数学Ⅱ演習（足助・09夏）
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- みひな りゅうとう
- 5 years ago
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1 II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w e z e w 5 z, w C, z, w arg zw arg z + arg w 2π 2 < arg z < π, < arg w < π,,,, z,, w, zw, arg zw arg z + arg w 2πZ 6 z C e z e re z, e z 2 n Z e z+2π n e z 3 e z e z e z 7 z C f z : R 2 R 2 g z : R 2 R 2 x x R x 2 2 re zx + x 2 re zx f z x im zx + + x 2, g z x x 2 im zx + x 2 zx + x 2, zx + x 2 n z n n!
2 A z M 2 R x R 2 f z x A z x A z z M 2 R z A z, w C A z A w 3 M 2 R 4 f z g z f z a x, x 2 R 2, f z x + x 2 f z x + f z x 2, b x R 2, λ R, f z λx λf z x, g z 8 x y 9 v, w R x 2 y 2, v w 2 v w v w x y + x 2 y 2, v v, v,, v, w C 2 R 2 v w x y + x 2 y 2 2 v w x y + x 2 y 2 v w 2 3 v C 2 v v v v w R 2, w C 2 w w 4 v C 2, λ C, λv λ v z c r, c, r C {z C z c r}, c, r C 2 z + z + p, p >, p > p R 3 z z + c, c R 4 z im z +
3 II z a + b, z a b, zz a 2 + b 2, a b z 2 z a + b, w a 2 + b 2, zw a a 2 b b 2 + a b 2 + a 2 b, z a 2 + b 2, w a 22 + b 2 2, zw a a 2 b b a b 2 + a 2 b 2 a 2 a 22 + b 2 b a 2 b b 2 a 22 zw z w 3 z, w 2, z + w z + w z + w 2 z + w 2 a + a b + b 2 2 a 2 + b 2 + a b a 2 + b 2 a 22 + b 2 2 a a 2 + b b 2 a 2 + b 2 a 22 + b 2 2, <, a a 2 + b b 2 2 a 2 + b 2 a b 2 2 a 2 a 2 2 2a a 2 b b 2 + b 2 b 2 2 a b 2 a 2 b 2 4 t R t t, tt t 2 t 2 z a + b z a b, z + z 2a, z z 2 b z + z 2 3 a rez, z z 2 b imz θ, θ 2, θ R argz, argw, argzw, e θ zw zw z z w w e θ +θ 2 e θ 2π, θ + 2πn θ + θ 2 n Z argzw argz + argw 2πn 2π 2, P, Qz, P w, Q zw < argz < π, < argw < π, < argzw < 2π, argzw argz + argw POQ P OQ argz, zw z z w z w OQ OQ OP OP w OPQ OP Q, : w 6 z a + b a, b R, e θ cos θ + sin θ, e z e a e b e x cos b + sin b e rez cos 2 b + sin 2 b e rez 2 n Z, e z+2π n e z e z+2π n e z cos 2πn + sin 2πn e z 3 e z e x, e z e x e x e z e x+z x z e z e z a 2 + b 2 a 22 + b 2 2
4 7 z a + b, z a b rez a, imz b zx + x 2 ax bx 2 + ax bx bx + ax 2, f z x 2 a b bx + ax 2 b a 2 a rez z + z 2, imz b z z z + z 2, A z + z z 2 2 z + z z + z deta z a 2 + b 2 z 2 a b, X a 2 + b 2 z + z z + z b a z z + z, z + z 2 2 A z X a 2 + b 2 a b b a a b b a, X A z x, X A z a, b a, b, w z, A z A w { a b 3 M a, b R}, A M b a 4 b b,, z + z z + z A z 2 2 z + z z + z 2 2 z + z z + z 2 z 2 2 2, w z z + z z + z f z x 2 8 a x b x s t, x 2 s2 fx + fx 2 x x 2 fλx, s t, t, s 2, t 2 R, 2 re zs + t + re zs 2 + t 2 im zs + t + re zs 2 + t 2 re z{s + s 2 + t + t 2 } im z{s + s 2 + f t + t 2 } z x + x 2 re zλx + λ x 2 im zλx + λ re zx + x λ 2 x 2 im zx + x 2, z z, g z λf z x
5 9 v, v, v 2 < 2 v, v x x + x 2 x 2 x 2 + x 2 2, x x 2 v a 3 w a, b R, a a, b b, w a b 2 + b 2, w 4 λv λx λx + λx 2 λx 2 λλx x + x 2 x 2 λ v z x + y x, y R z c, r / R, r < z C r, z c x re c + y im c x re c 2 + y im c 2 r 2 c, r r c 2 3 z + w z + w, w w z w z + w, z + + z z + z 2 p 2 p < 2 z + + z p x 2 + y 2 + x y 2 p x 2 + y 2 p x y 2, p 2 x y 2, x 2 + y 2 p 2 2p x y 2 + x y 2 2p x y 2 p 2 + 4x, p 2 + 4x 4p 2 {x y 2 } p 4 + 8p 2 x + 6x 2 4p 6x 2 + 4p 2 y 2 p 4 4p 2 y p 2 x 2 p 2 /4 + y 2 p 2 p > 2 /4 p 2, y x, y x p > 2, p 2 < p, p 2 < p2 4,, x2 p 2 /4 + y 2 p 2 /4,, p, p 2 4
6 3 4 z z + >, c < z C c, z c z + x + y c x + + y x 2 + y 2 c 2 {x y 2 } c 2 x c 2 x + c 2 + c 2 y 2 c x c, c 2, x c2 c y2 x + 2 c2 + c c 2 + y c 2 2c 2 c 2 + c2 2c, c2 c 2 c z im z + x + y x 2 + y 2 y + x 2 + y 2 y + 2 y + x 2 4y y x 2 4y, im z, c 2 2, 3 2, 4 im z
7 II I 2 9/4/28 9/5/7 2 L R 2 a, a 2, v, v 2 { } x L a x x + a 2 x 2 2 { } x x v t R, t x 2 x 2 v 2 a v L a 2 v 2 a a 2 v L v 2 22 P R 3 a, a 2, a 3 P x x 2 a x + a 2 x 2 + a 3 x 3 a L a 2 a 3 x 3 v, w P t, s tv + sw P 2 v, w P u P u tv + sw t, s tv + sw t s 3 v, w P u P u tv + sw t, s P P x x 2 a x + a 2 x 2 + a 3 x 3 x 3 x x 2 t, s R, x x 2 t v v 2 + s w w 2 x 3 x 3 v 3 w 3
8 R 3 a a P P {u R 3 a u } 23 R 3 R 3 L a, a 2 R 3 a a, a 2 a 2 a 2 a 3 a 22 a 23 L x x 2 a x + a 2 x 2 + a 3 x 3, a 2 x + a 22 x 2 + a 23 x 3 x 3 L {u R 3 a u a 2 u } 2 a, a v L x x 2 t R, x x 2 t v v 2 x v v R 3 v L 2 v 2 v 3 a, a 2 R 3 L { u R 3 a u a 2 u } a, a R θ, ϕ R T θ R 3 y- θ- S ϕ R 3 z- ϕ- xyz- z- x y x x R 3 T θ x S ϕ x x, x 2, x 3 x 2 x 3 x 3 v 3 a 2 a 3 v 2 v 3
