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- ゆりな ももき
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1 ver Web
2 n (1) (2) (3) 61 2
3 Schmidt 67
4 1 11 X Y X Y X Y f f; X Y 111 X = Y = R X x Y x 2 f : R R (x x 2 ) x x X Y X x Y e x f : X Y (x e x ) 113 f : X Y 1) x, x X x x f(x) f(x ) f 2) y Y f(x) = y x X f 3) f f 4
5 G (x, y) x y G (x, y) x y 1) x, y, z G (x y) z = x (y z) 2) x G x e = e x = x e G 3) x G x y = y x = e y G 2) e G e e 2) e = e e = e 2) e G G 1 G 1 3) y G x G y y 3) y = y 1 G = y (x y) = (y x) y = 1 G y = y 3) y G x G x G ; 1) g G (g 1 ) 1 = g, 2) g, h G (g h) 1 = h 1 g R 1 x R x x Z = {0, ±1, ±2, ±3, } 0 x Z x Z 125 X X X S(X) S(X) σ, τ σ τ X X S(X) X 1 X S(X) σ S(X) σ 1 S(X)
6 6 1 1) S(X) σ, τ, ρ (σ τ) ρ = σ (τ ρ) 2) S(X) σ σ 1 X = 1 X σ = σ 3) S(X) σ σ σ 1 = σ 1 σ = 1 X S(X) 1 X σ S(X) σ 125 X n S(X) n 3 n G 1) x G x x 1 G G 2) g G G G x x g x g x [ ] 1) (x 1 ) 1 = x 122 2) x, x G x g = x g g 1 x = x x x g y G x = x g 1 G x g = y x x g x x g x g x 127 G, H x, y G f(x y) = f(x) f(y) f : G H G H 128 f : G H 1) f(1 G ) = 1 H 2) x G f(x 1 ) = f(x) 1
7 13 7 [ ] 1) 1 G 1 G = 1 G f f(1 G ) f(1 G ) = f(1 G ) f(1 G ) 1 f(1 G ) = f(1 G ) 1 f(1 G ) = 1 H 2) x x 1 = 1 G f 1) f(x) f(x 1 ) = f(1 G ) = 1 H f(x) 1 f(x 1 ) = f(x) A a, b A a + b A ab A A 1) A (a, b) a + b 0 2) a, b, c A a(bc) = (ab)c 3) 1 A a A 1a = a1 = a 4) a, b A ab = ba 5) a, b, c A a(b+c) = ab+ac 3) 1 A 0a = (0 + 0)a = 0a + 0a a A 0a = 0 1 = 0 A = {0} Z = {0, ±1, ±2, ±3, } Q R C 133 A X a 0 + a 1 X + a 2 X a n X n (a i A, n = 0, 1, 2, 3, )
8 8 1 A[X] f(x) = i 0 a ix i, g(x) = i 0 b ix i A[X] f(x) + g(x) A[X] f(x)g(x) A[X] f(x) + g(x) = (a i + b i )X i, f(x)g(x) = a i b j X k i 0 k 0 i+j=k A[X] A[X] A- 134 ı Z[ı] = {a + bı a, b Z} Z[ı] 135 A a A ab = 1 b A a A A A A A A Z Z[ı] Z = {±1}, Z[ı] = {±1, ±ı} A A[X] A[X] = A 137 A A = {0 a A} A 0 A 138 Q R C ı Q(ı) = {a + bı a, b Q} Z
9 2 K K Q R C 21 K a ij (1 = 1, 2,, m, j = 1, 2,, n) m n a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn K (m, n)- a ij A (i, j)- i a i1, a i2,, a in A i j a 1j, a 2j,, a mj A j A (i, j)- A ij K (m, n)- M m,n (K) 0 (m, n)- O m,n (n, n)- n n n n A A 11, A 22,, A nn A 1 0 n n I n I n = K n M n (K) 9
10 (n, m)- A, B M m,n (K) A + B M m,n (K) (A + B) ij = A ij + B ij (i = 1, 2, m, j = 1, 2,, n) A + B (i, j)- A (i, j)- B (i, j)- 1) A, B, C M m,n (K) (A + B) + C = A + (B + C), 2) A, B M m,n (K) A + B = B + A, 3) A M m,n (K) A + 0 mn = A λ K A M m,n (K) A λ λ A M m,n (K) (λ A) ij = λ A ij (i = 1, 2, m, j = 1, 2,, n) λ A A λ 1) λ K A, B M m,n (K) λ (A+B) = λ A+λ B, 2) λ, µ K A M m,n (K) (λ + µ) A = λ A + µ A, 3) A M mn (K) 0 A = 0 mn 23 A M lm (K) B M mn (K) AB M ln (K) m (AB) ij = A ik B kj (i = 1, 2, l, j = 1, 2,, n) k=1 A (l, m)- B (m, n)- AB AB (l, n) ) A, B, C AB BC (AB)C = A(BC), 2) A M m,n (K) I m A = AI n = A, 3) A M lm (K) B, C M mn (K) A(B + C) = AB + AC,
11 24 11 n M n (K) A, B M n (K) A+B M n (K) AB M n (k) [ ] [ ] A =, B = M 2 (R) [ ] [ ] AB =, BA = AB BA 232 (m, n)- A A t A t A (n, m)- t A (i, j)- A (j, i)- (l, m)- A (m, n)- B t (AB) = t B t A 233 n 0 n 0 a 11 a 12 a 13 a 1n 0 a 22 a 23 a 2n 0 0 a 33 a 3n : a nn n n n A M n (K) AB = BA = I n n B M n (K) A n A M n (K) n AB = BA = I n (21) n B M n (K) A n B M n (K) AB = B = I n B = B I n = B (AB) = (B A)B = I n B = B B A A 1
12 ) I n M n (K) I 1 n = I n 2) A M n (K) A 1 M n (K) (A 1 ) 1 = A 3) A, B M n (K) AB M n (K) (AB) 1 = B 1 A 1 K [ ] 2 0 A = 0 1 [ ] M 2 (Q) A = M 2 (Z) 0 1 [ ] A = M 2 (Z) n A M n (K) AX = Y A = I n n X, Y M n (K) A X = Y = A n X A = I n X m X m = 0 A A 1 = I n + X + X X m 1
