(u i ) i I u i. u i u i. (u i ) i I = E

Dimension: px

Commencer à balayer dès la page:

Download "(u i ) i I u i. u i u i. (u i ) i I = E"

Thomas Lessard
il y a 5 ans
Total affichages :

1 E K A B E (A) = (B) A (B) B (A) E K (u i ) i I E (u i ) i I (u i ) i I u i u i u i u i u i u i a = (1, 2, 1) b = (1, 3, 2) c = (1, 1, 0) d = (3, 8, 5) R 3 (a, b) = (c, d) E K (u i ) i I E I (u i ) i I E E E u i (u i ) i I = E {0} ( ) = {0}

2 n N e 1 = (1, 0,..., 0) e 2 = (0, 1, 0..., 0) e n = (0,..., 0, 1) K n (e i ) 1in K n (1, X, X 2,..., X n ) K n [X] (X n ) n N K[X] (u 1,..., u n ) K E x E (λ 1,..., λ n ) K n x = n i=1 λ iu i (u i ) i I K E x E (λ i ) K (I) x = i I λ iu i E K u 1,..., u n E (u 1,..., u n ) u i (λ 1,..., λ n ) K n, n λ i u i = 0 E = i 1, n, λ i = 0 i=1 (u 1,..., u n ) u i (u 1,..., u n ) u i

3 (u 1,..., u n ) K E (λ 1,..., λ n ) K n n λ i u i = 0 E λ i i=1 n N e 1 = (1, 0,..., 0) e 2 = (0, 1, 0..., 0) e n = (0,..., 0, 1) K n (e i ) 1in K n (1, X, X 2,..., X n ) K[X] n (f i ) 1in R R I λ i f i = 0 n λ i f i (x) = 0 x I R i=1 i=1 (, ) R R E K (u i ) i I E I (u i ) i I u i (λ i ) i I K (I), i I λ i u i = 0 E = i I, λ i = 0 (u i ) i I u i (u i ) i I u i (u i ) i I (λ i ) i I (µ i ) i I λ i u i = µ i u i λ i = µ i i I i I i I (X n ) n N K[X]

4 E K (u i ) i I E I (u i ) i I E E (1, i) R C n N e 1 = (1, 0,..., 0) e 2 = (0, 1, 0..., 0) e n = (0,..., 0, 1) K n (e i ) 1in K n K N (1, X, X 2,..., X n ) K n [X] K n [X] (X n ) n N K[X] (e 1,..., e n ) K E x E x (e 1,..., e n ) n n (λ 1,..., λ n ) K n x = λ i e i i=1

5 (e i ) i I K E x E x (e i ) i I (λ i ) i I K (I) x = λ i e i i I F G K E F F G G B F G F G B F G F G R 3 P x = 0 Q y = 0 ((0, 1, 0), (0, 0, 1)) P ((1, 0, { 0), (0, 0, 1)) x = 0 Q P Q P Q y = 0 ((0, 1, 0), (0, 0, 1), (1, 0, 0), (0, 0, 1)) F 1,..., F p K E F 1,..., F p F 1,..., F p B F 1,..., F p p i=1 F i B p i=1 F i E F E F (e 1,..., e n ) (f 1,..., f p ) ((e 1, 0 F ),..., (e n, 0 F ), (0 E, f 1 ),..., (0 E, f p )) E F E F E F (e i ) i I (f j ) j J (e i, 0 F ) i I (0 E, f j ) j J E F E 1,..., E p E 1,..., E p (e 1 1,..., e1 n 1 ),..., (e p 1,..., ep n p ) k 1, p i 1, n k f k i = (0 E1,..., 0 Ei 1, e k i, 0 E i+1,..., 0 Ep ) (f 1 1,..., f1 n 1 ),..., (f p 1,..., fp n p ) p E p E 1,..., E p E 1,..., E p (e 1 i ) i I 1,..., (e p i ) i I p k 1, p i I k f k i = (0 E 1,..., 0 Ei 1, e k i, 0 E i+1,..., 0 Ep ) p (f 1 i ) i I 1,..., (f p i ) i I p E p

6 K n K n (u 1,..., u p ) K n i 1, p a i b i u i (u 1,..., u p ) (a 1,..., a n ) (b 1,..., b n ) (2, 3, 1, 2) ( 2, 1, 0, 0) (1, 0, 0, 0) R 4 (3, 2, 1, 1) (0, 2, 1, 4) (0, 0, 2, 3) (0, 0, 0, 0) R 4 K n K n K n ((1, 2, 1), (1, 3, 2), (1, 1, 0)) ((2, 1, 3, 4), (1, 3, 2, 0), (2, 3, 1, 1)) K n F K n F F F F

7 n 1 K n K[X] (P 1,..., P n ) d = 1in P i X d+1 P i E K L E G E L E G K E K E E E E F = ((1, 5, 7), (2, 6, 8), (3, 1, 15), (1, 11, 1)) ((1, 5, 7), (2, 6, 8)) F E K n E n E K E E K E 0 1 2

8 E K n = n K n [X] = n + 1 K[X] E E K K E C C = 1 R C = 2 E C E R R E = 2 C E E K n E n E n n n n

9 n n n E n B n B E B E B E E n B n B E B B ((0, 1, 2), (1, 2, 0), (2, 0, 1)) R 3 ( p p ) E 1,..., E p K E k E k = p E k F E F F E F = E E = F R 2 R 2 R 3 R 3

10 1 (E) y + ay = 0 a C(I, K) S (E) I K S K I 1 2 (a, b) K 2 (E) y + ay + by = 0 S (E) R K S K R 2 (a, b) K 2 S (u n ) n N K u n+2 + au n+1 + bu n = 0 n N S K N 2 F G E F = G F G G F F = G a = (1, 2, 1) b = (1, 3, 2) c = (1, 1, 0) d = (3, 8, 5) R 3 (a, b) = (c, d) E n E E n 1 F E F E F G E F G F G = F + G F G E F + G F + G = F + G F G

11 F G 3 R 5 F G {0} F G E F G (F + G) = F + G F G E F G E F + G = E F G = {0 E } F + G = E F G E E F G F G E F + G = E F G = {0 E } F + G = E F = {(x, y, z) R 3 x + 2y + 3z = 0} G = ((0, 1, 0)) F G R 3 F 1,..., F p E ( p ) p F k F k F 1,..., F p F 1,..., F p E E = p F k F 1,..., F p p F k = E E K F E (F) F F

12 F p F p E n F n F (n, p) F 1 F 2 E ( F 1, F 2 ) (F 1 F 2 ) F 1 + F 2 F p E F = F F E F = E F E K n F K n (F) F R 4 ((1, 2, 3, 0), ( 4, 6, 12, 2), ( 3, 6, 12, 3), ( 2, 4, 6, 0), ( 2, 2, 3, 1))

Documents pareils

Solutions particulières d une équation différentielle...

Solutions particulières d une équation différentielle... Solutios particulières d ue équatio différetielle......du premier ordre à coefficiets costats O cherche ue solutio particulière de y + ay = f, où a est ue costate réelle et f ue foctio, appelée le secod