9 2 x R 3 fx S ϕ T θ x x x fx 3 v, w R 3 T θ v T θ w v w 4 v, w R 3 fv fw v w 5 a R 3 a P {x R 3 a x } R 3 fp { x R 3 y P, x fy } fp P f fp 6 a a R 3 P 5 25 a 2 a 3 P x x 2 R 3 x 3 x 3 θ ϕ P fp θ ϕ f f P P 5 A M n C, B M n C, AB A, B 2 A M n R, A GL n C, A GL n R B M n C A, AB BA E n x 2 x 3 26 A A A 2 A q A A 2 A 22 A 2q A p A p2 A pq t A A ij
10 II 2 L x, x 2, t R x tv, x 2 tv 2, a v ta v + a 2 v 2 t, a 2 v 2 22 a a a 2, x P a x a 3 v, w P a v a w, a tv + sw ta v + sa w tv + sw P 2 u u u 2 u 3, v v v 2 v 3, w w w 2 w 3, u, v, w P, a u + a 2 u 2 + a 3 u 3 a v + a 2 v 2 + a 3 v 3 2 a w + a 2 w 2 + a 3 w 3 3 u tx + su s, t u tv + sw u 2 tv 2 + sw 2 u 3 tv 3 + sw t v 2 u v u 2 v 2 w v w 2 s v 2 w v w 2 3 v or w or v // w, v 2 u v u 2, v 2 u v u 2 s v 2 w v w 2, s v 2u v u 2, u w 2u w u 2 s, t v 2 w v w 2 v 2 w v w 2, tv + sw s, t, s, t s t v, t w s, t, v s t w v // w, w P,!s, t R 2 u tv + sw u, w, u \// w tv + sw t s! 3 2, w or v or v // w, v 2 u v u 2 s, t w, v, v \// w, u R 3 s, t, 2 23 u R 3 u x x 2, x 3 a u a x + a 2 x 2 + a 3 x 3, a 2 u a 2 x + a 22 x 2 + a 23 x 3
11 2 { a x + a 2 x 2 + a 3 x 3, a 2 x + a 22 x 2 + a 23 x 3 2 x x 2, x 3 x 2 a 2a 3 a a 23 x, x 3 a a 22 a 2 a 2 x 22 a 2 a 23 a 22 a 3 a 2 a 23 a 22 a 3, v a 2a 23 a 22 a 3 a 2 a 3 a a 23 L x x 2 t R, x x 2 tv a a 22 a 2 a 2 x 3 x 3 x x 3 v L x 2 t R, x 2 tv x 3 x 3 x x a, x 2, L x 2 a 2x + a 22 x 2 + a 23 x 3 x 3, a 2 a // a 2, k a ka 2, a x +a 2 x 2 +a 3 x 3 ka 2 x +a 22 x 2 +a 23 x 3, 2, L a, a 2 a, a 2, a \// a 2, R 2 a x 2,,, 24 cos θ sin θ x x cos θ y sin θ, sin θ cos θ y x sin θ + y cos θ T θ x x cos θ + x 3 sin θ x 2, S ϕ x x cos ϕ x 2 sin ϕ x sin ϕ + x 2 cos ϕ x sin θ + x 3 cos θ x 3 x cos θ + x 3 sin θ cos ϕ x 2 sin ϕ x cos θ cos ϕ x 2 sin ϕ + x 3 sin θ cos ϕ 2 fx x cos θ + x 3 sin θ sin ϕ + x 2 cos ϕ x cos θ sin ϕ + x 2 cos ϕ + x 3 sin θ sin ϕ x sin θ + x 3 cos θ x sin θ + x 3 cos θ 3 v v v 2, w w w 2, v 3 w 3 T θ v T θ w v cos θ + v 3 sin θw cos θ + w 3 sin θ + v 2 w 2 + v sin θ + v 3 cos θ w sin θ + w 3 cos θ v w cos 2 θ + v 3 w 3 sin 2 θ + v w 3 sin θ cos θ + v 3 w sin θ cos θ + v 2 w 2 + v w sin 2 θ + v 3 w 3 cos 2 θ v w 3 sin θ cos θ v 3 w sin θ cos θ v w + v 2 w 2 + v 3 w 3 v w a 2 x 3 x 2 4 v, w 3 3 S ϕ v S ϕ w, fv fw S ϕ T θ v S ϕ T θ w T θ v T θ w v w
12 5 fa fx a x, P Q { x R 3 fa x } fp Q, T θ x x cos θ x 3 sin θ x 2, S ϕ x x cos ϕ + x 2 sin ϕ x sin ϕ + x 2 cos ϕ, x sin θ + x 3 cos θ x 3 f x T θ S ϕ x, ff x x, x Q y f x P, fy x fp Q fp Q fp Q fp Q T θ x, S ϕ x T θ x, S ϕ x θ, ϕ θ, ϕ 6 a, P { x R 3 a x } a cos θ a 3 a, sin θ a2 + a 2 2 a θ, cos ϕ a a2 + a, sin ϕ a 2 ϕ, 2 2 a2 + a 2 2 sin θ cos ϕ fa a sin θ sin ϕ a a 2 a, 5 P fp, cos θ a 3 f cos θ cos ϕ cos θ sin ϕ sin θ a a a 3 a a 2 + a 2 2 a 3, f sin ϕ cos ϕ a 2 a 2 a a 2 a2 + a AB re A + im Are B + im B re A re B im A im B + im A re B re A im B re AB re A re B im A im B, im AB im A re B re A im B 2 A M n R, AB BA E n B M n C re A A, im A O n, { A re B re B A En, A im B O n 2 2 re B, re B A im B re B O n im B O n B M n R re X X, im X X 26 t A t A 2 t A p t t A t 2 A 22 t A p2 A t A t q A 2q t A pq
13 II I 3 9/5/8 9/5/9 K R C x y x y 4 z z w x y 5 2 z 3 b a, b R 2 6 a w 32 rank x,, x n a x + + a n x n c, a ij, c i R a m x + + a mn x n c m x C n v v x n C n C n re v R n v n im v R n a x + + a n x n, a m x + + a mn x n
14 c 2 C n c m R n c 3 c m R n C n C n R n 2 C n x c 34 A M m,n K v K n c K m x n c m Av c x,, x n K a m+, x + + a m+,n x n c m+ m n A GL n K v A c y,, y n t a n+, A y t c y n a n+,n c n+ A c v a n+, a n+,n c n+ y y n A c a n+, a n+,n c n+ A c 2 a n+, a n+,n c n+
15 A c 3 m n V, W V V W A c a m+, a m+,n c m+ A c 35 x,, x n K a x + + a n x n c, a ij, c i K a m x + + a mn x n c m x K n, w c c m Vw x n v V w v 2 V w2 v + v 2 V w +w 2 2 V w 3 v V w T v : V w V T v u u v T v u V T v
16 II /2 2, , 2 3 3,3 3 x, y, z , , ,2 2 3,3 3 x, y, z, w b 2 6 a b a , b 2 6 a b + a 2, b + b b a 2 a 2, α, β x α, y 2 4α 2β, z α, w β a 2,,
17 ,4 a 2 4 a 2 4 2, α x α, y 2 4α, z α, w, b, b, a 2, x α, y 2 4α 2β, z α, w β α, β b, a 2, x α, y 2 4α, z α, w α , 2, , , rank /4 /4 /4 9/8 3/8 / , 2,2 2, rank 2
18 33 A a ij, c c, c m Av A re v + A im v c R m A re v c, A im v re v R n, im v R n 2 R n C n, R n C n,, v C n re v R n, C n R n, C n, R n 3 2 R n C n, v C n, im v, v R n im v, im v R n, c im v R n, C n, R n 34 A c v a n+, a n+,n c n+, Av c,a GL n K v A c, an+, a n+,n A c c n+, an+, a n+,n E A c a n+, a n+,n c n+ a n+, a n+,n A { A E A c } a n+, a n+,n c n+ a n+, a n+,n A A c a n+, a n+,n c n+ a n+, a n+,n A y y n, y y n A c an+, a n+,n c n+, t a y n+, A t c a y n+,n n c n+,, y y n