13 3 K K Q R C 31 n 311 1) τ S(X) σ σ τ σ τ σ S(X) S(X) 2) σ σ 1 S(X) S(X) [ ] 1) σ, σ S(X) σ τ = σ τ τ 1 (σ τ) τ 1 = (σ τ) τ 1 (σ τ) τ 1 = σ (τ τ 1 ) = σ 1 X = σ (σ τ) τ 1 = σ σ = σ σ σ τ ρ S(X) α = ρ τ 1 S(X) σ τ = (ρ τ 1 ) τ = ρ (τ τ 1 ) = ρ 1 X = ρ σ σ τ σ τ σ 2) σ S(X) (σ 1 ) 1 = σ σ, σ S(X) σ 1 = σ 1 σ = σ ρ S(X) σ = ρ 1 S(X) σ 1 = ρ σ σ 1 n X = {1, 2, 3,, n} S(X) S n n n S n σ 1, 2, 3,, n σ(1) = i 1, σ(2) = i 2, σ(3) = i 3,, σ(n) = i n ( ) n σ = (31) i 1 i 2 i 3 i n 13
14 14 3 σ X = {1, 2, 3,, n} X (i 1, i 2, i 3,, i n ) (1, 2, 3,, n) (1, 2, 3,, n) (i 1, i 2, i 3,, i n ) (31) S n σ S n n! (31) (i 1, i 2, i 3,, i n ) (1, 2, 3,, n) ( ) j 1 j 2 j 3 j n σ = n σ ( ) σ n = j 1 j 2 j 3 j n 1 X 1 n ( ) n 1 n = n (31) ( ) ( ) σ =, τ = S ( ) ( ) σ τ =, τ σ = n S n σ τ τ σ S n n X 1, X 2,, X n P (X 1, X 2,, X n ) P (X 1, X 2,, X n ) = (X i X j ) P (X 1, X 2,, X n ) 1 i<j n = (X 1 X 2 )(X 1 X 3 )(X 1 X 4 ) (X 1 X n ) (X 2 X 3 )(X 2 X 4 ) (X 2 X n ) (X 3 X 4 ) (X 3 X n ) (X n 1 X n ) P (X 1, X 2,, X n ) 1 σ S n P (X σ(1), X σ(2),, X σ(n) ) = sign(σ) P (X 1, X 2,, X n ) sign(σ) = ±1 σ
15 ) σ, τ S n sign(σ τ) = sign(σ)sign(τ) 2) sign(1 n ) = 1 3) σ S n sign(σ 1 ) = sign(σ) [ ] 1) P (X σ τ(1),, X σ τ(n) ) = sign(σ τ)p (X 1,, X n ) P (X σ τ(1),, X σ τ(n) ) = P (X σ(τ(1)),, X σ(τ(n)) ) = sign(σ)p (X τ(1),, X τ(n) ) = sign(σ)sign(τ)p (X 1,, X n ) sign(σ τ) = sign(σ)sign(τ) 2) 3) σ σ 1 = 1 n 2) sign(σ σ 1 ) = sign(1 n ) = 1 1) sign(σ σ 1 ) = sign(σ)sign(σ 1 ) sign(σ 1 ) = sign(σ) 1 = sign(σ) 313 n n n = 2 ( ) ( ) σ sign(σ) 1 1 n = 3 σ ( ) ( ) ( ) sign(σ) σ ( ) ( ) ( ) sign(σ) n A M n (K) det A = σ S n sign(σ)a 1,σ(1) A 2,σ(2) A n,σ(n)
16 16 3 A [ ] a b det = ad bc, (32) c d a b c det d e f = ael bdl ceg afh + bfg + cdh (33) g h l 2 3 a b c d e f g h l a b c d e f g h l n 322 n A t A A det( t A) = det A det A = σ S n sign(σ)a σ(1),1 A σ(2),2 A σ(n),n [ ] σ S n ( ) ( ) 1 2 n j 1 j 2 j n σ = = i 1 i 2 i n 1 2 n ( ) σ n = j 1 j 2 j n A 1,σ(1) A 2,σ(2) A n,σ(n) = A 1,i1 A 2,i2 A n,in = A j1,1a j2,2 A jn,n = A σ 1 (1),1A σ 1 (2),2 A σ 1 (n),n
17 σ σ 1 S n S n σ S n τ = σ 1 S n 312 sign(σ) = sign(σ 1 ) det A = σ S n sign(σ)a 1,σ(1) A 2,σ(2) A n,σ(n) = τ S n sign(τ)a τ(1),1 A τ(2),2 A τ(n),n n a 11 a 12 a 1n a 21 a 22 a 2n A = n a n1 a n2 a nn a i = [a i1, a i2,, a in ] (i = 1, 2,, n) (34) a 1 a 2 A = A n a j = a 1j a 2j a nj a n (j = 1, 2,, n) A = [a 1, a 2,, a n ] 323 1) λ K a 1 a r 1 det λa r = λ det a r+1 a n a 1 a r 1 a r a r+1 a n
18 18 3 n λ λ 2) a 1 a r 1 det a r + a r = det a r+1 a n a 1 a r 1 a r a r+1 a n + det a 1 a r 1 a r a r+1 a n n [ ] 1) n (34) r λa rj = σ S n sign(σ)a 1,σ(1) λa r,σ(r) a n,σ(n) = λ = σ S n sign(σ)a 1,σ(1) a r,σ(r) a n,σ(n) 2) n (34) r a rj = a rj + a rj = σ S n sign(σ)a 1,σ(1) a r,σ(r) a n,σ(n) = sign(σ)a 1,σ(1) a r,σ(r) a n,σ(n) σ S n + sign(σ)a 1,σ(1) a r,σ(r) a n,σ(n) σ S n = 324 τ S n det[a τ(1), a τ(2),, a τ(n) ] = sign(τ) det[a 1, a 2,, a n ] τ τ
19 32 19 [ ] [a τ(1), a τ(2),, a τ(n) ] (i, j)- a i,τ(j) = σ S n sign(σ)a 1,τ σ(1) a 2,τ σ(2) a n,τ σ(n) (35) 311 σ τ σ S n S n σ S n ρ = τ σ S n σ = τ 1 ρ (35) ρ S n sign(τ 1 ρ)a 1,ρ(1) a 2,ρ(2) a n,ρ(n) (36) 312 sign(τ 1 σ) = sign(τ)sign(σ) (36) sign(τ) ρ S n a 1,ρ(1) a 2,ρ(2) a n,ρ(n) ) λ K det[a 1,, a r 1, λa r, a r+1,, a n ] =λ det[a 1,, a r 1, a r, a r+1,, a n ] n λ λ 2) det[a 1,, a r 1, a r + a r, a r+1,, a n ] = det[a 1,, a r 1, a r, a r+1,, a n ] + det[a 1,, a r 1, a r, a r+1,, a n ] n 326 τ S n det a τ(1) a τ(2) a 1 a 2 = sign(τ) det a τ(n) a n τ τ
