PC PSI PT JEAN-MARIE MONIER GUILLAUME HABERER CÉCILE LARDON MÉTHODES ET EXERCICES. Mathématiques. méthodes et exercices. 3 e.

Transkript

1 PC PSI PT MÉTHODES ET EXERCICES JEAN-MARIE MONIER GUILLAUME HABERER CÉCILE LARDON Mathématiques méthodes et exercices 3 e édition

2 Conception et création de couverture : Atelier 3+ Dunod, rue Laromiguière, Paris ISBN

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4 f : I C I R+ C. +, 0, a, a R x α f(x) x x x 1 f K E, A n(k) f(x) =λx λ K, x E {0}. f χf f E χf, ( ) 5 4 A = 2(R). 6 5 ( E ) f(p ) = λp P 0, P, P P. E f A χa A χa(λ) = 5 λ λ =(λ2 25)+24=λ 2 1=(λ + 1)(λ 1), A 1 1 R (A) ={ 1, 1} 1 ( x X = 2,1(R) y) X (A, 1) AX = X { 5x +4y = x x = y, 6x +5y = y ( 1 (A, 1) = ( ) (A, 1)=1, 1) X (A, 1) AX = X { 5x +4y = x 6x +4y =0, 6x +5y = y ( 2 (A, 1) = ( ) (A, 1)=1. 3) A 2(R)

5 I R (fn : I R)n N (Ak)k N I I k N Ak = I k N (fn)n Ak (fn)n (fn : I R)n N (Ak)k N I Ak = I k N (fn)n Ak (fn)n k N I (fn)n N f I n N fn f (fn)n N (gn)n N I (fngn)n N I I (fn)n N I g (fng)n N fn C.U. n f gn C.U. g fn + gn f + g n n (fn)n N f I n N fn C 1 I f C 1 I (fn)n N f I n N fn C 1 I (f n)n N I (Pn :[0;1] R)n1 R f f (fn : R R)n N C 1 (f n)n N R (fn)n N R y = f(x) Γ y = f(x), f(x) = 3 x 2 (x 6). { x(t) =t 2 1 Γ y(t) =2t 3 +3t 2 1. x(t) =t t Γ y(t) = t t. y = f(x) x Γ y = f(x), f(x) = x, f : R R, x f(x) = x. f C R f (n) (x) (n, x) N R. n N {0, 1} f (n) (x) =0, x R. C 1 (a, b) R 2 a b, f :[a; b] C C 1 [a; b]. b f(x) λx x a λ + 0. C 0 f :[0;+ [ R( C (xn)n N ]0 ; + [ xn n N, f(xn) =0 ). k N, f (k) (0) = 0. n 0 (x, y) R 2 x (1 + x) 0 <x<y < y (1 + y). 1 4 S = AA, n(r). S n +. λ R (S). V S λ. R (S) R+. XSX X n,1(r). L1,L2,L3 A L1 2 =1 L2 = ( a b c ) (L1 L2) = 0 L2 2 2 =1, L3 = L1 L2. Ω 3(R). (Ω). (Ω) =1 1. (Ω) =1 f f = E3 f f ΩX = X, X 3,1(R). θ f (Ω)=1+2 θ, θ [x, f(x), I] x E3 I I f (Ω) = 1 Ω f f f (M, N) ( n(r) ) 2 ( fa(m) fa(n) ) =(M N). S. ( A ) A = n ( A )=1, X n,1(r) {0} AX = λx. XSX C = AB BA. C n(r), C 2 n(r), C 4 + n. n(r). X X 3 = n. ( AA) = 2 (A) = 1. χ AA AA X (A). Y n,1(r), λ R, k N 1 1, k = t k 1 t XHnX X = x1 n,1(r). A A, A 1 + B. S = PDP 1, P n(r),d = (λ1,..., λn) n(r). P =(pij)ij. n i {1,..., n}, sii = λkp 2 ik. f λk p 2 ik, 1 i n. S ++ (X + λy )A(X + λy ) 0. k=1 0 xn f : x x. S + n S / ++ n n. S = A A S.