19 2 y y n A c an+, a n+,n c n+, A c a n+, a n+,n c n+ n + i y i i, 2,, n, A c A c 3 v A c, a m+, a m+,n c m+ A c, V W A c V W, X X a m+, a m+,n c m+ A c Y Y, P, Q GL m+ K X P X QY Y Q P X Y, Q P, X Y, V W, X Y 35 w v c c m x x n, w 2, v 2 c x c m x n, v V w, v 2 V w2 v + v 2 V w,w 2 a x + + a n x n c, a m x + + a mn x n c m a x + + a n x n c 2 a m x + + a mn x n c m + 2 a x + x + + a n x n + x n c + c 2 a m x + x + + a mn x n + x n c m + c m v + v 2 V w,w 2 2 V w, v V w v, V w, v v K n, v V w V w
20 3 v V w, v x x n a x + + a n x n c, 3 a m x + + a mn x n c m, u x x n, T v u, a x + + a n x n c 4 a m x + + a mn x n c m 34, a x x + + a n x n x n 2 a m x x + + a mn x n x n, T v u V, v V u v+v, v+v V w+ V w, T v u v+v v v T v u, u, u 2 V w T v u T v u 2 u v u 2 v u u 2 T v u T v u
21 II I 4 9/6/ 9/6/2 K R C 4 x,, x n {v K n v λ v + + λ s v s + d} n 8 x 2 + 3x 4 + x 7 + 3x 8, x 3 + 2x 4 + x 6 x 8, x 5 + 2x 6 + x 8 3 n 8 x 2 + 3x 4 + x 7 + 3x 8 4, x 3 + 2x 4 + x 6 x 8 6, x 5 + 2x 6 + x n 5 x + x 3 + x 5, x 2 2x 3, x 4 3 x 5 4 n 5 x + x 3 + x 5, x x 2 2x 3 + x 4, x2 3 2 x 3 + x 4 x 5 42 i ii iii i ii iii , V, W f : V W v, v 2 V fv fv 2 v v 2, 2 w W v V w fv
22 3 f f 4 f : V W a f b f c f d f 5 f : V W 45 x + 3x 2 + 7x 3, 2x + 4x 2 + 6x 3 2, 5x + x 2 + 6x A M m,n K, B M n,m K AB E m BA E n A, B Er O 2 n m r < n Ẽ r M n K Ẽr r,n r O n r,r O n r,n r AB Ẽr BA Ẽr A, B 47 R[ t ] { t } f R[ t ] f t, ϕf R[ t ] ϕft tft ft + t ϕft t + t 2, f R[ t ], ψf R[ t ] ft f ψft ft + 2t ψft 2 t f R[ t ] ψϕf f 2 ϕψf f 48 R n [ t ] n t R n [ t ] R[ t ] ψ : R n [ t ] R n [ t ] 47 A M n+ R f R n [ t ] ft a + a t + + a n t n b a b A a b n a n b,, b n ψf b + b t + + b n t n A
23 47 A, B M n K AB E n BA E n n R[ t ] + t, + 2t 3t , 3 ϕ ψ 48 ϕ, ψ, A, B BA AB
24 II v K 8 λ, λ 2, λ 3, λ 4, λ 5 K, v λ + λ λ λ 4 + λ v K 5 λ, λ 2 K, v λ 2 + λ v K 8 λ, λ 2, λ 3, λ 4, λ 5 K, v λ + λ λ λ 4 + λ , v K 5 λ, λ 2 K, v λ + λ 2
25 , 3 2 2, 2/3 4/3 /3 2/ ,2 2 i, ii 3, iii,, , 5 2 2, 3 2 3, , , i, ii 5, / iii,
26 3 i i 2 i 2i i i i i 4 i i 2 4 3i + 4i 2 + i + 2i 2i i/ 3 6i/ 4 + 3i/ i 6 + 2i 4 + i 7 6i/ + 2i/ 2 i/ 4 + 9i 2i 5 i 2 + i/ i/ 3 4i/ 3 3 i 2 + i 4 2i, 3, 2 2, , i 6 + 2i 4 + i i, 4 + 9i 2i 5 i 3 i 2 + i 4 2i ii 3, i 2 i 3 4 i 4 + 2i 2 4 3i + 4i i 2i 9 + 8i/ 7 6i/ 2 + i/ iii i 2 i i + 4i i 4 + 2i 2 + i i,, , 5 2 2, 3 2 2, , , ,,
27 , ,, A O A O A O 3 A, B, O B O B, A B O B ,, 2, , , p42-43,p83 44 fv fv 2 v v 2 2 w W, v, v 2 V w fv 3 f v, v 2 V fv fv 2, v v 2 f w W, v V w fv 4 a f : R R, fx Arctan x, b f : R R, fx x 3 x, c f : R R, fx sin x, d f : R R, fx x R R, a N N, fx 2x, 5 f, f f,, f a d
28 , , , , x, x , x , 3, 2 2, ,3, x, x 2 2, x 3 46 m < n, A E m O m,n m, B Em O n m,m O n m,m O n m,n m, Em O AB E m, BA m,n m E n Er O 2 A r,n r AB O n r,r O n r,n r Er O, B r,n r E n r,r O n r,n r Er O r,n r O n r,r O Ẽr, BA n r,n r, Er O r,n r E n r,r O n r,n r Ẽr
29 47 tft f ψφft ψtft ft f R[t] ψφf f t ft f ft f 2 φψft ψ t t t ft f f f R[t] φψf f 48 ft a + a t + + a n t n, ψf a + a 2 t + + a n t n, b a b a 2 b n a n b n A a ij, a i n+ a k a i,k i,, n + a n+ k f, a, a,, a n R, { j i a ij j i, A A M n R, A M n+ R
30 II I 5 9/6/5 9/6/6 K R C 5 2 A b 52 A M n K A A c d M n K A GL n K A En b L c d, U A LU b d 53 A M n K A k k A k M k K A k k k-th principal minor A,, A n A U GL n K L GL n K A LU LU n U, L A 2 n A GL n+ K A b A A A c d M n K A A n A A L U L n LU L LU A LU A LU L U LU L L U U E n L L, U U 54 LU
31 55 Vandermonde n 2 x x 2 x n det x x 2 x n x j x i nn 2 x i x j i<j n i<j n n n n x x 2 x n x j x i i < j n i, j i<j n x j x i x,, x n 56 A M m,n K A a,, a n K m a,, a n rank A n ranka,, a n n λ,, λ n K a n λ a + + λ n a n A n A a,, a n A A a n rank A n A A A 57 A M m,n K i < i 2 < < i r j < j 2 < < j r A i i r j j r M r K A i,,i r ;j,,j r r i < i 2 < < i r j < j 2 < < j r det A i,,i r ;j,,j r rank A r 58 A M n K A à A GL n K det à 2 A GL n K à det An 2 A 3 n > 2 A GL n K à O n rank A rank A < n 57 à O n rank A n 56 à rank A < n 2 A à A M nk n 2 A M n K {A n } n,2,, lim A n n A