20 n A 2 2 det A = 0 [ ] A i j i < j τ S n k k i, j τ(k) = j k = i i k = j τ i j sign(τ) = 1 sign(σ) = 1 σ S n A n σ(ρ) = 1 ρ S n {σ τ σ A n } A n τ 311 σ σ τ A n A n τ S n A n A n τ A n A n τ = det A = σ A n a 1,σ(1) a 2,σ(2) a n,σ(n) + σ A n sign(σ τ)a 1,σ τ(1) a 2,σ τ(2) a n,σ τ(n) a 1,σ τ(1) a 2,σ τ(2) a n,σ τ(n) = a τ 1 (1),σ(1)a τ 1 (2),σ(2) a τ 1 (n),σ(n) A i j τ a 1,σ(1) a 2,σ(2) a n,σ(n) det A = σ A n a 1,σ(1) a 2,σ(2) a n,σ(n) = 0 σ A n a 1,σ(1) a 2,σ(2) a n,σ(n) n A, B det(ab) = (det A)(det B) [ ] n A, B a 11 a 12 a 1n b 1 a 21 a 22 a 2n A =, B = b 2 a n1 a n2 a nn b n
21 33 21 b i B i AB n k a 1=1 1,k 1 b k1 n k AB = 2 =1 a 2,k 2 b k2 n k a n=1 n,k n b kn 323 n n n det(ab) = a 1,k1 a 2,k2 a n,kn det k 1 =1 k 2 =1 k n =1 b k1 b k2 (37) b n,kn k 1, k 2,, k n 327 det b k1 b k2 b kn = 0 (37) a 1,σ(1) a 2,σ(2) a n,σ(n) det σ S n 326 b σ(1) b σ(2) b σ(n) σ S n sign(σ)a 1,σ(1) a 2,σ(2) a n,σ(n) det B = det A det B 33 3 (33) a, b, c a b c [ e det d e f = a det h g h l ] [ f d b det l g ] [ f d + c det l g ] e h
22 22 3 b, e, h a b c [ d det d e f = b det g g h l ] [ f a + e det l g ] [ c a h det l d ] c f (i, j)- det (i j (2, 2)- ) n 331 a 11 a 1,n 1 a 1n det a 11 a 1,n 1 a n 1,1 a n 1,n 1 a n 1,n = det a nn a n 1,1 a n 1,n a nn a 11 a 1,n 1 0 det a 11 a 1,n 1 a n 1,1 a n 1,n 1 0 = det a nn a n 1,1 a n 1,n 1 a n1 a n,n 1 a nn [ ] 322 n A det A = σ S n sign(σ)a 1,σ(1) A n 1,σ(n 1) A n,σ(n) A n,σ(n) 0 σ(n) = n det A = σ sign(σ)a1,σ(1) A n 1,σ(n 1) a nn
23 33 23 σ σ(n) = n σ Sn σ ( ) 1 2 n 1 n σ = S n i 1 i 2 i n 1 n (i 1, i 2,, i n 1 ) (1, 2,, n 1) ( ) σ 1 2 n 1 = S n 1 i 1 i 2 i n 1 n 1 P (x 1, x 2,, x n 1, x n ) = P (x 1, x 2,, x n 1 ) (x i x n ) sign(σ) = sign(σ ) (i 1, i 2,, i n 1 ) (1, 2,, n 1) det A = i=1 τ S n 1 sign(τ)a 1,τ(1) A n 1,τ(n 1) a nn 332 a 11 a 12 a 1n 0 a 22 a 2n det = a 11a 22 a nn, 0 0 a nn a a 12 a 22 0 det = a 11a 22 a nn n a n1 a n2 a nn a 11 a 12 a 1n a 21 a 22 a 2n A = a n1 a n2 a nn
24 24 3 n a 1n a 2n a nn = a 1n a 2n a nn 325 det A = n i=1 det a 11 a 1,n 1 0 a i 1,1 a i 1,n 1 0 a i1 a i,n 1 a i,n a i+1,1 a i+1,n 1 0 a n1 a n,n 1 0 i i + 1 i + 1 i det a 11 a 1,n 1 0 a i 1,1 a i 1,n 1 0 a i1 a i,n 1 a i,n a i+1,1 a i+1,n 1 0 a n1 a n,n 1 0 = ( 1) n i det a 11 a 1,n 1 0 a i 1,1 a i 1,n 1 0 a i+1,1 a i+1,n 1 0 a n1 a n,n 1 0 a i1 a i,n 1 a i,n 331 det A = n i=1 ( 1) i+n a in det ( A i n ) (38)
25 34 25 n A n k k k + 1 k + 1 k + 2 k n ( 1) n k (38) ( ) n det A = ( 1) n k ( 1) i+n A i k a ik det i=1 333 n A 1 k n ( ) n det A = ( 1) i+k A i k A ik det i=1 334 n A 1 k n ( ) n det A = ( 1) k+j A k j A kj det j= k k n n 1 34 n A M n (K) AB = BA = I n n B M n (K) AB = I n 328 det A, det B K, det A det B = 1 det A K
26 n A M n (K) n à M n(k) ( ) (Ã) ij = ( 1) i+j A j i det (i, j = 1, 2,, n) à A A A A 342 n A M n (K) à M n(k) Aà = ÃA = (det A)I n [ ] Aà (i, j)- (AÃ) ij = = n A ik (Ã) kj k=1 ( n ( 1) j+k A j k A ik det k=1 ) j A A j i i j 0 i = j A (AÃ) det A i = j ij = 0 i j Aà = (det A)I n (ÃA) ij = = n k=1 k=1 (Ã) ika kj ( n ( 1) k+i A k i A kj det = det(a i j ) det A i = j = 0 i j )
27 35 (1) 27 ÃA = (det A)I n 343 n A M n (K) det A K A A 1 = (det A) 1 Ã [ ] A det A K det A K B = (det A) 1 Ã B K n 342 AB = BA = I n A M n (K) B = (det A) 1 Ã A 35 (1) x 1, x 2,, x n n a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 a n1 x 1 + a n2 x a nn x n = b n (39) a ij b j K n a 11 a 12 a 1n a 21 a 22 a 2n A = M n(k) a n1 a n2 a nn n A = [a 1, a 2,, a n ], a j = a 1j a 2j a nj
