MÉTHODES ET EXERCICES

Mathématiquesméthodes et exercices

JEAN-MARIE MONIERGUILLAUME HABERERCÉCILE LARDON

PC | PSI | PT

édition3e

© Dunod, 201

ISBN 978-2-10-07 910-2

Conception et création de couverture : Atelier 3+

5 rue Laromiguière, 75005 Pariswww.dunod.com

f : I −→ CI

R+

C.

+∞, 0, a, a ∈ R

xαf(x)+∞ 0

∫ →+∞

1

x

xx

fKE,

An(K)

f(x) = λxλ ∈ K, x ∈ E − {0}.

fχf f Eχf ,

E(f(P ) = λP

P �= 0),

P,

P

P.

E

f A➟

A =(−5 4−6 5

)∈

2(R).χA A

χA(λ) =∣∣∣∣−5− λ 4−6 5− λ

∣∣∣∣ = (λ2− 25) + 24 = λ2− 1 = (λ+ 1)(λ− 1),

A −1 1

R(A) = {−1, 1}1

X =(xy

)∈

2,1(R)X ∈

(A,−1) ⇐⇒ AX = −X

⇐⇒{−5x+ 4y = −x−6x+ 5y = −y ⇐⇒ x = y,

(A,−1) =(

(11

))

(A,−1) = 1,

X ∈(A, 1) ⇐⇒ AX = X

⇐⇒{−5x+ 4y = x−− 6x+ 5y = y ⇐⇒ − 6x+ 4y = 0,

(A, 1) =(

(23

))

(A, 1) = 1.A

2(R)

I

R(fn : I −→ R)n∈N

(Ak)k∈N

I

⋃k∈N

Ak = I

k ∈ N (fn)n

Ak (fn)n

I(fn : I −→ R)n∈N

(Ak)k∈N

I

⋃k∈N

Ak = I

k ∈ N (fn)n

Ak (fn)n

I(fn)n∈N

f I

n ∈ N fn

f

(fn)n∈N (gn)n∈N

I(fngn)n∈N

I

(fn)n∈N

Ig

(fng)n∈N

Ifn C.U.−→

n∞ f gn C.U.−→n∞ g

fn + gn −→n∞ f + g

(fn)n∈N

f I

n ∈ N fnC 1

If

C 1I(fn)n∈N

f I

n ∈ N fnC 1

I(f ′

n)n∈N

I

(Pn : [0 ; 1] −→ R)n�1

R

ff

(fn : R −→ R)n∈N

C 1

(f ′n)n∈N

R

(fn)n∈N

R

y = f(x)

Γy = f(x),

f(x) =3√

x2(x− 6).

Γ

{x(t) =

t2 − 1

y(t) =2t

3 + 3t2 − 1.

Γ

⎧⎪⎨⎪⎩x(t) =

t t

y(t) =t

t.

y = f(x)

Γy = f(x),

f(x) =x

1 +1x

.

,

f : R −→ R, x �−→ f(x) =x.

fC∞ R

f(n)(x)

(n, x) ∈ N∗ × R.

n ∈ N− {0, 1}f(

n)(x) = 0,x ∈ R.

C1

(a, b) ∈ R2

a � b, f : [a ; b]−→ C

C1 [a ; b].

∫ b

a

f(x)λx x −→

λ −→ +∞0.

C∞

0

f : [0 ; +∞[ −→ RC∞

(xn)n∈N

]0 ; +∞[

xn −→n∞

0

(∀n ∈ N, f(xn) =0).

∀k ∈ N, f(k) (0) = 0.

(x, y) ∈ R2

0 < x < yx

y<

(1 + x)

(1 + y).

1 4

S = AA,n(R).

S ∈ +n . λ ∈

R(S).V

S

λ.R(S) ⊂ R+.

XSXX ∈

n,1(R).L1, L2, L3

A ||L1||2 = 1

L2 =(a b c

)(L1 |L2) = 0

||L2||22 = 1,

L3 = L1 ∧ L2.Ω ∈

3(R).(Ω).

(Ω) = 1 −1.