32 II 5 A, B A a ij, B b ij, i > j a ij, b ij A + B i, j a ij + b ij, i > j a ij + b ij n, AB i, j a ik b kj, k < i a ik, j < k b kj, i > j k n a ik b kj, A, B A + B AB k, 2 A a ij GL n K, k, k k i k /a kk ii l k a kl l k i /a kk, ii, l < k E n a kl, a kl l > k, k,, n, A E n A, A a ij GL n K, i > j a ij A ã ã n deta ã n ã nn A k, l M n K Ãkl, a a,k a,k a,l a,l+ a,n a k, a k,k a k+, a k+,k à kl a l, a l,l a l+, a l+,l+ a l+, k < l, i > j a ij, a k+,k,, a l,l, à kl,, ã kl det Ãkl A A a nn
33 A b A A 52 A LU b c d c cb + d, 53 { b A b d cb + d { b A b d d cb n U, L A 2 n A M n+ K, A n A A b A, A c d LU L U L L cu d cu L, U b b, L, U A b, L U A, n + LU c d 2 LU A LU L U LU L L U U 5,,, En b, U, U En b, U En b U U En b b, U U b b L L U U E n L L, U U LU, 53, LU, ,, , LU, ,, 4 3, n 2 det x x x 2 x 2
34 2 n, x x 2 x n+ x x n+ x 2 x n+ x n det x x 2 x n+ det x 2 2 x n+ x x n+ x n+ n n n x x 2 x n+ x n n x n+ x n n n 2 x n+ x n+ k,, n, k n + x x n+ x 2 x n+ x n x n+ x 2 2 x n+ x x n+ x 2 2 n x n+ n+2 x 3 3 x n+ x x n+ x 3 3 n x n+ x n n x n+ x n n 2 x n+ x n n n x n+ x + x n+ x 2 + x n+ x n + x n+ n x x x n+ + x n+ x x 2 x n+ + x n+ x 2 2 n + x n x n+ + x n+ n x k x n+ k n n n x n k k x n+ x n k k 2 x n+ x n k k n x n+ k, k,, n, k x k x n+ n x x 2 x n x n+ x k x x 2 x n k n x n x 2 n x n k 2,, n, k k x n+ n x n+ x k x j x i x j x i nn+ 2 x i x j k i<j n i<j n+ n + i < j n + i, j N 2, n + 2 k nn +, 2 i<j n+ n r k n! r!n r! 56 A A a n, A a a n A A a a n, k,, n λ k,,, λ k,n K, a k λ k,a + + λ k,n a n A i k k A A a n rank A n, n k i k, a n l a n,l, k n k a n,ik n λ k a n,il λ l,k k,, n a n λ a + + λ n a n l 57 i < i 2 < < i r, j < j 2 < < j r deta i,,i r ;j,,j r, ranka i,,i r ;j,,j r r A Ai,,i r;j,,j r P Q R rank, ranka r
35 ranka r, A, i k k i < i 2 < < i r, A, i k k j < j 2 < < j r i k, j k k r deta i,,i r ;j,,j r 58 Aà detae n, deta detaã detadetã detdetae n det deta n deta deta, detã detan 2 à à detãe n A, Aà à deta à detãa detan A deta, à detan 2 A 3 A rank i ranka n 2 57, ranka < n, i < i 2 < < i n, j < j 2 < < j n deta i,,i n ;j,,j n, i, j n A i, j à Õn O n ii ranka n 5,7,detA i,,i n ;j,,j n i < i 2 < < i n, j < j 2 < < j n i n, i i k k, n i N i A a a n, 56, λ i,, λ in K, a i λ i a i + + λ in a in a k j K n a k,j, A i, j ã i,j, ã i,j i +j deta i,j a in,j, i k, j ã ik,j, ã ik,j i k+j deta i,j a ik,j a ik +,j a i,j a in,j i+j deta i,j a i,j a i+,j λ i a i,j + + λ in a in,j a in,j i k+j deta i,j a ik,j a ik +,j λ i a i,j a in,j i k λ ik j deta i,j a ik,j a ik +,j a i,j a in,j i k+ i k i λ ik j deta i,j a in,j ik i sgn k + i k n 2 n n i k i i k+ i k i λ ik, j deta i,j a in,j i k, j, fi k, gj, fi i, ã i j figj, fi g fi g2 fi gn fi 2 g fi 2 g2 fi 2 gn ã ã n detã det fi g fi g2 fi gn ã n ã nn fi n g fi n g2 fi n gn,, rankã < n 2 i, à O n
36 II I 6 9/6/29 9/6/3 K R C V K- V K- V V K- V 6 M m,n K K K- 2 K[ t ] t K K[ t ] {a + a t + + a n t n a,, a n K} n K n [ t ] t n K K n [ t ] {a + a t + + a m t m a,, a m K, m n} K[ t ] K n [ t ] K- K[ t ] K n [ t ] 3 V {{a n } n,2, a n K, a n+2 3a n+ + 2a n } a, b V a {a n }, b {b n } c n a n + b n a+b V a+b {c n } a V, λ K a {a n } d n λa n λa V λa {d n } V K- 62 V C- W V v, v W v+v V λ R λv λ C V C W R- v V w v W V w W W v w v W 63 V, W K- f : V W K- v, v 2 V, fv + v 2 fv + fv 2 2 v V, λ K, fλv λfv V W
37 64 n d W {a n n dt + a d n n n dt + + a d n dt + a } a,, a n R d n w a n dt + a d n n n dt + + a d n dt + a f R[ t ] d n f wf a n dt + a d n f n n dt + + a df n dt + a f w : R[ t ] R[ t ] R- d n 2 w a n dt + a d n n n dt + + a d n dt + a w d n b n dt + b d n n n dt + + b d n dt + b W w + w a n + b n dn dt n + + a + b d dt + a + b λ R λw λa n dn dt n + λa n dn dt n + + λa d dt + λa f R[ t ] w + w f wf + w f, λwf λwf 3 W R- W R n+ 65 V K 3 W, W 2, W 3 W + W 2, W + W 3, W 2 + W 3, W + W 2 + W 3 V K 3 W, W 2, W 3 2 V K 3 W, W 2, W 3, 66 V K[x] W, W 2 W W 2, W + W 2 W x +, W 2 x 2 2 W x, W 2 x 2 3x W x 2 3x + 2, W 2 x 2 4
38 67 R 2 R 2 { } { } x x W x R 2 W 2 x R { } { } t t 3 3 W 3 t 2 t R 4 W 4 t 3 t R { } 68 R 2 x W W x R R 2 U, U 2, R 2 W U W U 2 U U 2 69 V K- W, W 2, W 3 V W + W 2 W 3 W W 3 + W 2 W 3 2 W W 2 + W 3 W + W 3 W 2 + W 3 6 V K- V {f : V K, K- } f, g V f + g : V K v V f + gv fv + gv f + g V 2 f V λ K λf : V K v V λfv λfv λf V 3 V K- V V V 6 V, W K- f : V W K- g W f g f gv gfv, v V f g V f g f g 2 f : W V K-
