 y1 < , 8 y2 <<

Flatten @ Transpose @ Y1 D .Dl.Y1 D @@ 1 DD

- 4 y1 ²

€€€€€€€€€€€€€€ •!!!! 3 + 4 y2 ²

€€€€€€€€€€€€€€ •!!!! 3

N @ % D Š 52

-2.3094 y1 ² + 2.3094 y2 ² Š 52

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