6 t R x 1 9 ( 6 x x I (a, b) R 2 x [a; b] I fn(x) n0 I =[0;1[ fn : I R, x x n I =[0;1[ fn : I R, x x n I =[ 1;1], fn : x x + 1 n +1 x + 1 n +2, + fn(x) = x +1 x n=0 n N 0 fn + gn fn + gn. I = R fn : I R, x 1 n 2 g : I R, x x fn I n1 fng n1 I n N 0 fng fn g. fn fn [1 ; + [ n1 1 (n +1) x 1 1, Rn n +1 n +1 ]1 ;+ [ =1 Rn n Rn(x) fn+1(x) = Rn 0 fn [1 ; + [ n n1 f : x 3 x 2 (x 6) R R {0, 6} x R {0, 6}, f x(x 4) (x) = ( 3 x 2 (x 6) ) 2 f x f (x) f(x) t x (t) x y y (t) f(x) =x x = x ( y ) 2 + o ( 1 x 2 )) = x 2 4 ( 1 ) x + o, x Γ D y = x 2, + Γ D y 1 O x Γ D 1 O 1 x A 1 x y C 1 R x (t) =2t, y (t) =6t 2 +6t =6t(t +1). x y +, x y 1 0, { x (t) =0 Γ t R, y (t) =0 t =0, Γ

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9 K K R C n,p (K) K = R C

10 E K A, B, C E A +(B C) (A + B) (A + C). B C B, A +(B C) A + B, B C C, A +(B C) A + C A +(B C) (A + B) (A + C). x A +(B C) a A, y B C x = a + y a A y B x = a + y A + B a A y C x = a + y A + C x (A + B) (A + C) F, G E E F G = {0} F + G = E F F G G F G F G E F G = {0} F + G = E (E) = (F )+ (G), E n N n(r) n(r) n(r) n(r) n(r) n(r) n(r) n(r) n(r) n(r) n(r) n(r) 0 n(r) α R, A,B n(r) (αa + B) =α A + B = αa + B, αa + B n(r) n(r) n(r) n(r) n(r) 0 n(r) α R, A,B n(r) (αa + B) =α A + B = α( A)+( B) = (αa + B), αa + B n(r) n(r) n(r)

11 {0} n(r) n(r) A n(r) n(r) A = A A = A, 1 A = A, 2A =0 2A =0, A =0 2 n(r) n(r) {0} n(r) n(r) ={0} n(r)+ n(r) n(r) M n(r) S n(r), A n(r) M = S + A S n(r), A n(r) M = S + A M = (S + A) = S + A = S A. 1 2 S = 1 2 (M + M), A = 1 2 (M M). S = 1 2 (M + M), A = 1 2 (M M). S = 1 2 ( M + M) =S, S n(r) A = 1 2 ( M M) = A, A n(r) E = R 4 a =(1, 1, 1, 1), b =(1, 2, 3, 0), c =(1, 1, 1, 4), d =(1, 2, 3, 0), F = (a, b), G = (c, d). F G E S + A = 1 2 M + 1 M M 1 M = M, (S, A) 2 n(r)+ n(r) = n(r) n(r) n(r) n(r) n(r) (a, b) (c, d) F =(a, b) F G =(c, d) G F G =(a, b, c, d) E (F G)=4= (E), F G (α, β, γ, δ) R 4 α + β + γ + δ =0 α +2β γ +2δ =0 αa + βb + γc + δd =0 α +3β + γ 3δ =0 α +4γ =0 α = 4γ α = 4γ γ =0 β 3γ + δ =0 β = γ + δ δ =0 2β 5γ +2δ =0 2γ +2δ =0 α =0 3β 3γ 3δ =0 3γ +4δ =0 β =0. F G E F G E E