28 28 3 b 1, b 2,, b n n b 1 b 2 b = (39) Cramer 351 det A K (39) x r = (det A) 1 det[a 1,, a r 1, b, a r+1,, a n ] (r = 1, 2,, n) [ ] (39) x 1 a 1 + x 2 a x n a n = b det[a 1,, a r 1, b, a r+1,, a n ] [ ] n = det a 1,, a r 1, x k a k, a r+1,, a n = k=1 b n n x k det[a 1,, a r 1, a k, a r+1,, a n ] k=1 k r det[a 1,, a r 1, a k, a r+1,, a n ] = 0 det[a 1,, a r 1, b, a r+1,, a n ] = x r det A det A K (det A) 1 Cramer (39) x 1, x 2,, x n n x = x 1 x 2 x n
29 35 (1) 29 n n (39) Ax = b (310) A 343 det A K A A 1 M n (K) (310) A 1 Ax = I n x = x x = A 1 b 343 x 1 x 2 b 1 = (det b 2 A) 1 Ã (311) x n b n (39) (311) Cramer Cramer det[a 1,, a r 1, b, a r+1,, a n ] 333 r ( n ( 1) i+r A i r b r det i=1 ) n r b 1 b 2 Ã b n
30 4 K K Q R C 41 (m, n)- A M m,n (K) I) A 0 II) A I) A 0 II) A A A A A A = (a 1, a 2,, a n ) 1 2 (a 1, a 2, a 3,, a n ) 2 1 (a 1 + a 2, a 2, a 3,, a n ) 2 1 (a 1 + a 2, a 1, a 3,, a n ) 2 1 (a 2, a 1, a 3,, a n ) 2 1 (a 2, a 1, a 3,, a n ) 30
31 (m, n)- A M m,n (K) [ ] I r (41) 1 r 0 r = 0 [ ] A 0- r = 0 A 0- A 0 (1, 1)- 1 (1, 1) A 1 b 12 b 1n b 21 b 22 b 2n b m1 b m2 b mn (42) 1 b 12 b 13 b 1n 2 3 n 1 b 21 b 31 b m1 2 3 m (42) C 0 (43) C (m 1, n 1)- C [ ] (41) I r 1 0 (43) 0 0 [ ] I r (41) 1 r A A (m, n)- A M m,n (K) A l l l A l A l 412 (m, n)- A M m,n (K) (41) A r A r 0
32 32 4 [ ] (m, n)- r r 0 (P r ) A i 0 λ B l C C B i det C A l C B i det C A l λ A i λ j i j B B l C C B j i j det C A l C B j i det C A l A l λ A l 0 A B l 0 B A A B A l 0 B l 0 A B (P r ) A (P r ) (41) r (41) (P r ) A (P r ) 413 (m, n)- A M m,n (K) (41) 1 r A rank(a) 42 (i, j)- 1 0 n E (n) ij M n (K) 0 λ K
33 i n E (n) i (λ) = I n + (1 λ)e (n) ii = λ (i (i, i)- λ 1 n λ K 1 i, j n (i j) E (n) ij (λ) = I n + λe ij 1 = 1 λ 1 1 (i (j 1 (i, j)- λ 0 E (n) i (λ) E (n) ij (λ) n det E (n) i (λ) = λ, det E (n) ij (λ) = 1 E (n) i E (n) i (λ)e (n) i (µ) = E (n) i (λµ), E (n) (λ)e(n) (µ) = E(n) (λµ) (1) = E (n) (0) = I n ij E (n) i (λ) 1 = E (n) ij i (λ 1 ), E (n) ij ij ij (λ) 1 = E (n) ij ( λ) 421 (m, n)- A M m,n (K)
34 34 4 1) 0 λ K 2) λ K E (m) i (λ)a = A i λ, AE (n) i (λ) = A i λ E (m) ij (λ)a = A i j λ, AE (n) ij (λ) = A j i λ A A [ ] 422 n A M n (K) 1) A 2) det A 0 3) rank(a) = n 4) A [ ] 1) 2) K K A det A 0 4) 1) 1) 3) A K 343 det A rank(a) = n 3) 4) rank(a) = n A I n 421 P 1,, P r, Q 1,, Q s P 1 P r AQ 1 Q s = I n P 1,, P r Q s,, Q 1 A = Pr 1 P 1 1 Q 1 s Q 1 1
35 42 35 A (m, n) A, B M m,n (K) 1) rank(a) = rank(b), 2) A B, 3) P AQ = B m P n Q 424 (m, n)- A M m,n (K) rank(a) = r [ ] I r 0 P AQ = 0 0 m P n Q 423 (m, n) A, B M mn (K) rank(a) = rank(b) B = P AQ P, Q P, Q m + n [ ] [ ] [ ] [ ] [ ] P 0 A 1 m Q 0 P AQ P B P = = 0 I n I n 0 0 I m Q 0 Q 0 [ ] [ ] P 0 Q 0 m + n 0 I n 0 I m m + n [ ] A I m (44) I n 0 m n 425 (m, n) A, B M m,n (K) A B A m + n (44) m n [ ] B P Q 0
36 36 4 P, Q m n B = P AQ A M n (K) n 422 A P i A = P 1 P 2 P r A 1 = Pr 1 P2 1 P1 1 A I n (n, 2n)- (A, I n ) Pr 1 P2 1 P1 1 (A, I n ) = (I n, A 1 ) (45) P 1 i (45) (A, I n ) (I n, A 1 ) A A =
37 43 37 (A, I 3 ) = / /2 1/2 1/ /2 1/2 1/ / /2 1/2 1/ /2 1/2 1/ /2 1/2 1/2 A A 1 =
38 (2) x 1, x 2,, x n K m a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 a m1 x 1 + a m2 x a mn x n = b m (46) a ij, b i K 1) (46) 2) (46) a 11 a 12 a 1n a 21 a 22 a 2n A =, x = x 1 x 2 b 1, b = b 2 a m1 a m2 a mn (46) x n b m Ax = b (47) 441 (46) rank(a, b) = rank(a) a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 (A, b) = a m1 a m2 a mn b m (m, n)- A m b (m, n + 1)-
39 44 (2) 39 [ ] (46) x 1 = λ 1, x 2 = λ 2,, x n = λ n A A = (a 1, a 2,, a n ) λ 1 a 1 + λ 2 a λ n a = b (A, b) 1 n λ 1 λ n n + 1 n (A, b) (A, 0) rank(a) = r A [ ] I r 0 (m, n)- 0 0 (A, 0) A (A, 0) [ ] I r 0 (m, n + 1)- 0 0 rank(ab) = r = rank(a) rank(a, b) = rank(a) = r 424 [ ] E r 0 P AQ = 0 0 m P n Q [ ] Q 0 = 0 1 Q n + 1 [ ] [ Q 0 P (A, b) = [P AQ, P b] = 0 1 E r P b ] (48) c 1 c 2 P b = i > r c i = 0 c i 0 r < i m (48) i c i j i j c m
