k2 J1, nK Lk P Rn 1[X] i2J1, nK P(ai

Universit´e du Littoral Cˆote d’Opale Ann´ee universitaire 2019–2020

Master MEEF 1`ere ann´ee

DS 1 - Pr´eparation `a l’´ecrit d’Alg`ebre

Dur´ee : 5 h le 3 octobre 2019 `a 8h30 Documents non autoris´es Calculatrices personnelles autoris´ees

n

N R

m n Jm, nK k m6k6n

I R n2 N Cⁿ(I)

I n n

n p M^n,p(R) n

p Mn,n(R) Mn(R)

R[X] R

n Rⁿ[X]

n

n 2 a1, . . . , an

k2J1, nK

Lk(X) = Y

16i6n, i6=k

X ai

ak ai

.

k2 J1, nK Lk P R^{n 1}[X]

i2J1, nK

P(ai) =

(0 i6=k, 1 i=k.

F :

⇢ R^{n 1}[X] ! Rⁿ

P 7! (P(a1), . . . , P(an)).

F

(e1, . . . , en) Rⁿ k2J1, nK P R^{n 1}[X] F(P) =ek

F F

f R R

P 2R^{n 1}[X] k2J1, nK

P(ak) = f(ak) P f

a1, . . . , an

f a1, . . . , an

L1, . . . , Ln f a1, . . . , an

B

n

Z R

R²

R² d

R² O=

✓0 0

◆

e1=

✓1 0

◆

e2=

✓0 1

◆

Z² R² R

R

4 3 2 1 1 2 3 4 5 6 7

4 3 2 1 0 1 2 3 4 5 6 7

Z

B = (e⁰₁, e⁰₂) R² B Z R

e⁰₁, e⁰₂ 2R

X R X =ae⁰₁+be⁰₂ a, b2Z

C= (e1, e2) R² C Z R

e⁰₁ =

✓a1

b1

◆

e⁰₂=

✓a2

b2

◆

R² A=

✓a1 a2

b1 b2

◆ . X 2R² x, y2 R X =xe⁰₁+ye⁰₂

X =A

✓x y

◆ .

(e⁰₁, e⁰₂) Z R (a1, a2, b1, b2)2Z⁴

2

(x1, x2, y1, y2)2Z⁴

x1e⁰₁+y1e⁰₂=e1 x2e⁰₁+y2e⁰₂=e2. B =

✓x1 x2

y1 y2

◆

AB =I2

det(A)2{ 1,1}

(a1, a2, b1, b2)2Z⁴ det(A)2{ 1,1}

A A ¹

(e⁰₁, e⁰₂) Z R

e⁰₁=

✓a1

b1

◆

R

e⁰₁ Z R a1 b1

a1 b1

e⁰₂ R (e⁰₁, e⁰₂) Z R

Z R

✓7 10

◆

f :R² ! R² C = (e1, e2) R²

A

f(R) ✓ R A

f(R) =R Imf

f

f ¹(R)✓R

A A ¹

det(A)2{ 1,1}

A det(A)2{ 1,1}

(f(e1), f(e2)) Z R

f(R) =R

n

N R

m n Jm, nK k m k n

n Mⁿ R n n

In n

z z

X ^k k 0 Rⁿ X x1, . . . , xn Rⁿ

k X ^k x₁^k , . . . , xn^k X ^k k 0

X i J1, nK x_i^k k 0 xi

A^k k 0 Mn R A ai,j 1 i,j n Mn R

k A^k a_i,j^k 1 i,j n A^k k 0

A i, j J1, nK a_{i,j k}^k 0 ai,j

G i j j

i G i j

1 n

i j

1 i, j n ai,j i j

i j ai,j 0

i j ai,j A

k N P ^k p₁^k , p₂^k . . . , p_n^k ₁, pn^k 1 i n

p_i^k i k

k p₁^k . . . pn^k 1 k P ^{k 1} P ^k A

k P ^k A

k P ⁰

P ^k k 0

P p1, . . . , pn P A P p1 . . . pn 1

3

G

1OO✏✏ WW ⌫⌫

GG

⌥⌥

@@4^^

2 ^{oo //} 3

1

P ⁰ 1 0 0 0 .

U

0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

.

A U

U² U³

↵k k 0 k k 0

k

U^k

↵k k k k

k ↵k k k

k k ↵k k

k k k ↵k

. k

↵k 1 3 k k 1 ↵k 2 k

k k 2 2 k 1 3 k

k k

3^k 1 ^k

4 ↵k

3^k 3 1 ^k 4

k P ^k

P ^k k 0 P ^k

k

G

1 ^{oo //}

OO✏✏ ^^ 2

OO✏✏@@

5OO✏✏ oo // 6

OO✏✏

@@8 oo // 7_^^

4 ^{oo //} 3

1

P ⁰ 1 0 0 0 0 0 0 0 . X 1,3,6,8 Y 2,4,5,7

A

1 1 1 1 1 1 1 1 A.

X Y

Y X

P ^k X

k P ^k

Y k

P ^k k 0

n

P ^k k 0

P p1, . . . , pn Rⁿ p1, . . . , pn

p1 . . . pn 1 P A P

X x1, . . . , xn Rⁿ X

i J1, nK xi 0 x1 . . . xn 1

A Mn R A

A

A Mn R A

A 1 1

1 1

.

k P ^k