39 II 6 A a ij, B b ij, C c ij M m,n K, λ, µ K A + B a ij + b ij, K λa λa ij, A + B, λa M m,n K, A + B + C a ij + b ij + c ij a ij + b ij + c ij A + B + C, 2 A + B a ij + b ij b ij + a ij B + A, 3 a ij A O, A O + A + a ij a ij A, 4 A A a ij, A + A a ij + a ij O, 5 λ + µa λ + µa ij λa ij + µa ij λa + µa, 6 λa + B λa ij + b ij λa ij + λb ij λa + λb, 7 λµa λµa ij λµa ij λµa, 8 A a ij a ij A 8, M m,n K K- 2 p, q, r K n [t] pt a n t n + + a, qt b n t n + + b, rt c n t n + + c, λ, µ K p + qt a n + b n t n + + a + b, λpt λa n t n + + λa, p + q, λp K n [t], p + q + rt {a n + b n + c n }t n + + {a + b + c } {a n + b n + c n }t n + + {a + b + c } p + q + rt p + q + r p + q + r, 2 p + qt a n + b n t n + + a + b b n + a n t n + + b + a q + pt p + q q + p, 3, p K n [t] + p p, 4 p K n [t], p p, p + p t {a n + a n }t n + + {a + a }, 5 λ + µpt λ + µa n t n + + λ + µa λa n + µa n t n + + λa n + µa n λp + µpt λ + µp λp + µp, 6 λp + qt λa n + b n t n + + λa + b λa n + λb n t n + + λa + λb λp + λqt λp + q λp + λq, 7 λµpt λµa n t n + + λµa λµa n t n + + λµa λµpt λµp λµp, 8 pt a n t n + + a a n t n + + a pt p p 8, K n [t] K- K[t] 3 a, b, c V a {a n }, b {b n }, c {c n }, λ, µ K a + b + c {a n + b n + c n } {a n + b n + c n } a + b + c, 2 a + b {a n + b n } {b n + a n } b + a, 3 {}, a + a a, 4 a a a, a + a {a n + a n } {}, 5 λ + µa {λ + µa n } {λa n + µa n } λa + µa 6 λa + b {λa n + b n } {λa n + λb n } λa + λb 7 λµa {λµa n } {λµa n } λµa 8 a { a n } {a n } a 8, V K-
40 62 v, v W V, W v + v V, V C-, W W, R C V C-, λ R W λv V, W W W R- 64 f, g R[t], λ R, f + g, λf R[t], wf +g a n d n f + g dt n + +a f +g wλf a n d n λf dt n + + a λf λ a n d n f dt n + + a f a n d n f dt n + + a f d n g + a n dt n + + a g wf +wg R[t], λwf R[t] w : R[t] R[t] R 2 w + w f a n + b n dn f dt n + a d n f + b f a n dt n + b d n f n dt n + a f + b f d n f a n dt n + + a d n f f + a b dt n + + b f wf + w f, λwf λa n dn f dt n + + λa d n f f λ a n dt n + + a d n λf f a n dt n + + a λf λwf 3 w, w, w W w a n d n dt n + + a, w b n d n dt n + + b, w c n d n dt n + + c, λ, µ K w+w +w {a n +b n +c n } dn dt n +{a +b +c } {a n +b n +c n } dn dt n +{a +b +c } w+w +w, 2 w + w a n + b n dn dt n + a + b b n + a n dn dt n + b + a w + w, 3 a,, a n w, w W w, 4 w W, w a n dn dt n + + a, w+w {a n + a n } dn dt n +{a + a }, 5 λ + µw λ + µa n d n dt n + + λ + µa λa n + µa n dn dt n + + λa + µa λw + µw, 6 λw + w λa n + b n dn dt n + + λa + b λa n + λb n dn dt n + + λa + λb λw + λw, d n 7 λµw λµa n dt n + + λµa λµa n dn dt n + + λµa λµw 8 w a n dn dt n + + a d n a n dt n + + a w 8, V R- 65 W + W 2 {w + w 2 w W, w 2 W 2 } λ + λ 2 λ, λ 2 K,, W + W 3 λ + λ 2 λ, λ 2 K,, W 2 + W 3 λ + λ 2 λ, λ 2 K λ + λ 2 + λ 2 λ, λ 2 K µ + µ 2 µ, µ 2 K,, λ µ µ 2, λ 2 µ 2
41 W + W 2 + W 3 λ + λ 2 + λ 3 λ, λ 2, λ 3 K λ + λ 2 + λ 3 + λ 3 λ, λ 2, λ 3 K µ + µ 2 + µ 3 µ, µ 2, µ 3 K,, K 3 λ µ, λ 2 µ 2 µ 3, λ 3 µ 3 2 W + W 2 λ + λ 2 λ, λ 2 K,, W + W 3 λ + λ 2 + λ 3 λ, λ 2, λ 3 K µ + µ 2 + µ 3 µ, µ 2, µ 3 K,, K 3, λ µ µ 2 µ 3, λ 2 µ 2, λ 3 µ 3 W 2 + W 3 λ + λ 2 + λ 3 λ, λ 2, λ 3 K λ + λ 2 + λ + λ 3 λ, λ 2, λ 3 K µ + µ 2 µ, µ 2 K, λ k K, λ 2 µ k, λ 3 µ 2 + k W + W 2 + W 3 W + W 2 + W 3 W + W 3 K 3 66 p W W 2, p λ x + λ 2 x 2 λ, λ 2 K { { λ λ 2 λ p W W 2 {} λ 2λ 2 λ 2, W + W 2 {λ x + + λ 2 x 2 λ, λ 2 K} {λ + λ 2 x + λ 2λ 2 λ, λ 2 K} {µ x + µ 2 µ, µ 2 K} K [x] λ 2µ + µ 2 3, λ 2 µ µ p W W 2, p λ x λ 2 x 2 3x + 2 λ, λ 2 K λ 2 { λ λ 3λ 2 p W W 2 {} λ 2 λ 2λ 2, W + W 2 { λ x + λ 2 x 2 3x + 2 λ, λ 2 K } {x λ 2 x + λ 2λ 2 λ, λ 2 K} {x µ x + µ 2 µ, µ 2 K} {f K 2 [x] f } λ 2µ + µ 2, λ 2 µ
42 3 p W W 2, p λ x 2 3x + 2 λ 2 x 2 4 λ, λ 2 K λ λ 2 { λ 3λ p W W 2 {} λ 2 2λ 4λ 2, W + W 2 { λ x 2 3x λ 2 x 2 4 λ, λ 2 K } {x 2λ + λ 2 x λ + 2λ 2 λ, λ 2 K} {x 2µ x + µ 2 µ, µ 2 K} {f K 2 [x] f2 } λ 2µ µ 2 3, λ 2 µ + µ 2 3 λ, K µ, K, 67 R 2 R- x x x, x R, w, w W, W Ø, x + x w, w W, w + w λx W w W, λ R, λw W W R 2 2 W 2, 2 / W 2 2, W 2 R W 3, 2 / W 2 2, W 3 R 2 t 3 s 4 t, s R, w t 3 w 3 s 3 W 4, W 4 Ø, t w, w W 4, w + w 3 + s 3 3 t t 3 + s s t 3 + s 3 3 W 4 λt 3 3 λt w W 4, λ R, λw λt λt 3 W 4 W 4 R 2 68 { { y y2 U y y R}, U 2 y y 2 R}, U U 2, 2 W U W U 2, W U W + U R 2, W U 2 W + U 2 R 2 69 W W 3 + W 2 W 3 {x + y x W W 3, y W 2 W 3 }, x W W 3, y W 2 W 3, x W, y W 2 x + y W + W 2, x, y W 3 W 3 V x + y W 3 W + W 2 W 3 W W 3 + W 2 W 3 V K 3, W W + W 2 W 3, W 2, W 3,,, W W 3 + W 2 W 3