40 40 4 i c j i r + 1 r + 1 n + 1 (A, b) [ ] E r rank(a, b) = r + 1 rank(a, b) = rank(a) c = 1 c c r x = x 1 x 2 [ ] = Q c 0 x n [ ] [ ] [ ] P Ax = P AQQ 1 E r 0 c c x = = = P b P 1 Ax = b (46) (46) Av = 0 n v Ker(A) (m, n)- A Ker(A) = x = v = x 1 x 2 x n v 1 v n Av = 0 (49) x 1, x x 2 = x n (46) A(x x) = Ax Ax = b b = 0 x x = v Ker(A) x = x + v (v Ker(A))
41 44 (2) 41 x = x 1 x 2 (46) v Ker(A) x n x = x + v = x 1 x 2 x n Ax = A(x + v) = Ax + Av = b x (46) 442 x = x 1 x 2 (46) (46) {x + v v Ker(A)} (m, n)- A M m,n (K) Ker(A) 1) Ker(A) 0 2) v Ker(A) λ K λv Ker(A) 3) u, v Ker(A) u + v Ker(A) Ker(A) 443 rank(a) = r n Q = (q 1, q 2,, q n ) GL n (K) (q k K n ) x n Ker(A) = { n r } λ k q r+k λ k K k=1
42 42 4 Ker(A) λ 1 q r λ n r q n (λ i K) r = n Ker(A) = {0} [ ] rank(a) = r 424 [ ] I r 0 P AQ = 0 0 m P n Q Q 1 [ ] I r 0 P A = Q 1 (410) 0 0 P n v Av = 0 P Av = 0 (410) y = Q 1 v [ ] I r 0 y = y 1 y n y 2 y = y 1 [ ] I r 0 y = y r n v Av = 0 0 v = Qy, y = 0 λ 1 λ n r r λ 1,, λ n r K Q = (q 1, q 2,, q n ) v = Qy = λ 1 q r+1 + λ 2 q r λ n r q n n i=1 λ i q i = i=1 µ i q i (λ i, µ i K) (411)
43 44 (2) 43 λ 1 = µ 1, λ 2 = µ 2,, λ n = µ n n u, v u = λ 1, v = µ 1 λ n µ n (411) Qu = Qv Q Q 1 u = v Q = (q 1, q 2,, q n ) 425 [ ] I r 0 P AQ = (r = rank(a)) 0 0 P GL m (K), Q GL n (K) [ ] [ ] I r 0 P 0 A Q = I n I n Q (m, n) A M m,n (K) n I n (m + n, n) [ ] A m I n I r Q Q n 443 n 443 (m, n) A M m,n (K) Ker(A) 57
44 44 4 n x 1, x 2,, x n a 11 x 1 + a 12 x a 1n x n = 0 a 21 x 1 + a 22 x a 2n x n = 0 a m1 x 1 + a m2 x a mn x n = 0 (412) x 1 = x 2 = = x n = 0 (412) (412) 444 (412) x 1 = x 2 = = x n = 0 a 11 a 12 a 1n a 21 a 22 a 2n rank < n a m1 a m2 a mn [ ] a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn 412 x = Ker(A) 412 Ker(A) rank(a) < n x 1 x n
45 n a 11 a 12 a 1n a 21 a 22 a 2n A = M n(k) 451 λ K a n1 a n2 a nn Av = λv, v 0 n v λ A n v λ A 452 t t n ti n A M n (K[t]) A χ A (t) = det(ti n A) 453 λ K A λ A [ ] λ x 1, x 2,, x n (A λi n )x = 0, x = 444 rank(a λi n ) < n A λi n n 422 det(a λi n ) = 0 det(a λi n ) = ( 1) n χ A (λ) χ A (λ) = 0 x 1 x 2 x n 46
46 5 K K Q R C V V K K- 1) u, v V u + v V (a) u, v V u + v = v + u (b) u, v, w V (u + v) + w = u + (v + w) (c) u V u + o = u o V (d) u V u + u = o u V 2) α K u V u α αv V (a) α, β K u V (αβ)u = α(βv) (b) u V 1u = u (c) α, β K u V (α + β)u = αu + βu (d) α K U, v V α(u + v) = αu + αv V K-V 1) (c) o V o V 1) (c) o = o + o = o + o = o 46
47 ) (c) o V K- V 1) (d) u V u V u V 1) (d) u = u + o = u + (u + u ) = (u + u) + u = (u + u ) + u = o + u = u u V u V u u = ( 1)u 512 x = x 1 x 2 x n, y = u + ( 1)u = (1 + ( 1))u = 0 u = o y 1 y 2 y n K n = x 1 x 2 x n x + y x α αx x i K Kn α K x, y αx 1 x 1 + y 1 x + y = x 2 + y 2, αx = αx 2 x n + y n αx n K- 513 K (m, n) M m,n (K) K- K n = M n,1 (K) [a, b] (a < b) C([a, b]) φ, ψ C([a, b]) φ + ψ C([a, b]) φ C([a, b]) λ R λφ C([a, b]) (φ + ψ)(t) = φ(t) + ψ(t), (λφ)(t) = λ φ(t) (aleqt b) C([a, b]) R-
48 K- V W V W V K- 1) W 2) w, w W w + w W 3) α K w W αw W 522 K- V 1) V {o} V K- 2) W V K-V W [ ] 1) 2) W w W 0 K o = 0 w W W K- V K-V W W K ) W V w W u = ( 1)u V w 523 K- V {v 1, v 2,, v r } { r } v 1, v 2,, v r K = α i v i α i K V K- [ ] W = v 1,, v r K V v, w v 1,, v r K r r v = α i v i, w = β i v i (α i, β i K) i=1 i=1 i=1