12 E K 5 F, G E F G = {0}, (F )=2, (G) =3. F G E (F +G) = (F )+ (G) (F G) =2+3 0 =5= (E), F + G = E F G = {0} F + G = E F G E E 1,..., E N E (x 1,..., x N ) E 1... E N N x i =0 = ( i {1,..., N}, x i =0 ) ( N ) N E i = (E i ) E 1,..., E N E R R R F 1 F 2 F 3 E ] ;1] ] ; 1] [1 ; + [ [ 1;+ [ F 1,F 2,F 3 E A R F = { f E ; x A, f(x) =0 } E F E 0 F α R, f,g F x A, (αf + g)(x) =αf(x) + g(x) =0, αf + g F }{{}}{{} =0 =0 F E F 1,F 2,F 3 E f 1 F 1,f 2 F 2,f 3 F 3 f 1 + f 2 + f 3 =0 x ] ; 1] F 1 F 2 f 1 (x) =0 f 2 (x) =0 f 3 (x) = ( f 1 (x)+f 2 (x) ) =0. x ] ; 1], f 3 (x) =0. F 3 x [ 1;+ [, f 3 (x) =0. x R, f 3 (x) =0, f 3 =0 f 1 =0 f 2 =0 F 1,F 2,F 3

13 (f a) a R a R { 0 x a f a : R R, x 1 x>a N N, a 1,..., a N R λ 1,..., λ N R N λ k f ak =0. k=1 i {1,..., N} λ i 0 f ai = 1 λ k f ak. λ i k i a R f a R {a} f a a 1 λ k f ak a i λ i k i f ai a i i {1,..., N}, λ i =0, (f ai ) 1iN (f a) a R (f a) a R (f a) a R a R f a : R R, x (x + a) a R x R, f a(x) =(x + a) = a x a x, f a =( a) +( a). f a f 0,f 1,f 2, (f 0,f 1,f 2 ) (f a) a R (f a) a R

14 H E E H E H 1 H H E (H) = (E) 1, E H 0 R E E R H E D E (1) 1 H D = {0} 0 0 u =(u n) n N E l u v = u (l) u = v +(l), v H, (l) D. H + D = E D H E H E { 1/2 } H = f E ; f =0 0 R E = C([0 ; 1], R). 1/2 ϕ : E R, f f 0 E 1/2 ϕ(1) = 1= H = (ϕ) E n N H = C n 1 [] E = C n[] E C H E (E) =n +1, (H) =n (H) = (E) 1 H E

15 0. N N, E K p 1,..., p N E N p i =0. i {1,..., N}, p i =0. i {1,..., N} E p i E (p i )= (p i ). ( N ) N N 0= p i = (p i )= (p i ). }{{} 0 i {1,..., N}, (p i )=0 i {1,..., N}, p i =0. A n,p (K) (A) =r P ( n (K), ) Q p (K) A = P n,p,r Q, r 0 n,p,r = 0 0 n,p (K). n, p, q, r N,A p,q(k), B n,r(k) (B) (A) (P, Q) n,p(k) q,r(k), B = PAQ. = (B) (A) a = (A), b = (B) R p(k), S q(k) A = R p,q,as T n(k), U r(k) B = T n,r,b U b a ( ) ( )( )( ) b 0 b 0 a 0 b 0 J n,r,b = = = n,p,b p,q,a q,r,b. B = T n,r,b U = T n,p,b p,q,a q,r,b U =(T n,p,b R 1 )(R p,q,as)(s 1 q,r,b U). P = T n,p,b R 1 n,p(k) Q = S 1 q,r,b U q,r(k), B = PAQ = (P, Q) n,p(k) q,r(k) B = PAQ (B) = ( (PA)Q ) (PA) (A).

