)
rank ) 15 ʃ a 1 x + b 1 y + c 1 z = 0 | a2x + b2y + c2z = 0
(x,y,z) α1, b1, c1, α2, b2, c2 [1]  a 1  = a 2 = 0 b1y + c1z = 0 b2y + c2z = 0 b1c2 — b2c1  ≠ 0 rank [2]  (a1,a2) ≠  (0,0) α1  ≠  0 rank b2a1)y + (c1a2 − c2a1)z = 0 y = z = 0, x
(b1a2 - 0 y, z [2-a]  b1a2 — b2a1 = 0 (c 1a2 − c 2a1)z = 0        z = 0
● b 1 = 0 ● b1 ≠ 0
[2-b]  c1a2 = c2a1 = 0 [2-a] ● c 1 = 0
● c1 ≠ 0 y = 0, x : z = c 1 : (−a 1 ) y = [2-c]  [2-a]  [2-b] y : z = (c1a2 - c2a1) : (b2a1 - b1a2) x = (b1c2 − b2c1)t ) (c 1 a2 − c 2 a1)t, z = (b 2 a1 − b 1 a2)t  (t ● b1c2 − b2c1 = 0 ● b1c2 — b2c1 ≠ 0 x = 0, y : z = (c1a2 − c2a1) : (b2a1 − b1a2) x : y : z = (b1c2 − b2c1) : (c1a2 − c2a1) : (a1b2 − a2b1) 11
16  (a × b) × c + (b × c) × a + (c × a) × b = 0 a = (a1,a2,a3), b = (b1,b2,b3), c = (c1,c2,c3) a × b = (a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1) (a xb)×c = (c 3 (a b 1 − a 1b )− c 2 (a 1b 2 − a b 1 ), c 1 (a 1b 2 − a b 1 )− c 3 (a b 3 − a b 2 ), c 2 (a b 3 − a b 2 )− c 1 (a 3b 1 − a 1b 3 )) (b×c)×a = (a3(b3c1-b1c3)-a2(b1c2-b2c1), a1(b1c2-b2c1)-a3(b2c3-b3c2), a2(b2c3-b3c2)-a1(b3c1-b1c3)) (c×a)×b = (b3(c3a1−c1a3)−b2(c1a2−c2a1), b1(c1a2−c2a1)−b3(c2a3−c3a2), b2(c2a3−c3a2)−b1(c3a1−c1a3)) 0 Jacobi (b × c) × a = (a · b)c − (c · a)b
B) C) 3 (2)  3
??
det(a 1 , a 2 , a 3 )
R3
C
2 det 2 n = 2 18
(1)  (af + cd − be)² (3) −(a − b) 4 (2)  (a − x)(b − y)(c − z) (4)  4(x³ + y³ + z³ − 3xyz) 19 (1)
(x,y,z) = (t, −t, −t)  (t ) 1 (2)  A ∈ M(3,C) A x = 0 rank(A) = 3 – s
Cramer [2]  rank(A) = 2 A. (a ;j )
1 ≤ j 1 < j 2 ≤ 3, a 1 j 1  ≠ 0, a 2 j 2  ≠ 0 0
T (a,b,c)̸= (0,0,0)) ax 1 + bx 2 + cx 3 = 0 ( x E R 3 IIB 20　R² / cos θ sin θ (s cos( ϕ + (s cos ϕ, s sin ϕ)  (s > 0,  ϕ ∈ [0,2π)) (x,y) ∈ R² (rx,ry) θ), s sin(ϕ + θ)) 21 A ∈ M(n,C) E n A. P 1 , . . , P t , Q 1 , . . , Q s P t P t − 1 . .  P 2 P 1 AQ 1 Q 2 . . Q s − 1 Q s = E n 22  4 (1, 1, 2),  (− 1, 2, 3),  (2, 4, 5),  (0, 1, 1) 44 (2 0 2してθ ) l ) A(1,1,2),  B(−1,2,3),  C(2,4,5),  D(0,1,1)          a := AB = (−2,1,1),  b := AC = (1,3,3),  c := AD = (−1,0, −1) |a × b| = |a||b|sin θ  ( θ   a   b        ) a, b SS φ                      c     a × b (π/2) − ϕ   |c|sin((π/2) − ϕ) = |c|cos ϕ SS c S
77 23  {1,2,3,4,5,6,7} = AU BU C A \ ( B U C ) = {1}      C !!(A N B ∩ C ) = 1 B \ ( A ∪ C ) 20 C C C' = {2,5,6,7},  {3,4,6,7},  {2,3,4,5,6} 20 __ 1 ∈ A \ ( B ∪ C ) 3 [1] C = {2,5,6,7} 0 Ç B\(AUC) ⊆ {3,4}               B\(AUC) = {3}, {4}             ANB B \ ( A U C ) = {3,4} An B !(An BnC) = 1 An BnC = {7} 7 A N B [2]  C = {3,4,6,7} 0 Ç B \ ( A U C ) ⊆ {2,5}               B \ ( A U C ) = {2} B \ ( A U C ) = {5} A ∩ B = {2,3} An BnC = {3} B\(AUC) = {2,5}                 AnB 7 AnBuc' = {7} 7 C [3] C = {2,3,4,5,6}             B \(AUC) = {7}              ANB 2 !!(An BnC) = 1 C = {2,5,6,7}, {3,4,6,7} σ  E  S3 24   f (x 1,x 2,x 3)                 x 1,x 2,x 3 σf(x 1, x 2, x 3) := f(x σ(1), x σ(2), x σ(3) ) (1)  f(x1,x2,x3) = (x1 − x2)(x1 − x3)(x2 − x3)           σf(x1,x2,x3) S'3 (2)  σf(x1,x2,x3) = f(x1,x2,x3)
14 sgn(σ)   +1 A3
S n
A3 #An = 1 #Sn 25
(1) 0
26
?? ) 27
(rank)
28 4 S4 24 | 24 . . . ...(1,2), (1,3), (1,4), (2,3), (2,4), (3,4)し...(1,2,3), (1,3,2), (1,2,4), (1,4,2), (1,3,4), (1,4,3), (2,3,4), (2,4,3) 4 ...(1,2,3,4), (1,2,4,3), (1,3,2,4), (1,3,4,2), (1,4,2,3), (1,4,3,2) 2 ...(1,2)(3,4), (1,3)(2,4), (1,4)(2,3) . . 29 28 4 × 4
A = (α έ ε )
28 4 | det A = a11a22a33a44 — a12a21a33a44 — a13a22a31a44 — a14a22a33a41 — a11a23a32a44 — a11a24a33a42 — α11α22α34α43 + α12α23α31α44 + α13α21α32α44 + α12α24α33α41 + α14α21α33α42 + α13α22α34α41 + α14022031043 + α11023034042 + α11024032043 − α12023034041 − α12024031043 − α13024032041 — α13021034042 — α14023031042 — α14021032043 + α12021034043 + α13024031042 + α14023032041 4×4 etc.) 30
7
31 A ≠ E A³ = E A ∈ M(2, Z) ←⇒ det A = ±1 (⇒)   B ∈ M (2, Z ) det( E 2 ) = 1 AB = BA = E2 det(A) det(B) = det(AB) = det( A ), det( B ) ∈ Z det( A ) = ±1 )