49 53 49 r v + w = (α i + β i )v i W i=1 α K αv = r (αα i )v i W i= K- V K- W f : V W f V W K- 1) v, v V f(v + v = f(v) + f(v ) 2) α K v V 1) f(o) = ok- f V W 2) v V f( v) = f(v)k v V 0 v = o f(o) = f(0 v) = 0 f(v) = o v = ( 1)v f( v) = f(( 1)v) = ( 1)f(v) = f(v) 532 (m, n) A M m,n (K) f A : K n K m f A (x) = Ax f A K- K n K- K m K-f A A K-
50 n m {v 1, v 2,, v n } K m K- f : K n x 1 x 2 x n n x j v j K m 535 K- V K- W K- f : V W j=1 Im(f) = {f(v) W v V }, Ker(f) = {v V f(v) = o} Im(f) W K-Ker(f) V K- Im(f), Ker(f) K- f [ ] Im(f) W w, w Im(f) w = f(v), w = f(v ) v, v V f(v + v ) == w + w w + w Im(f) α K f(αv) = αw αw Im(f) Im(f) W K- f(o) = o V Ker(f) Ker(f) V v, v Ker(f) f(v) = f(v ) = o f(v + v ) = o v + v Ker(f) α K f(αv) = αf(v) = o αv Ker(f) Ker(f) V K- 536 K- V K- W K- f : V W 1) f Im(f) = W 2) f Ker(f) = {o} [ ] 1) 2) f v Ker(f) f(v) = o f(o) = o
51 54 51 f(v) = f(o) f v = o Ker(f) Ker(f) = {o} v, v V f(v) = f(v ) f(v v) = f(v ) f(v) = o v v Ker(f) Ker(f) v v = o v = v f 537 K- V K- W K- f : V W f V W K- f : V W K- V, W K K- f : V W f f 1 : W V f 1 K- V, W K K- f : V W, g : W V f g = id W, g f = id V K- V {v 1, v 2, v r } λ 1 v 1 + λ 2 v λ r v r = o λ i K λ 1 = λ 2 = = λ r = 0 {v 1, v 2,, v r } K {v 1, v 2,, v r } K {v 1, v 2,, v r } K K- V {v 1, v 2,, v r } x 1 x 2 x = Kr f(x) = x r r x i v i V (51) K- f : K r V x 1 x r i=1 x 2 Ker(f) = Kr x 1v 1 + x 2 v x r v r = o
52 ) {v 1, v 2,, v r } K (51) K- f 542 K- V 1) {v 1, v 2,, v r } V K v i o (i = 1, 2,, r) 2) v 1 V {v 1 } K v 1 o [ ] 1) v 1 = o λ 1 = 1 λ 2 = = λ r = 0 λ 1 v 1 + λ 2 v λ r v r = o {v 1, v 2,, v r } K 2) {v 1 } K v 1 o v 1 o 0 λ 1 K λ 1 v 1 = o λ 1 K v 1 = o {v 1 } K R R- R 3 1) {v 1, v 2 } R 3 R {v 1, v 2 } 2) {v 1, v 2, v 3 } R 3 R {v 1, v 2, v 3 } K K n r v 1, v 2,, v r v j K n v j = a 1j a 2j a nj (a ij K) K λ 1, λ 2,, λ r λ 1 v 1 + λ 2 v λ r v r = o
53 54 53 λ j a 11 λ 1 + a 12 λ a 1r λ r = 0 a 21 λ 1 + a 22 λ a 2r λ r = 0 a n1 λ 1 + a n2 λ a nr λ r = 0 {v 1, v 2,, v r } K λ 1 = λ 2 = = λ r = rank(v 1, v 2,, v r ) = r 544 {v 1, v 2,, v r } K n K rank(v 1, v 2,, v r ) = r (v 1, v 2,, v r ) v 1, v 2,, v r (n, r) 545 K- V r K r V K dim K V r dim K V = dim K V < K- V K- K- V dim K V = ) K- V dim K V K- K n K n dim K K n = n [ ] K n n e 1 = 0, e 2 = 0,, e n = (e 1, e 2,, e n ) = I n rank(e 1, e 2 m, e n ) = n 544 {e 1 e 2 e n } K
54 54 5 dim K K n n r > n K n r v 1, v 2,, v r rank(v 1, v 2,, v r ) n < r 544 {v 1, v 2,, v r } K dim K K n = n 547 K- V, W K- f : V W 1) f dim K V dim K W 2) f dim K V dim K W f K- dim K V = dim K W [ ] 1) {w 1, w 2,, w r } W K f f(v i ) = w i v i V {v 1, v 2,, v r } V K α 1 v 1 + α 2 v α r v r = o (α i K) K- f f(o) = o α 1 w 1 +α 2 w 2 + +α r w r = o α 1 = α 2 = cdots = α r = 0 dim K V dim K W 2) {v 1, v 2,, v r } V K w i = f(v i ) {w 1, w 2,, w r } W K α 1 w 1 + α 2 w α r w r = o (α i K) f K- f(α 1 v 1 + α 2 v α r v r ) = o = f(o) f α 1 v 1 + α 2 v α r v r = o α 1 = α 2 = cdots = α r = 0 dim K V dim K W K- V {v 1, v 2,, v n } {v 1, v 2,, v n } V K 1) {v 1, v 2,, v n } K
55 ) V = v 1, v 2,, v n K 552 K- V o V K [ ] dim K V = n n 1 K {v 1, v 2,, v n } V v V {v 1, v 2,, v n, v} K α 1 v 1 + α 2 v α n v n + α n+1 v = o α i K i α i 0 α n+1 = 0 {v 1, v 2,, v n } K α n+1 0 v = ( α 1 /α n+1 )v ( α n /α n+1 )v n v v 1, v 2,, v n K V = v 1, v 2,, v n K {v 1, v 2,, v n } V K K- V {v 1, v 2,, v n } f : K n x 1 x 2 x n n x i v i V f K- K n V K- Im(f) = v 1, v 2,, v n K, Ker(f) = x 1 x 2 x n 536 i=1 x 1v 1 + x 2 v x n v n = o 1) f V = v 1, v 2,, v n K 2) f {v 1, v 2,, v n } K