43 2 W W 2 + W 3 {x + y x W W 2, y W 3 }, x W W 2, y W 3, x W, y W 3 x + y W + W 3, x W 2, y W 3 x + y W 2 + W 3 W W 2 + W 3 W + W 3 W 2 + W 3 V, W, W 2, W 3, W W 2 + W 3 W 3, W + W 3 W 2 + W 3 K 3 6 v, v V, f + gv + v fv + v + gv + v fv + fv + gv + gv fv + gv + fv + gv f + gv + f + gv, v V, λ K, f + gλv fλv + gλv λfv + λgv λfv + gv λf + gv f + g : V K K- f + g V 2 v, v V, λfv + v λfv + v λfv + gv λfv + λfv λfv + λfv, v V, µ K, λfµv λfµv λµfv µλfv µλfv λf : V K K- λf V 3 f, g, h V, λ, µ K, v V f + g + hv f + gv + hv fv + gv + hv fv + g + hv f + g + hv f + g + h f + g + h, 2 f + gv fv + gv gv + fv g + fv f + g g + f, 3 o V ov, f V, o + fv ov + fv fv o + f f 4 f V, f V f v fv, f + f v fv + f v fv + fv ov f + f o, 5 λ + µfv λ + µfv λfv + µfv λfv + µfv λf + µfv λ + µf λf + µf, 6 λf + gv λf + gv λfv + gv λfv + λgv λfv + λgv λf + λgv λf + g λf + λg, 7 λµfv λµfv λµfv λµfv λµfv λµf λµf, 8 fv fv fv f f 8, V K- 6 f : V W, g : W K f g : V K, v, v V, f gv + v gfv + v gfv + fv gfv + gfv f gv + f gv, v V, λ K, f gλv gfλv gλfv λgfv λf gv f g : V K K- f g V 2 v V g, g W, f g + g v g + g fv gfv + g fv f gv + f g v f g + f g v, f g + g f g + f g, g W, λ K, f λgv λgfv λgfv λf gv, f λg λf g f : W V K-
44 II I 7 9/7/3 9/7/4 K R C 7 t, t 2, t 3, t 4 K K 4 a, a 2, a 3 a, a 2, a 3 A a a 2 a 3 V a, a 2, a 3 V {w K 4 t t 2 t 3 t 4 v K 3, w Av} 2 A A V {w K 4 v K 3, w A v} 3 dim V ranka a 2 a 3 4 V 72 K[x] x K- K- K[x] {a + a x + + a n x n a,, a n K} f, g K[x] f + g f g f K[x] λ K λf f λ f, g K[x] f + g f g f + g f + gx fx + gx f + g f + g f K[x] λ K λ f f λ λ f λ fx λfx λ f λf + 73 f : K n+ K n [x] a a f a + a x + + a n x n a n f K- f
45 2 g : K n+ K n [x] a a g a + a + a x + a + a + a 2 x a + + a n x n a n g K- g 74 ϕ : K 2 K 3, ψ : K 3 K 3 2x + 3y x ϕ x + y, ψ x y x + z 2x + y 3z y y z x y x ψ ϕ y 2 ψ ϕ ϕ ψ A, B, C A, B, C A CB 75 f : R 3 R 2 v x y x R 3 fv n + y + z x + y z n f n 76 ϕ : R 3 R ϕ t ϕ s ϕ t s ϕ t ϕ s 2 n > m f : K n K m f 3 n < m f : K n K m f 77 R 3 v, v 2, v 3 fv, fv 2, fv 3 R 3 R 3
46 v, v 2, v 3 f a v 2 3, v 2 5, v b v 3, v 2 3 2, v A v v 2 v 3 f A A f A 78 f R n [x] ϕf R 2n [x] ϕfx fx 2 ϕ R n [x] R 2n [x] ϕ 2 ϕ 3 ψ : R 2n [x] R n [x] ψ ϕ id Rn [x] 79 ψ ϕ ψ id R2n [x] f : K 8 K 3 x x f 2 x 8 x 2 + 3x 4 + x 7 + 3x 8 x 3 + 2x 4 + x 6 x 8 x 5 + 2x 6 + x 8 V { v K 8 fv }, W { w K 3 v K 8, w fv } V Ker f W Im f V W 2 g : C 5 C 3 g x x 2 x 3 x 4 x 5 x + x 3 + x 5 x 2 2x 3 x 4 3 x 5 V W V W
47 7 V, W U, U 2 V f : U W f 2 : U 2 W V U U 2 v V v u + u 2 u U u 2 U 2 fv f u + f 2 u 2 f V W well-defined 2 V U + U 2 f 7 t, t 2, t 3 K K 3 W W 2 W,, W 2, t t 2 W + W 2 t, t 2, t 3 K 3 W W 2 W + W 2 2 t, t 2, t 3, t 4, t 5 K K 5 W W 2 W 3 t W, t 2, W 2 W 3 t 3 t 4 W + W 2 + W 3 t, t 2, t 3, t 4, t 5 72 f R f x R f x fx f x R f x fx t 3 t 5 73 V {f R[x] f } W {f R[x] f } V W R[x] 2 R[x] V W 3 R[x] V W f R[x] g V h W f g + h g h f 2
48 II 7 V a, a 2, a 3 {λ a + λ 2 a 2 + λ 3 a 3 λ, λ 2, λ 3 K} λ a a 2 a 3 λ 2 λ, λ 2, λ 3 K { Av v K 3} { w K 4 v K 3, w Av } λ 3 2 A A, P GL 3 K A AP v P v, A v AP v Av, v P v, v K v K { w K 4 v K 3, w A v } { w K 4 v K 3, w Av } V 3 A A t 3 t t t 3 t 2 t 4, t 4 t 2 V v + v 2 + v 3 t 3 t v, v 2, v 3 K V K 3, t 4 t 2 t t 3 t 2 t 4 V v + v 2 v, v 2 K V K 2 {, A A t 3 t ranka 2 t t 3, t 2 t 4 ranka 3 otherwise t 4 t 2 { 2 t t 3, t 2 t 4 dimv ranka a 2 a 3 3 otherwise 4 t t 3, t 2 t 4 V,, t t 3 t 2 t 4 V,, t 3 t t 4 t 2 72 f, g K[x] fx a n x n + + a, gx b n x n + + b, f + gx a n + b n x n + + a + b a n x n + + a + b n x n + + b fx + gx, f + gx fx + gx f + gx f + gx f + g f + g, λfx λa n x n + + λa λa n x n + + a λ fx, λ fx λ fx λfx λ fx λf λ f 73 p K n [x], a n {p x n }, f : K n [x] K n+ f p f f id Kn[x], f f id K n+ f K-, f f f a n f n /n! a a, a n
49 a 2 p K n [x],a n, g : K n [x] K n+ g a a p, a n a n g g id Kn [x], g g id K n+ g K-, g g g 74 2x + 3y 2x + 4y x ψ φ ψ x + y 3x + 4y y y 3x + 2y 2 CB A f, λ R, λ fλv n x n + λy + λz λx λfv n + λy + λz λ λx + λy λx + λy n x n λx n n n, v, v R 3 v x y, v y, z z x + x v, v R 3, fv + v + y + y + z + z x + y + z x x + x + y + y + + y + z x + y x + y fv + fv, λx + λy + λz x + y + z v R 3, λ K, fλv λ λfv f λx + λy x + y x,f n 76 φ, t φ, s φ, φ : R R 3, φ t, φ s φ s t t φ φ s/t s t φ s, φ s s/t, 2, φ φ : R 3 R 2 f A a ij M m,n K f, fx fy x y, Av w f, v v v n K n, Av v n > m, A, j,, j k k m ranka