(Ω) = 1

f

f =E3

ff

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

ff

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.χAA AA

X ∈(A).

Y ∈n,1(R),

∀λ ∈ R, (X + λY )A(X + λY ) � 0.

∀k ∈ N∗, 1

k=∫

1

0tk−1

tXHnX

X =

⎛⎝ x1

xn

⎞⎠ ∈

n,1(R).A

A,A−1 +B.

S = PDP−1,

P ∈n(R), D =

(λ1, ..., λn) ∈n(R).

P = (pij )ij .

∀i ∈ {1, ..., n}, sii =n∑

k=1

λkp2ik .

fλk

p2ik , 1 � i � n.

S ∈ ++n

f : x �−→ − x.S ∈ +

n S /∈ ++n .S = A A

S.

x ∈ I(a, b) ∈ R

2 x ∈ [a ; b] ⊂ I

∑n�0

fn(x)

I = [0 ; 1[fn : I −→ R, x �−→ xn

I = [0 ; 1[fn : I −→ R, x �−→ xn

I = [−1 ; 1],fn : x �−→

√x+

1

n+ 1−√

x+1

n+ 2,

+∞∑n=0

fn(x)=√x+ 1−

√x

n ∈ N 0 � ||fn + gn||∞ � ||fn||∞ + ||gn||∞.

I = R fn : I −→ R, x �−→1

n2g : I −→ R, x �−→ x

∑n�1

fn

I

∑n�1

fng

I

n ∈ N

0 � ||fng||∞ � ||fn||∞||g||∞.

||fn||]1;+∞[

∞= 1

0

n

∑n�1

fn

[1 ; +∞[

Rn

n |Rn(x)| � |fn+1(x)| =1

(n+ 1)x�

1

n+ 1,

||Rn||∞ �1

n+ 1

||Rn||∞ −→n∞

0

∑n�1

fn[1 ; +∞[

f : x �−→ 3√x2(x− 6)

R

R− {0, 6}∀x ∈ R− {0, 6}, f ′(x) = x(x− 4)(

3√x2(x− 6)

)2fx

f ′(x)

f(x)

−∞0

46

+∞+−

++

+∞

−∞+∞

+∞

−∞−∞00

−2 3√4−2 3

√4

+∞+∞0

+∞f(x) = x 3

√1− 6

x= x(1− 1

36

x− 1

9

(6

x

)2+ o(1

x2

))= x− 2− 4

x+ o(1

x

),

Γ

D | y = x−2,

+∞ −∞ ΓD

x

y

O 14

6

1

−2 3√4

D

Γ

x y

C1R

t ∈ R

x′(t) = 2t, y ′(t) = 6t2 + 6t = 6t(t+ 1).

x y +∞ −∞,

x

y −1 0,

t

x′(t)

x

y

y ′(t)

−∞−1

0+∞

−−

++∞+∞

−1−1 +∞+∞0

−∞−∞ 00

−1−1 +∞+∞+−

+

x

y

O−1

1

−1A

Γ

{x′(t) = 0y ′(t) = 0 t ∈ R,t = 0,

Γ

K

K R C

n,p(K) K = R C

➟

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 EE

F ∩ G = {0} F +G = E

F F G GF ∪ 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)

� {0} ⊂ n(R) ∩ n(R)

� A ∈ n(R) ∩ n(R)

A = A A = −A,

A = −A, 2A = 01

22A = 0, A = 0

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)

S +A =1

2M +

1

2M +

1

2M − 1

2M = M, (S,A)

n(R) + n(R) = n(R)

n(R) n(R) n(R)

n(R)

E = R4

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

(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) ∈ R4

αa+ βb+ γc+ δd = 0 ⇐⇒

⎧⎪⎪⎪⎨⎪⎪⎪⎩α+ β + γ + δ = 0

α+ 2β − γ + 2δ = 0

α+ 3β + γ − 3δ = 0

α+ 4γ = 0

⇐⇒

⎧⎪⎪⎪⎨⎪⎪⎪⎩α = −4γ

β − 3γ + δ = 0

2β − 5γ + 2δ = 0

3β − 3γ − 3δ = 0

⇐⇒

⎧⎪⎪⎪⎨⎪⎪⎪⎩α = −4γ

β = γ + δ

−2γ + 2δ = 0

−3γ + 4δ = 0

⇐⇒

⎧⎪⎪⎪⎨⎪⎪⎪⎩γ = 0

δ = 0

α = 0

β = 0.

