数学Ⅱ演習（足助・09夏）

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 2 2 3 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!

A z M 2 R x R 2 f z x A z x A z z M 2 R 2 2 2 z A z, w C A z A w 3 M 2 R 4 f z g z 3 8 7 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 +

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 2 2 + a b 2 + a 2 b 2 a 2 a 22 + b 2 b 2 2 + a 2 b 2 2 + b 2 a 22 zw z w 3 z, w 2, z + w z + w z + w 2 z + w 2 a + a 2 2 + b + b 2 2 a 2 + b 2 + a 2 2 + b 2 2 + 2 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 2 2 + 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

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 2 2 2 deta z a 2 + b 2 z 2 a b, X a 2 + b 2 z + z z + z b a z 2 2 2 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 2 2 3 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

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 + 2 + y 2 p x 2 + y 2 p x + 2 + y 2, p 2 x + 2 + y 2, x 2 + y 2 p 2 2p x + 2 + y 2 + x + 2 + y 2 2p x + 2 + y 2 p 2 + 4x, p 2 + 4x 4p 2 {x + 2 + 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

3 4 z z + >, c < z C c, z c z + x + y c x + + y x 2 + y 2 c 2 {x + 2 + y 2 } c 2 x 2 2 + c 2 x + c 2 + c 2 y 2 c x c, c 2, x 2 2 + c2 c 2 + + y2 x + 2 c2 + c c 2 + y 2 2 2 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

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 2 2 3 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

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 2 22 2 3 v L x x 2 t R, x x 2 t v v 2 x 3 3 2 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 2 22 2 3 4 R 2 2 3 2 24 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

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

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 3 4 5 6 6 5 45 45 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

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, 22 2 3 4 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

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 2 2 + a 2 a2 + a 2 2 25 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

II I 3 9/5/8 9/5/9 K R C 3 2 3 5 x y 5 2 4 x 8 5 2 6 3 2 8 y 4 z 3 5 4 6 z 2 2 7 5 5 w 5 3 2 2 2 x 4 3 2 y 5 2 z 3 b a, b R 2 6 a w 32 rank 2 4 3 5 2 2 2 3 2 3 2 2 5 6 4 3 33 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

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+

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

II 3 2 3 5 5 4 6 2 5 5 3 7 4 6 2 5 5 2 3 2 3 /2 2,2 2 5 5 2 3 4 6 2 5 5 3 7 2 3 5 6 23 23, 2 3 3,3 3 x, y, z 2 5 2 4 8 6 3 2 8 3 4 5 2 7 5 5 5 5 2 4 8 4 26 29 9 5 3 7 9 3 8 34 53 4 26 29 35 239 274 33 44 8 8 4 3 3 4 34 34, 4 3 2 3 4 4,4 4 5 2 4 8 33 4 32 5 9 5 3 7 9 3 5 2 4 8 4 26 29 9 5 3 7 7 5 8 34 53 4 26 29 33 44 35 239 274 2 4 2 2,2 2 3,3 3 x, y, z, w 3 2 2 2 4 3 2 3 5 2 b 2 6 a 3 2 3 4 2 2 2 8 4 b + 3 3 2 a + 4 6 2 2,2 2 3 2 3 2 2 2 4 5 2 b 2 6 a 4 2 2 b + a 2, b + b b a 2 a 2, α, β x α, y 2 4α 2β, z α, w β a 2,,

4 2 2 3 4 4 2 2 3,4 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 32 2 4 3 5 2 2 5 6 5 3 5 6 5 3 4 3 2 23 3 2 7 2 6 6 2 5 3 2 4 3 5 2 6 5 3 4 3 2 7 6 6 2 5, 2,2 2 2 3 3,3 3 23 3 2 7 2 6, rank 3 6 2 5 2 2 2 3 2 3 6 4 3 2 2 8 9 3 8 9 2 /4 /4 /4 9/8 3/8 /8 3 3 2 2 3 2 2 2 3 2 3 8 9 2 2 2 8 9 3 3, 2,2 2, rank 2

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

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

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

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 8 2 2 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 2 2 3 2 2 4 3 3 4 2 2 3 2 4 3 + 4 43, 5 2 5 4 6 2 2 5 5 3 2 4 6 3 2 5 5 2 3 5 3 2 5 4 6 3 2 5 44 V, W f : V W v, v 2 V fv fv 2 v v 2, 2 w W v V w fv

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 3 4 46 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

47 A, B M n K AB E n BA E n n R[ t ] + t, + 2t 3t 2 2 3 2, 3 ϕ ψ 48 ϕ, ψ, A, B BA AB

II 4 3 3 2 2 v K 8 λ, λ 2, λ 3, λ 4, λ 5 K, v λ + λ 2 3 2 + λ 3 2 + λ 4 + λ 5 3 2 2 3 v K 5 λ, λ 2 K, v λ 2 + λ 2 3 3 3 3 4 2 6 2 2 v K 8 λ, λ 2, λ 3, λ 4, λ 5 K, v λ + λ 2 3 2 + λ 3 2 + λ 4 + λ 5 3 + 4 6 2 4, 2 3 2 2 3 2 2 + 2 v K 5 λ, λ 2 K, v λ + λ 2