50 v + a,j v j + + a,jk v jk v + a,j v j + + a,jk v jk v j,, v jk v l l j,, j k, v, f 3 f A a ij M m,n K f, w K m, v K n w fv, V {w K m v K n, w Av} K m v w f, v K n,w K m 7 2, A v n w m V n < m, A, i i k k, w ik v k k,, r w i a i, v + + a i,r v r r ranka w ik v k k,, r w i a i, v + + a i,r v r w K m w V, V K m f 77 a , , , f, fv, fv 2, fv 3 b f , f, fv f 4v 2 v 3 4fv 2 fv f P, P A P v v 2 v 3 E P A GL n K, P A 78 f f, g R n [x], φf + gx f + gx 2 fx 2 + gx 2 φfx + φgx φf + φgx, f R n [x], λ K, φλfx λfx 2 λfx 2 λφfx λφfx λφfx φ : R n [x] R 2n [x] 2 f, g R n [x], fx a n x n + + a, gx b n x n + + b φfx φgx fx 2 gx 2 fx 2 gx 2 a n b n x 2n + + a b x 2 + a b φf φg a i b i i,, n f g, φ
51 3 fx a 2n x 2n + + a R 2n [x], ψ : R 2n [x] R n [x] ψfx a 2n x n + a 2n 2 x n + + a 2 x + a, ψ φ id Rn [x], gx x 2 + x g R 2n [x], φ ψgx x 2, φ ψg g φ ψ id R2n[x] 79 V fv,, dimv V v K 8 λ, λ 2, λ 3, λ 4, λ 5 K, λ + λ 2 + λ λ 4 + λ 5 f A M 3,8 K, W { w K 3 v K 8, w Av } A W A A E 3 O 3,5, v W { w K 3 v K 8, w A v } v v 2 v 3 v, v 2, v 3 K K3 dimw 3 2 2, V v C 5 λ, λ 2 C, λ + λ 2 3 dimv 2 v v 8, g B M 3,5, B B E 3 O 3,2,, W C 3, dimv 3 7 U + U 2, v u + u 2, v V u, u 2, f, f 2, u, u 2 fu, fu 2 v fv f u + f 2 u 2, f : V W well-defined v, v V, λ K u, u U, u 2, u 2 U 2 v u + u 2, v u + u 2, v + v u + u + u 2 + u 2 u + u U, u 2 + u 2 U 2, λv λu + λu 2 λu U, λu 2 U 2, v, v V, fv + v f u + u + f 2 u 2 + u 2 f u + f u + f 2 u 2 + f 2 u 2 f u + f 2 u 2 + f u + f 2 u 2 fv + fv, v V, λ K, fλv f λu + f 2 λu 2 λf u + λf 2 u 2 λf u + f 2 u 2 λfv f : V W 2 V W K 3, U,, U 2,, f, f 2 f u u, f 2 u 2 2u 2 v, u u 2, u 2 u, u, u U, u 2, u 2 U 2, v u + u 2 u + u 2, 2 fu + fu 2, fu + fu 2 f well-defined
52 7 W W 2, W + W 2 2 W W 2 {}, W + W 2 W 3 {} t W + W 2,, t 2, W 3 t 3,, t 4 t 5 t 4 t 5, W + W 2 + W 3,,, t 4 t 5 73 V V Ø, f, g V, f + g x f x + g x fx gx fx + gx f + gx f + g V, f V, λ K, λf x λf x λ fx λfx λfx λf V V R[x] W 2 fx a n x n + + a f V, f x fx a a 2 a 2m V { a 2m x 2m + + a 3 x 3 + a x a, a 3,, a 2m R }, f W, f x fx a a 3 a 2m W { a 2m x 2m + + a 2 x 2 + a a, a 2,, a 2m R } V W {a n x n + + a a a a n } {} V + W, V + W {a n x n + + a a, a,, a n R} R[x] V W R[x] 3 fx a n x n + + a, fx f x 2a 2m x 2m + + a 3 x 3 + a x, fx + f x 2a 2m x 2m + + a 2 x 2 fx f x + a, g, h gx, hx 2, g V, h W, f g + h, fx + f x 2
2 (2016 3Q N) c = o (11) Ax = b A x = c A n I n n n 2n (A I n ) (I n X) A A X A n A A A (1) (2) c 0 c (3) c A A i j n 1 ( 1) i+j A (i, j) A (i, j) ã i
[ ] (2016 3Q N) a 11 a 1n m n A A = a m1 a mn A a 1 A A = a n (1) A (a i a j, i j ) (2) A (a i ca i, c 0, i ) (3) A (a i a i + ca j, j i, i ) A 1 A 11 0 A 12 0 0 A 1k 0 1 A 22 0 0 A 2k 0 1 0 A 3k 1 A rk
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5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6
2000年度『数学展望 I』講義録
2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53
II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K
II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F
2 7 V 7 {fx fx 3 } 8 P 3 {fx fx 3 } 9 V 9 {fx fx f x 2fx } V {fx fx f x 2fx + } V {{a n } {a n } a n+2 a n+ + a n n } 2 V 2 {{a n } {a n } a n+2 a n+
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(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y
![(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y](/thumbs/101/151465237.jpg "(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y")
(2018 2Q C) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c b
1W II K =25 A (1) office(a439) (2) A4 etc. 12:00-13:30 Cafe David 1 2 TA appointment Cafe D
1W II K200 : October 6, 2004 Version : 1.2, kawahira@math.nagoa-u.ac.jp, http://www.math.nagoa-u.ac.jp/~kawahira/courses.htm TA M1, m0418c@math.nagoa-u.ac.jp TA Talor Jacobian 4 45 25 30 20 K2-1W04-00
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meiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F
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(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y
![(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y](/thumbs/98/136512264.jpg "(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = ( ) a c b d (a c, b d) P = (a, b) O P ( ) a p = b P = (a, b) p = ( ) a b R 2 {( ) } R 2 x = x, y")
(2016 2Q H) [ ] R 2 2 P = (a, b), Q = (c, d) Q P QP = a c b d (a c, b d) P = (a, b) O P a p = b P = (a, b) p = a b R 2 { } R 2 x = x, y R y 2 a p =, c q = b d p + a + c q = b + d q p P q a p = c R c b
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
December 28, 2018
e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................