56 56 5 3) f {v 1, v 2,, v n } V K {v 1, v 2,, v n } V K K= K n K- V K dim K V = n 553 K- V V K 554 K- V {v 1, v 2,, v n, v n+1 } (n 1) 1) v 1, v 2,, v n, v n+1 K = v 1, v 2,, v n K, 2) v n+1 = α 1 v 1 + α 2 v α n v n α i K [ ] 1) 2) v n+1 v 1, v 2,, v n+1 K = v 1, v 2,, v n K 2) 1) v 1, v 2,, v n K v 1, v 2,, v n, v n+1 K v v 1,, v n, v n+1 K n+1 v = λ i v i (λ i K) = i=1 n (λ ; + λ n+1 α i )v i i=1 v v 1,, v n K 555 K- V {v 1, v 2,, v n } 1) V K {v 1, v 2,, v r } V K V K {v 1,, v r, v r+1,, v n } 2) V = v 1, v 2,, v m K (v i V ) V K {v 1, v 2,, v m } V K [ ] 1) {v 1, v 2,, v r } V K {v 1,, v r,, v n } V
57 55 57 K {v 1,, v r,, v n } V K V = v 1,, v r,, v n K v V {v 1,, v r,, v n, v} K α 1 v α r v r + + α n v n + α n+1 v = o α i K i α i 0 α n+1 = 0 {v 1,, v r,, v n } K α n+1 0 v = ( α 1 /α n+1 )v ( α r /α n+1 )v r + + ( α n /α n+1 )v n v v 1,, v r,, v n K 2) {v 1, v 2,, v m } K V S V K S = {v 1, v 2,, v n } (n m) K K α 1 v 1 + α 2 v α n v = o α i K i α i 0 α n 0 v n = ( α 1 /α n )v 1 + ( α 2 /α n )v n + + ( α n 1 /α n )v n V = v 1, v 2,, v n 1 K S 556 n K- V n {v 1, v 2,, v n } 1) {v 1, v 2,, v n } V K 2) {v 1, v 2,, v n } K 3) V = v 1, v 2,, v n K [ ] 1) 2), 3) 2) 1) {v 1, v 2,, v n } K 555 1) {v 1, v 2,, v n } V V K n = dim K V 553
58 58 5 {v 1, v 2,, v n } V K 3) 1) 555 2) {v 1, v 2,, v n } V K n = dim K V 553 {v 1, v 2,, v n } V K 557 K- K- V, W W V dim K V = dim K W < V = W [ ] dim K W = dim K V = n {w 1, w 2,, w n } W K 556 {w 1, w 2,, w n } V K V = w 1, w 2,, w n K = V K- K n 558 K- K n n {v 1, v 2,, v n } K n K n (v 1, v 2,, v n ) det(v 1, v 2,, v n ) 0 [ ] 556 {v 1, v 2,, v n } K- K n K {v 1, v 2,, v n } K 544 {v 1, v 2,, v n } K rank(v 1, v 2,, v n ) = n (v 1, v 2,, v n ) n rank(v 1, v 2,, v n ) = n (v 1, v 2,, v n ) K- V K- W K- f : V W V K Im(f) K dim K Im(f) = dim K V dim K Ker(f)
59 56 59 [ ] Im(f) = {o} dim K Im(f) = 0 Ker(f) = V Ker(f) = {o} 536 f V Im(f) K- 547 dim K Im(f) = dim K V Im(f) {o} Ker(f) {o} V K {v 1, v 2,, v n } V = v 1, v 2,, v n K Im(f) = f(v 1 ), f(v 2 ),, f(v n ) K 555 2) Im(f) K Im(f) K {w 1,, w r } r = dim K Im(f) w i Im(f) w i = f(v i ) v i V Ker(f) K {u 1,, u s } s = dim K Ker(f) {v 1,, v r, u 1, cdots, u s } V K α i, β j K α 1 v α r v r + β 1 u β s u s = o (52) f s j=1 β ju j Ker(f) f(v i ) = w i r i=1 α iw i = o {w 1,, w r } α 1 = = α r = 0 (52) s j=1 β ju j = o {u 1,, u s } β 1 = = β s = 0 {v 1,, v r, u 1,, u s } K v V f(v) Im(f) f(v) = r i=1 α iw i α i K u = v r i=1 α iv i V u Ker(f) u = s j=1 β ju j β j K v = r α i v i + i=1 s β j u j v 1,, v r, u 1,, u s K j=1 {v 1,, v r, u 1,, u s } V K dim K V = r + s = dim K Im(f) + Ker(f) (m, n) A M m,n (K) K- f A : K n K m f A (x) = Ax 534 Ker(f A ) = Ker(A) (m, n) A M m,n (K) K- f A dim K Ker(f A ) = n rank(a), dim K Im(f A ) = rank(a)
60 60 5 [ ] rank(a) = r dim K Ker(f A ) = n r K- f A : K n K m dim K Im(f A ) = r 443 Ker(f A ) = Ker(A) = q r+1, q r+2,, q n K Q = (q 1, q,, q n ) n 558 {q 1, q 2,, q n } K n K {q r+1, q r+2,, q n } K {q r+1, q r+2,, q n } Ker(f A ) K dim K Ker(f A ) = n r (m, n)- A M m,n (K) Ker(F A ) Im(f A ) K rank(a) = r [ ] I r 0 P AQ = 0 0 m P n Q 562 Q = (q 1, q 2,, q n ), (q i K n ) {q r+1,, q n } Ker(f A ) K Im(f A ) [ ] f A (x) = Ax = P 1 I r 0 Q 1 x (x K n ) 0 0 y = Q 1 x Q 1 x K n y K n P 1 = (v 1, v 2,, v m ) (v j K m ) y y 1,, y n [ ] f A (x) = P 1 I r 0 y = y 1 v y r vr 0 0 Im(f A ) = v 1,, v r K P 1 {v 1,, v r } K 558{v 1,, v r } Im(f A ) K 563 (m, n)- A M m,n (K) [ ] I r 0 P AQ = (r = rank(a)) 0 0
61 57 (3) 61 P, Q P 1 = (v 1,, v m ), Q = (q 1,, q n ) (v j K m, q i K n ) {v 1,, v r } Im(f A ) K {q r+1,, q n } Ker(f A ) K P, Q K- K n {v 1, v 2,, v m } dim K v 1, v 2,, v m K = rank(v 1, v 2,, v m ) (v 1, v 2,, v m ) {v 1, v 2,, v n } (n, m) [ ] A = (v 1, v 2,, v n ) M n,m (K) Im(f A ) = v 1, v 2,, v m K (3) a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 (53) a m1 x 1 + a m2 x a mn x n = b m (a ij K, b i K, x j K) a 11 a 12 a 1n a 21 a 22 a 2n A = M m,n(k), x = x 1 x 2 b 1 b 2 Kn, b = Km a m1 a m2 a mn x n b m (53) Ax = b A M m,n (K) K- f A : K n x Ax K m (54)