42 2 2 2 2 2 2 3 2 3, 3 2 2, 2/3 4/3 /3 2/3 2 3 2,2 2 i, 3 2 4 2 ii 3, 2 2 2 2 3 iii,, 2 2 2 3 3 2 3 3 2 4 2 4 3 2 4 5 3 3 2 3 2 3 2 7 8 2 3 2 4 2 6 9 4 8 3 6 3 3 5 2 3 3 5 2 8 28 3 7 8 2 4 3 2 33 5 5 7 5, 5 2 2, 3 2 3, 3 2 4 2,2 2 5 3,3 8 3 6 i, 8 28 3 7 5 ii 5, 3 2 4 3 /3 2 2 3 2 2 2 3 7 3 iii, 3 2 4 3 3 2 3

3 i i 2 i 2i i i i 3 4 4 + 2i 4 i i 2 4 3i + 4i 2 + i + 2i 2i 2 9 + 8i/ 3 6i/ 4 + 3i/ 24 + 3i 6 + 2i 4 + i 7 6i/ + 2i/ 2 i/ 4 + 9i 2i 5 i 2 + i/ 4 + 23i/ 3 4i/ 3 3 i 2 + i 4 2i, 3, 2 2,2 2 3 3,3 24 + 3i 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 2 2 + i 2i 9 + 8i/ 7 6i/ 2 + i/ iii i 2 i 3 4 2 4 3i + 4i i 4 + 2i 2 + i i,, 43 5 2 5 4 6 3 4 6 3 3 4 5 2 3 5 2 3 5 2 9 7 2 2 3 3 2 7 3 9 3 25 7 2 4 3 2 4 3 8 5 9 7 2 2 4 2 2 38 26 5 9 3, 5 2 2, 3 2 2, 3 3 4 2,2 2 5 3,3 3 25 7 8 5 9 3,,

2 5 2 5 3 3 4 6 3 3 7 9 2 3 5 9 7 9 7 2 5 5 5 2 5 4 6 4 6 2, 3 2 5 2 3 5 5 2 5 3 25 7 8 5 4 6 8 5 2 3 25 7 2 3 5 9 3 9 3,, 2 5 5 5 2 5 4 6 4 6 3 2 5 2 3 5 A O A O A O 3 A, B, O B O B, A B O B 3 2 2 3 2,, 2, 3 3 2 2 3 2 3 2 5 5 8 5 4 6 3 25 7, 2 5 5 4 6 3 2 5 9 3 3 2 5, 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

45 3 7 3 7 2 8 2 4 6 2 8 2 9 8 5 2, 5 9 5 6 5 9 3 7 3 7 2 2 4 6 2 44 2 22, 22 44 4 6 22 44 7 7 4 8 2 2 6 4 8 4 9 8 9 4, 9 9 5 4 6 9 9 3 3 2 4 2 4 2 2 4 2 9 4 5 2 5 9 5 4 5 9, 3 7 7 3 2 4 6 2 2 6 2 4 2 4 6 5 4 6 5 4 x, x 3 7 2 2, x 3 7 3 3 7 2 4 6 2 4 6 2 4 6 5 6 5 6 5 6 3 7 2 4 6 2 3 7 2 8 4 5 5 4 2 2 5 6 4 5 9 9 2 3, 3, 2 2,2 2 3 3,3, 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

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

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 2 2 2 2 2 4 6 3 2 2 2 4 6 3 4

55 Vandermonde n 2 x x 2 x n 2 2 2 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

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

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 54 2 2 LU, 53, 2 2 2 4 4 4 2 2 2 2 2 4 5 2 2 2 2 LU, 4 4 3 6 5 2,, 4 3 6 2 5, 2 2 2 4 6 2 4 3 2 6 5 2 2 2 3 LU, 3 3 2 2,, 4 3, 2 2 2 2 3 4 3 55 n 2 det x x x 2 x 2

2 n, x x 2 x n+ x x n+ x 2 x n+ x n+ 2 2 2 det x x 2 x n+ det x 2 2 x n+ x 2 2 2 2 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 2 2 2 x n+ x 2 2 n x n+ n+2 x 3 3 x n+ x 3 3 2 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 2 2 + x x n+ + x n+ x 2 2 2 + 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 2 2 2 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

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

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

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 + 2 3 W x 2 3x + 2, W 2 x 2 4

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-

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-

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

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 2 3 2 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

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 + 2 + 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 2 3 2 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 3 3 + s 3 3 3 t 3 + s 3 3 W 4 λt 3 3 λt w W 4, λ R, λw λt 3 3 3 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

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-

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

2 g : K n+ K n [x] a a g a + a + a x + a + a + a 2 x 2 + + 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

v, v 2, v 3 f a v 2 3, v 2 5, v 3 4 2 2 5 b v 3, v 2 3 2, v 3 2 5 2 2 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

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

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

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 2 3 2 3 2 4 3 4 A 3 2 75 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

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 23 3 9 7 2 4 2 3 23 3 9, 7 2 4 5, 6 2 3 2 6 2 3 2 23 3 9 4 23 3 9 7 2 4, 7 2 4 f, 6 2 3 5 6 2 3 fv, fv 2, fv 3 b f 3 4 3 2, f, 2 5 2 fv f 4v 2 v 3 4fv 2 fv 3 4 2 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

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 5 3 3 2 V v K 8 λ, λ 2, λ 3, λ 4, λ 5 K, λ + λ 2 + λ 3 2 + 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

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