F ∪ GE

F G E E

E K 5F,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

E1, ..., EN

E

(x1, ..., xN ) ∈ E1 × ...× EN

N∑i=1

xi = 0 =⇒ (∀i ∈ {1, ..., N}, xi = 0)

( N∑i=1

Ei

)=

N∑i=1

(Ei)

E1, ..., EN

E R

R R F1 F2 F3

E

]−∞ ; 1]

]−∞ ;−1] ∪ [1 ; +∞[ [−1 ;+∞[

F1, F2, F3

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)︸︷︷︸= 0

+ g(x)︸︷︷︸= 0

= 0, αf + g ∈ F

F E

F1, F2, F3 E

� f1 ∈ F1, f2 ∈ F2, f3 ∈ F3 f1 + f2 + f3 = 0

x ∈ ]−∞ ;−1]

F1 F2 f1(x) = 0 f2(x) = 0

f3(x) = −(f1(x) + f2(x))= 0.

∀x ∈ ]−∞ ;−1], f3(x) = 0.

F3 ∀x ∈ [−1 ;+∞[, f3(x) = 0.

∀x ∈ R, f3(x) = 0, f3 = 0

f1 = 0 f2 = 0

F1, F2, F3

➟

(fa)a∈R a ∈ R

fa : R −→ R, x �−→{0 x � a

1 x > a

N ∈ N∗, a1, ..., aN ∈ R λ1, ..., λN ∈ RN∑

k=1

λkfak = 0.

i ∈ {1, ..., N} λi �= 0

fai = − 1

λi

∑k �=i

λkfak .

a ∈ R faR− {a} fa a

− 1

λi

∑k �=i

λkfak ai

fai ai

∀i ∈ {1, ..., N}, λi = 0,

(fai )1�i�N

(fa)a∈R

(fa)a∈R

➟

(fa)a∈R a ∈ R

fa : R −→ R, x �−→ (x+ a)

a ∈ R

∀x ∈ R, fa(x) = (x+ a) = a x− a x,

fa = ( a) +(− a) .

fa

f0, f1, f2, (f0, f1, f2)

(fa)a∈R

(fa)a∈R

H EE

H E H1

H

H E (H) = (E)− 1,E

H

0

R E

E R H E

D E (1)1

� H ∩ D = {0} 00

� u = (un)n∈N ∈ E � u v = u− (�)

u = v + (�), v ∈ H, (�) ∈ D.

H +D = E

D H E

H E

H ={f ∈ E ;

∫ 1/2

0f = 0

}R

E = C([0 ; 1],R).

ϕ : E −→ R, f �−→∫ 1/2

0f

E

ϕ(1) =

∫ 1/2

01 =

1

2�= 0.

H = (ϕ) E

n ∈ N∗

H = Cn−1[ ]

E = Cn[ ]

E C H E

(E) = n+ 1, (H) = n

(H) = (E)− 1

H E

� 0.➟

N ∈ N∗, E K

p1, ..., pN EN∑i=1

pi = 0.

∀i ∈ {1, ..., N}, pi = 0.

i ∈ {1, ..., N} E piE (pi) = (pi).

0 =( N∑

i=1

pi

)=

N∑i=1

(pi) =

N∑i=1

(pi)︸ ︷︷ ︸�0

.

∀i ∈ {1, ..., N}, (pi) = 0

∀i ∈ {1, ..., N}, pi = 0.

A

n,p(K) (A) = r

P ∈ n(K), Q ∈ p(K) A = P n,p,rQ,

n,p,r =

(r 00 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,bU

b � a

Jn,r,b =

(b 00 0

)=

(b 00 0

)(a 00 0

)(b 00 0

)= n,p,b p,q,a q,r,b.