16 n N A, B, C, D n(k) (α, β) K 2 α β ( ) ( ) α n 0 A B J =, M =. 0 β n C D M J B =0 C =0. ( )( ) ( )( ) α n 0 A B A B α n 0 JM = MJ = 0 β n C D C D 0 β n ( ) ( ) αa αb αa βb = βc βd αc βd { αb = βb βc = αc { (α β)b =0 (α β)c =0 { B =0 C =0. E K F E n = (E) p = (F ) L F (E) E F L(E) L F (E) L(E) G F E L F,G (E) E F G L(E) L F,G (E) L(E) L F (E) L(E) 0 L F (E) α E, f,g L F (E) x F, (αf + g)(x) =αf(x) + g(x) F, }{{}}{{} F F F αf + g αf + g L F (E) L F (E) L(E) E B = (e 1,..., e n) F (e 1,..., e p) F f L(E), M = B (f) F f M ( ) A B M =, 0 C A p(k), B p,n p (K), C n p (K) p(k) p,n p (K) n p (K) L F (E) (A, B, C) f E ( ) A B B (f) =, 0 C ( L F (E) ) = ( p(k) p,n p (K) n p (K) ) = ( ) p(k) + ( p,n p (K) ) + ( n p (K) ) = p 2 + p(n p)+(n p) 2 = n 2 np + p 2. L F (E) L(E) ( L F (E) ) = ( L(E) ) 1. ( )

17 ( ) n 2 np + p 2 = n 2 1 np p 2 =1 p (n p)=1 }{{}}{{} { p =1 n p =1 { n =2 p =1. N N L F (E) L(E) n =2 p =1 L F,G (E) L(E) L F,G (E) = L F (E) L G (E), L F,G (E) L(E) E B =(e 1,..., e n) E = F G (e 1,..., e p) F (e p+1,..., e q) G f L(E), M = B (f) F G f M ( ) A 0 M =, 0 C A p(k), C n p (K) p(k) n p (K) L F,G (E) (A, C) f E ( ) A 0 B (f) =, 0 C ( L F,G (E) ) = ( p(k) n p (K) ) = ( ) p(k) + ( n p (K) ) = p 2 +(n p) 2 ( L F,G (E) ) = ( L(E) ) 1 = n 2 2np +2p 2. n 2 2np +2p 2 = n 2 1 2(np p 2 )=1, L F,G (E) L(E)

18 E,F K n N E 1,..., E n E E 1,..., E n E ( x 1 E 1,..., x n E n, x1 + + x n =0 = x 1 =... = x n =0 ). E 1,..., E n E (i, j) {1,..., n} 2 (, i j = Ei E j = {0} ). E 1,E 2,E 3 E E 1 (E 2 + E 3 )=(E 1 E 2 )+(E 1 E 3 ). E 1,E 2,E 3 E E 1 +(E 2 E 3 )=(E 1 + E 2 ) (E 1 + E 3 ). R E = R R E 1 E 2 E E R 3 p, q, r 4p +5q +6r = R 3. f,g E (f) (f) g f g f,g E A, B n (K), (AB) = (BA) A, B n (K), (AB) = (A) (B)

19 E K A, B, C E. A + ( B (A + C) ) = A + ( C (A + B) ). ( f a :[0;+ [ R, x 1 ) x + a a ]0 ;+ [ ( f a : R R, x (x a) ) a R. A 3,2 (R) B 2,3 (R) AB = C, C C = , , ? n N,X= {x 1,..., x n } n F = K X i {1,..., n}, i : F K, f f(x i ), x i. ( i ) 1in F. n N (A, B) ( n (C) ) 2 AB BA = n ( ) A B n, p N M = A 0 C n (K), B n,p (K), C p (K). M A C A C M 1 ( f a : R R, x x a 3/2) a R RR. E K[]. E E

20 n N. A n (K) n (K) K, X (AX) n (K) θ : n (K) n (K) ( ) A n (K), X n (K), θ(a) (X) = (AX) K n N,A,B,C n (C) A 2 = A, B 2 = B, C 2 = C. M = A + 2 B + 3 C M 2 = M. B = C =0. n, p N,A n,p (K), r= (A). U n,r (K), V r,p (K), A = UV. M M =(U V ), (M) (U)+ (V ). ( ) R M M =, S (M) (R)+ (S). ( ) A B M M =, C D A D (M) (A)+ (B)+ (C)+ (D). ( ) A B m, n, p N m n p n, M = C 0 n (K), A m,p (K), B m,n p (K), C n m,p (K). M (A) m + p n.. 1. n, p N,A=(a ij ) ij n,p (K). ( n ) A l = a ij, A c = 1jp 1in X =(x j ) 1jp p,1 (K) X 1 = p ( p ) a ij, x j, X = 1jp x j. AX 1 AX A l =, A c =. X p,1(k) {0} X 1 X p,1(k) {0} X