62 62 5 (53) f A (x) = b A = (a 1, a 2,, a n ), a j = a 1j a 2j a mj Km f A : K n x 1 x 2 n x j a j K m j=1 Im(f A ) = a 1, a 2,, a n K (53) f A (x) = b x K n b Im(f A ) b a 1, a 2,, a n K b a 1, a 2,, a n, b K x n a 1, a 2,, a n K a 1, a 2,, a n, b K ) b a 1, a 2,, a n K, 2) a 1, a 2,, a n K = a 1, a 2,, a n, b K, 3) dim K a 1, a 2,, a n K = dim K a 1, a 2,, a n, b K, 4) rank(a 1, a 2,, a n ) = rank(a 1, a 2,, a n, b) 571 (53) rank(a, b) = rank(a) 441 (53) (53) x 1 = u 1, x 2 = u 2,, x n = u n
63 58 63 u = u 1 u 2 Kn u n x K n (53) f A (x) = b f A (x u) = f A (x) f A (u) = f A (x) b = 0 x u = v Ker(f A ) rank(a) = r dim K Ker(f A ) = n r Ker(f A ) K {v 1, v 2,, v n r } (53) x 1 x 2 = u 1 u 2 + λ 1v 1 + λ 2 v λ n r v n r (λ i K) x n u n n r K- V, W K- f : V W V K [v 1, v 2,, v n } K- φ : K n V ( x 1 x 2 x n n x j v j ) W K {w 1, w 2,, w m } K- ψ : K m W ( y 1 y 2 y m j=1 m y i w i ) K- f : V W K- V, W K n, K m i=1
64 64 5 j = 1,, n f(v j ) W {w 1,, w m } K f(v j ) = m a ij w j (a ij K) i=1 v = n j=1 x jv j V f(v) = n x j f(v j ) = j=1 i=1 n m x j j=1 i=1 a ij w i (55) m n = a ij x j w i (56) j=1 a 11 a 12 a 1n a 21 a 22 a 2n A = M mn(k) (57) a m1 a m2 a mn K- f A (x) = Ax (x K n ) f φ(x) = ψ f A (x) (x K n ) K n φ V f A K m ψ f W K- f : V W f A : K n K m (57) {v 1,, v n }, {w 1,, w m } f f V, W K- f : V W K- U K- g : U V f g U W K- U K- {u 1,, u l } f, g f g k = 1,, l n m g(u k ) = b jk v j (f g)(u k ) = c ik w i (b jk, c ik K) (58) j=1 i=1
65 58 65 g, f g B = (b jk ) j,k M nl (K), C = (c ik ) i,k M ml (K) (f g)(u k ) = f(g(u k )) = = n m b jk j=1 i=1 n b jk f(v j ) j=1 m n a ij w i = a ij b jk w i (58) c ik = n j=1 a ijb jk 581 K- g : U V, f : V W A, B f g : U W AB i=1 j=1
66 6 K = R C R C K- V, : V V K, V 1) u, v, v V λ K u, v + v = u, v + u, v, u, λv = λ u, v, 2) u, v V u, v = v, u, 3) v V v, v 0 v, v = 0 v = o K- V, 1), 2) u, u, v V λ K u + u, v = u, v + u, v, λu, v = λ u, v 3) v = v, v v V 1) u V V K v u, v K- 612 x = x 1 x 2, y = y 1 y 2 Kn x n y n n x, y = x i y i i=1, K- K n 66
67 62 Schmidt [0, 1] φ, ψ φ, ψ = 1 0 φ(t)ψ(t)dt, R- C([0, 1]) Schmidt K- V, 621 V {v 1, v 2,, v n } 1 i = j v i, v j = 0 i j {v 1, v 2,, v n } 622 {v 1, v 2,, v n } V {v 1, v 2,, v n } K [ ] λ i K n i=1 λ iv i = o 1 j n v j n n 0 = v j, λ i v i = λ i v j, v i = λ j i=1 i=1 λ 1 = λ 2 = = λ n = {v 1, v 2,, v n } V K V {u 1, u 2,, u n } 1 k n v 1, v 2,, v k K = u 1, u 2,, u k K [ ] n n = 1 u 1 = v 1 1 v 1 n > 1 {u 1, u 2,, u n 1 } 1 k n 1 v 1, v 2,, v k K = u 1, u 2,, u k K (61)
68 68 6 n 1 v n = v n u j, v n u j j=1 v n o v n = o n 1 v n = u j, v n u j u 1, u 2,, u n 1 K = v 1, v 2,, v n 1 K i=j {v 1, v 2,, v n 1, v n } K u n = v n 1 v n v 1,, v n 1, v n K = u 1,, u n 1, v n K = u 1,, u n 1, v n K = u 1,, u n 1, u n K 1 i < n n 1 u i, v n = u j, v n u i, u j = 0 u i, u n = 0 j=1 K {v 1, v 2,, v n } u 1 = v 1 1 v 1 v 2 = v 2 v 2, u 1 u 1 u 2 = v 2 1 v 2 v 3 = v 3 u 1, v 3 u 1 u 2, v 3 u 2 u 3 = v 3 1 v 3 {u 1, u 2,, u n } 623 Schmidt 624 V, K V {u 1, u 2,, u n } {u 1, u 2,, u n } V K 625 K
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7 2010 11 22 1 7 http://www.sml.k.u-tokyo.ac.jp/members/nabe/lecture2010 nabe@sml.k.u-tokyo.ac.jp 2 1. 10/ 4 2. 10/18 3. 10/25 2, 3 4. 11/ 1 5. 11/ 8 6. 11/15 7. 11/22 8. 11/29 9. 12/ 6 skyline 10. 12/13