linearal1.dvi
19 4 30 I 1 1 11 1 12 2 13 3 131 3 132 4 133 5 134 6 14 7 2 9 21 9 211 9 212 10 213 13 214 14 22 15 221 15 222 16 223 17 224 20 3 21 31 21 32 21 33 22 34 23 341 23 342 24 343 27 344 29 35 31 351 31 352
20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................
211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s
![[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s](/thumbs/94/118579915.jpg "[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s")
[ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =
n ( (
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16 B (1) 3 (2) (3) 5 ( ) 3 : 2 3 : 3 : () 3 19 ( ) 2 ax 2 + bx + c = 0 (a 0) x = b ± b 2 4ac 2a 3, 4 5 1824 5 Contents 1. 1 2. 7 3. 13 4. 18 5. 22 6. 25 7. 27 8. 31 9. 37 10. 46 11. 50 12. 56 i 1 1. 1.1..
2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =
1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,
17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1,
17 ( ) 17 5 1 4 II III A B C(1 ) 1,, 6, 7 II A B (1 ), 5, 6 II A B (8 ) 8 1 I II III A B C(8 ) 1 a 1 1 a n+1 a n + n + 1 (n 1,,, ) {a n+1 n } (1) a 4 () a n OA OB AOB 6 OAB AB : 1 P OB Q OP AQ R (1) PQ
v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i
1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [
,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x
. P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +
Part () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
1 Ricci V, V i, W f : V W f f(v ) = Imf W ( ) f : V 1 V k W 1
1 Ricci V, V i, W f : V W f f(v = Imf W ( f : V 1 V k W 1 {f(v 1,, v k v i V i } W < Imf > < > f W V, V i, W f : U V L(U; V f : V 1 V r W L(V 1,, V r ; W L(V 1,, V r ; W (f + g(v 1,, v r = f(v 1,, v r
() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (
3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc
1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2
θ i ) AB θ ) A = B = sin θ = sin θ A B sin θ) ) < = θ < = Ax Bx = θ = sin θ ) abc θ sin 5θ = sin θ fsin θ) fx) = ax bx c ) cos 5 i sin 5 ) 5 ) αβ α iβ) 5 α 4 β α β β 5 ) a = b = c = ) fx) = 0 x x = x =
1 n A a 11 a 1n A =.. a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = 0 ( x 0 ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 1.1 Th
1 n A a 11 a 1n A = a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = ( x ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 11 Th9-1 Ax = λx λe n A = λ a 11 a 12 a 1n a 21 λ a 22 a n1 a n2
chap1.dvi
1 1 007 1 e iθ = cos θ + isin θ 1) θ = π e iπ + 1 = 0 1 ) 3 11 f 0 r 1 1 ) k f k = 1 + r) k f 0 f k k = 01) f k+1 = 1 + r)f k ) f k+1 f k = rf k 3) 1 ) ) ) 1+r/)f 0 1 1 + r/) f 0 = 1 + r + r /4)f 0 1 f
1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3
1. 1 A : l l : (1) l m (m 3) (2) m (3) n (n 3) (4) A 2 1 2 1 2 3 α, β γ α β + γ = 2 m l lm n nα nα = lm. α = lm n. m lm 2β 2β = lm β = lm 2. γ l 2. 3 4 P, Q R n = {(x 1, x 2,, x n ) ; x 1, x 2,, x n R}
X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
II R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k
II 231017 1 1.1. R n k +1 v 0,, v k k v 1 v 0,, v k v 0 1.2. v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4.
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0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................
x V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R
V (I) () (4) (II) () (4) V K vector space V vector K scalor K C K R (I) x, y V x + y V () (x + y)+z = x +(y + z) (2) x + y = y + x (3) V x V x + = x (4) x V x + x = x V x x (II) x V, α K αx V () (α + β)x
January 27, 2015
e-mail : kigami@i.kyoto-u.ac.jp January 27, 205 Contents 2........................ 2.2....................... 3.3....................... 6.4......................... 2 6 2........................... 6
LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University
LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2002 2 2 2 2 22 2 3 3 3 3 3 4 4 5 5 6 6 7 7 8 8 9 Cramer 9 0 0 E-mail:hsuzuki@icuacjp 0 3x + y + 2z 4 x + y
r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B
1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n
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18 10 01 ( ) 1 2018 4 1.1 2018............................... 4 1.2 2018......................... 5 2 2017 7 2.1 2017............................... 7 2.2 2017......................... 8 3 2016 9 3.1 2016...............................
18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C
8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,
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2S III IV K200 : April 16, 2004 Version : 1.1 TA M2 TA 1 10 2 n 1 ɛ-δ 5 15 20 20 45 K2-2S04-00 : C 2S III IV K200 60 60 74 75 89 90 1 email 3 4 30 A4 12:00-13:30 Cafe David 1 2 TA 1 email appointment Cafe
AI n Z f n : Z Z f n (k) = nk ( k Z) f n n 1.9 R R f : R R f 1 1 {a R f(a) = 0 R = {0 R 1.10 R R f : R R f 1 : R R 1.11 Z Z id Z 1.12 Q Q id
1 1.1 1.1 R R (1) R = 1 2 Z = 2 n Z (2) R 1.2 R C Z R 1.3 Z 2 = {(a, b) a Z, b Z Z 2 a, b, c, d Z (a, b) + (c, d) = (a + c, b + d), (a, b)(c, d) = (ac, bd) (1) Z 2 (2) Z 2? (3) Z 2 1.4 C Q[ 1] = {a + bi
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