21 E K F, G E E. G = { f L(E); (f) =F (f) =G }. G. E n = (E), p = (F ), B 1 =(e 1,..., e p ) F, B 2 =(e p+1,..., e n ) G, B =(e 1,..., e n ) E. θ : f B (f) (G, ) { ( ) } M 0 (H, ) H = 0 0 n (K); M p (K). n (K) n (K) n N {0, 1}. n (K) n (K). ( ) n, p N n B,B n,p (K), C p (K). = n + (C). 0 C n, p N,R n,p (K), S p,n (K). p + ( n + RS) =n + ( p + SR). ( ) A 0 n, p N,A n (K), B p (K). = (A)+ (B). 0 B ( ) ( ) A 0 B 0 n N,A,B n (K). 0 A 0 B A B n, ( p N,A,B ) ( n (K), ) U,V p (K). A B A 0 B 0 U V 0 U 0 V E K f L(E) F E (F ) (f) G F E (u, v) ( L(E) ) 2 (u f v) =F (u f v) =G. ( ) A B M =, A C D n (K), B n,p (K), C p,n (K), D p (K). M D CA 1 B M 1

22 X AXB =0 m, n, p, q N,A m,n (K), B p,q (K). E = { X n,p (K); AXB =0 }. E K n N,A n (K) B,C n (K) A = BC, B, C. ( ) A B n, p N M = C D A n (K) B n,p (K) C p,n (K) D p (K) (M) =n D = CA 1 B. K E K p N,F 1,..., F p E p F i = E. i {1,..., p} F i = E. GL(E) E K e = E, G GL(E) n = (G) p = 1 g. n g G h G, p h = p. p E. (g e) = (p). g G ( g G ) (g e) = 1 n (g). g G

23 (f a) a [0 ;+ [ (f a) a R (f 1,f 0,f 1 ) A, B (A, B) i {1,..., n} i F. ( i ) 1in j {1,..., n} f j : x i δ ij. ( ) X Y M = Z T a R, f a C 2 R {a} C 2 R. n = (E). (P 1,..., P n+1 ) (P 1 )... (P n+1 ), (Q 1,..., Q n+1 ) Q n+1 = P n+1 i {1,..., n}, (Q i ) < (P n+1 ), (P 1,..., P n) (P 1 ) <... < (P n) (S 1,..., S n) S n = P n i {1,..., n}, (S i )= (P n). A n(k), ϕ A : n(k) K, X (AX) n(k). θ (α, β, γ) Z 3 α + β 2+γ 3=0, α = β = γ =0. n,p,r. (M) =n. X p,1 (K), AX 1 A l X 1. j j n A l = a ij. X p,1 (K), AX A c X. 1 X = ε, ε p a i0 j a i0 j 0 ε j = a i0 j 1 a i0 j =0, p i 0 A c = a i0 j. G, G GL(E). F G, ( f ) G f M 0 B, M 0 0 p(k). ( ) M 0 A = H, M 0 0 p(k), f E, A B G. θ ϕ θ (G, ) (H, ).

24 H n(k). H n(k) =. H ( )( ) ( ) n B n B n 0 =. 0 C 0 p 0 C n + RS p + SR n + p... (u, v) ( L(E) ) 2 u f v = p, p F G.... ( ) X Y N =, MN = Z T n+p. ( )( )( n 0 A B n A 1 ) B CA 1 p C D 0 p ( ) A B = 0 D CA 1. B E K m,n,a p,q,b a = (A), b= (B) a b r = (A) < n ( ) M r r+1 (K) r Mr 0 N r = 0 0 n(k). ( )( )( n 0 A B n A 1 ) B CA 1 p C D 0 p ( ) A 0 = 0 CA 1. B D p. F 1,..., F p+1 E p+1 p F i = E, F p+1 E, F i E, p x, y E x / F p+1 y / F i, y x h G g g h G g h = g. g G g G p 2 x (g e), g G p(x) =x. x (p) g(x) = (g p)(x), g p = p.

25 n =2 n 3 (E 1 E 2 )+(E 1 E 3 ) E 1 (E 2 + E 3 ). E 1 +(E 2 E 3 ) (E 1 + E 2 ) (E 1 + E 3 ). E 1 E 2 E E 1 E 2 = {0} 0 f E f = g 1 + g 2 g 1 : x f(x)+f( x) f(x) f( x), g 2 : x, (g 1,g 2 ) E 1 E (p, q, r) 3= ( R 3)= (4p +5q +6r) =4 (p)+5 (q)+6 (r) =4 (p)+5 (q)+6 (r), (p), (q), (r) N ( E = ) R 2 f,g ( ) A =, B = n 2 A = B = n

26 x A + ( B (A + C) ). a A, b B (A + C) x = a + b. b B a A, c C b = a +c. x = a + b =(a + a )+c. a + a A. c C c =( a )+b A + B, c C (A + B). x A + ( C (A + B) ). A + ( B (A + C) ) A + ( C (A + B) ). (B, C) (C, B), A + ( B (A + C) ) = A + ( C (A + B) ). (A + B) (A + C). n N,a 1,..., a n ]0 ; + [ n λ 1,..., λ n R λ k f ak =0. k=1 x [0 ; + [, n λ k =0. x + a k=1 k [0 ; + [ R, λ k x R {a 1,...,a n}, =0. x + a k=1 k x a k k {1,..., n}, λ k =0. (f a) a ]0 ;+ [ a R x R, f a(x) = (x a) = a x a x, f a (f a) a R, 2 x R (f 1 + f 1 )(x) = (x +1)+ (x 1) =2 1 x =(2 1)f 0 (x), n f f 0 + f 1 =0, (f a) a R A = , B = 0 0 A = , B = (A, B) 3= (C) = (AB) (A) 2, (A, B) ( 1 0 ) ( 1 1 ) i {1,..., n}, i F i F K i α K, f, g F, i (αf + g) =(αf + g)(x i ) = αf(x i )+g(x i )=α i (f)+ i (g). n (α 1,..., α n) K n α i i =0. j {1,..., n} { f j : X K, x i 1 i = j 0 i j. n 0= α i f j (x i )=α j. ( 1,..., n) F. X n F = K X n F n ( 1,..., n) n F F. (A, B) ( ) 2 n(c) AB BA = n (AB BA) = ( n)=n. (AB BA) = (AB) (BA) =0, (A, B) ( ) 2 n(c) AB BA = n. ( ) A B (M) = = (A) (C) 0 C (M) 0 ( (A) 0 (C) 0 ) M A C

27 A C M ( M) 1 M M 1 X Y =. Z T ( )( ) ( ) MM 1 A B X Y n 0 = n+p = 0 C Z T 0 AX + BZ = n Z =0 AY + BT =0 T = C 1 CZ =0 C AX = n CT = p AY = BC 1 Z =0 A T = C 1 X = A 1 Y = A 1 BC 1. ( M 1 A 1 A = 1 BC 1 ) 0 C 1. n N a 1,..., a n R n λ 1,..., λ n R λ k f ak =0. k=1 i {1,..., n}. λ i 0. f ai = 1 λ k f ak. λ i 1kn, k i a R, f a C 2 R {a} C 2 R. f ai C 2 a i, f ai C 2 a i, i {1,..., n}, λ i =0. (f a) a R n = (E). n =1. n. E K[] n +1. E B =(P 1,..., P n+1 ). B, i {1,..., n +1}, (P i ) (P n+1 ). C = (Q 1,..., Q n+1 ) Q n+1 = P n+1 i {1,..., n} P i (P i ) < (P n+1 ) Q i = P i α i P n+1 (P i )= (P n+1 ), α i (P i α i P n+1 ) < (P n+1 ). α i P i P n+1 Q 1,..., Q n+1 P 1,..., P n+1. P n+1 = Q n+1 i {1,..., n} P i = Q i P i = Q i + α i Q n+1, P 1,..., P n+1 Q 1,..., Q n+1. (C) = (B) =E. (E) =n +1 C E n +1 C E. F = (Q 1,..., Q n), n R[]. F F =(R 1,..., R n) G =(R 1,..., R n,p n+1 ). E = F P n+1 K[] F F, G E. i {1,..., n}, R i (Q 1,..., Q n) (Q 1,..., Q n) < (P n+1 ) i {1,..., n}, (R i ) < (P n+1 ). G E n. n = (E). E E B =(P 1,..., P n) (P 1 ) <... < (P n). P i + P n i<n i {1,..., n} S i = P n i = n. i {1,..., n}, (S i )= (P n). S 1,..., S n P 1,..., P n. S i S n i<n i {1,..., n}, P i = S n i = n, P 1,..., P n S 1,..., S n. (E) =n C =(S 1,..., S n) n E, C E. E A n(k). ϕ A : n(k) K, X (AX) α K, X, Y n(k), ϕ A (αx + Y )= ( A(αX + Y ) ) = (αax + AY ) = α (AX)+ (AY )=αϕ A (X)+ϕ A (Y ). ϕ A n(k).

28 θ : n(k) n(k) A n(k), X n(k), θ(a)(x) = (AX). A n(k), θ(a) =ϕ A. θ α K, A, B n(k). X n(k) θ(αa + B)(X) = ( (αa + B)X ) = (αax + BX) =α (AX)+ (BX) = αθ(a)(x)+θ(b)(x) = ( αθ(a)+θ(b) ) (X), θ(αa + B) =αθ(a)+θ(b), θ. θ A (θ). θ(a) =0 X n(k), (AX) =0. A =(a ij ) ij. (i, j) {1,..., n}. a 1i 0= A ij )= (0) (0) = a ji, a ni i A j A =0. (θ) ={0}, θ θ : n(k) n(k) n(k) n(k) θ K A, B, C, M (M) = (A + 2 B + 3 C) = (A)+ 2 (B)+ 3 (C), ( ) (A) (M) + (B) 2+ (C) 3=0. } {{ } }{{} }{{} α β γ (α, β, γ) Z 3 α + β 2+γ 3=0. (α, β, γ) =(0, 0, 0). γ 3 α 2 +2β 2 +2αβ 2=3γ 2, 3γ 2 α 2 2β 2 αβ 0 2= Q, 2αβ 2 αβ =0. αγ =0 βγ =0. α 0, β =0 γ =0, α =0, α =0. βγ =0, β =0 γ =0 β =0 γ =0. α =0, β =0, γ =0. (B) =0 (C) =0, (B) = (B) =0 (C) = (C) =0, B =0 C =0. r = (A), P n(k), Q p(k) A = P n,p,rq, n,p,r = ( ) r 0 r,p r. 0 n r,r 0 n r,p r ( ) r (r ) n,p,r = 0 0 r,p r, n r,r A ( r A = P ) ( ) r 0 0 r,p r Q, n r,r }{{} V } {{ } U U n,r(k), V r,p(k). U 1,..., U p U V 1,..., V q V, (U 1,..., U p,v 1,..., V q) = (U 1,..., U p)+ (V 1,..., V q), (U 1,..., U p,v 1,..., V q) (U 1,..., U p)+ (V 1,..., V q), (M) (U)+ (V ). ( R ( ( (M) = = R ) = S) S) ( R S ) ( R)+ ( S)= (R)+ (S). ( ) ( ( A B A B (M) = + C D C) D) ( (A)+ (C) ) + ( (B)+ (D) ). M n = (M) (A)+ (B)+ (C). B m,n p (K) C n m,p (K), (B) n p (C) n m, n (A)+(n p)+(n m), (A) m + p n. x 1 X = p,1 (K) x p n p AX 1 = a ij x j n p a ij x j = p ( n ) a ij x j

29 p p A l x j = A l x j = A l X 1. X p,1 (K) {0}, A l = 1jp ( n j {1,..., p} A l = AX 1 X 1 A l. ) a ij, n a ij. X = j, j 1. a 1j X 1 =1 AX =, a nj p AX 1 = a ij = A l, AX 1 = A l. X 1 A l AX 1 = A l. X p,1 (K) {0} X 1 x 1 X = x p p,1 (K) n AX = a ij x j 1in p ( p ) a ij x j a ij X 1in 1in ( p ) = a ij X = A c X. 1in X p,1 (K) {0}, A c = 1in ( p i 0 {1,..., n} A c = 1 X = ε ε p j {1,..., p} ε j = AX X A c. ) a ij, p a i0 j. p,1 (K) a i0 j a i0 j a i0 j 0 1 a i0 j =0. X =1, X 1 X 0. AX = 1in ( p ) a ij ε j p a i0 jε j. j {1,..., p} a i0 jε j = a i0 j, a i0 j 0, a i0 j =0. p AX a i0 j = A c. AX X p,1 (K) A c. X AX = A c. X p,1 (K) {0} X G. f 1,f 2 G. (f 2 f 1 ) (f 2 )=F. z F. z F = (f 2 ), y E z = f 2 (y). E = F G, u F, v G y = u + v. z = f 2 (y) =f 2 (u + v) =f 2 (u)+f 2 (v). u F = (f 1 ), x E u = f 1 (x), v G = (f 2 ), f 2 (v) =0. ( z = f 2 f1 (x) ) = f 2 f 1 (x) (f 2 f 1 ). F (f 2 f 1 ). (f 2 f 1 )=F. (f 2 f 1 ) (f 1 )=G. ( x (f 2 f 1 ); f 2 f1 (x) ) =0, f 1 (x) (f 1 ) (f 2 )=F G = {0}, x (f 1 )=G. (f 2 f 1 ) G. (f 2 f 1 )=G. f 2 f 1 G. p F G. p L(E), (p) =F, (p) =G, p G. f G. x E, f(x) (f) =F, x E, p ( f(x) ) = f(x), p f = f. x E, x p(x) (p) =G = (f), x E, f ( x p(x) ) =0, x E, f(x) =f ( p(x) ), f = f p. p G.

30 f G. F G = (f) E, f : F (f) =F, x f(x) K g : E E, x f 1( p(x) ), p g (g) =f 1( p(e) ) = f 1 (F )=F. x E x (g) g(x) =0 f 1( p(x) ) =0 p(x) =0 x G, (g) =G. g G. x E (f g)(x) =f ( f 1( p(x) )) = f ( f 1( p(x) )) = p(x), f g = p. x E f(x) (f) =F, p ( f(x) ) = f(x), g ( f(x) ) = f 1( p ( f(x) )) = f 1( f(x) ). f = f p, f 1( f(x) ) = f 1( f ( p(x) )) = f 1( f ( p(x) )) = p(x). g f = p. g f = f g = p, f g (G, ). (G, ) f G, (f) ( =F ) (f) =G, M 0 f B, M 0 0 f f F. ( ) M 0 (M) = = (f) = (F )=p. 0 0 ( ) M 0 M p(k), H. 0 0 θ : G H, f B (f). ϕ A H, f E B (f) =A ( ) M 0 A = = 0 0 B (f) M p(k), (f) =F (f) =G, f G. θ ϕ ( )( ) M2 0 M1 0 f 1,f 2 G, θ(f 2 )θ(f 1 )= ( ) M2 M = 1 0 = θ(f f 1 ). θ (G, ) (H, ). (G, ) (H, ) θ : f B (f) (G, ) (H, ). H n(k). H n(k) =. H N n(k), N / H. N / H H n(k), n(k) =H KN. M H α K n = M + αn. M = n αn. N k N N k =0 ( k 1 ( n αn) (αn) p) = n α k N k = n p=0 ( k 1 (αn) p) ( n αn) = n α k N k = n, p=0 n αn n(k). M H n(k), H n(k) (0) (0) N 1 =,N 0 (0) 2 = (0) N 1 N 2 N 1 H N 2 H, H N 1 + N 2 H (0) N 1 + N 2 = 0, 0 (0) n(k) n(k).