B = T n,r,bU = T n,p,b p,q,a q,r,bU

= (T n,p,bR−1)(R p,q,aS)(S

−1q,r,bU).

P = T n,p,bR−1 ∈ n,p(K) Q = S−1

q,r,bU ∈ q,r(K),

B = PAQ

=⇒(P,Q) ∈ n,p(K)× q,r(K) B = PAQ

(B) =((PA)Q

)� (PA) � (A).

➟

n ∈ N∗ A,B,C,D ∈ n(K)(α, β) ∈ K2 α �= β

J =

(α n 00 β n

), M =

(A BC D

).

M J

B = 0 C = 0.

JM = MJ ⇐⇒(α n 00 β n

)(A BC D

)=

(A BC D

)(α n 00 β n

)

⇐⇒(αA αBβC βD

)=

(αA βBαC βD

)

⇐⇒{αB = βB

βC = αC⇐⇒

{(α− β)B = 0

(α− β)C = 0⇐⇒

{B = 0

C = 0.

E KF E n = (E)

p = (F )

LF (E)E F

L(E)

LF (E)L(E)

G FE

LF,G(E)E F G

L(E)

LF,G(E)L(E)

� LF (E) ⊂ L(E) 0 ∈ LF (E)

� α ∈ E, f, g ∈ LF (E)

∀x ∈ F, (αf + g)(x) = αf(x)︸︷︷︸∈F

+ g(x)︸︷︷︸∈F

∈ F,

F αf + g αf + g ∈ LF (E)

LF (E) L(E)

E B = (e1, ..., en) F(e1, ..., ep) F

f ∈ L(E), M = B(f)

F f M

M =

(A B0 C

),

A ∈ p(K), B ∈ p,n−p(K), C ∈ n−p(K)

p(K) × p,n−p(K) × n−p(K) LF (E)(A,B,C) f E

B(f) =(A B0 C

),

(LF (E))=

(p(K)× p,n−p(K)× n−p(K)

)=

(p(K))+

(p,n−p(K)

)+

(n−p(K)

)= p2 + p(n− p) + (n− p)2 = n2 − np+ p2.

LF (E) L(E)(LF (E))=

(L(E))− 1. (∗)

(∗) ⇐⇒ n2 − np+ p2 = n2 − 1 ⇐⇒ np− p2 = 1

⇐⇒ p︸︷︷︸∈N

(n− p︸ ︷︷ ︸∈N

) = 1 ⇐⇒{p = 1

n− p = 1⇐⇒

{n = 2

p = 1.

LF (E) L(E)n = 2 p = 1

LF,G(E)L(E)

LF,G(E) = LF (E) ∩ LG(E),LF,G(E) L(E)

E B = (e1, ..., en)E = F ⊕G (e1, ..., ep) F

(ep+1, ..., eq) G

f ∈ L(E), M = B(f)

F G f M

M =

(A 00 C

),

A ∈ p(K), C ∈ n−p(K)

p(K) × n−p(K) LF,G(E) (A,C)f E

B(f) =(A 00 C

),

(LF,G(E))=

(p(K)× n−p(K)

)=

(p(K))+

(n−p(K)

)= p2 + (n− p)2

= n2 − 2np+ 2p2.(LF,G(E))=

(L(E))− 1

⇐⇒ n2 − 2np+ 2p2 = n2 − 1 ⇐⇒ 2(np− p2) = 1,

LF,G(E) L(E)

E,F K n ∈ N∗ E1, ..., En E

E1, ..., En E

∀x1 ∈ E1, ..., ∀xn ∈ En,(x1 + · · ·+ xn = 0 =⇒ x1 = ... = xn = 0

).

E1, ..., En E

∀(i, j) ∈ {1, ..., n}2, (i �= j =⇒ Ei ∩ Ej = {0}).E1, E2, E3 E

E1 ∩ (E2 + E3) = (E1 ∩ E2) + (E1 ∩ E3).

E1, E2, E3 E

E1 + (E2 ∩ E3) = (E1 + E2) ∩ (E1 + E3).

R E = RR E1 E2

EE

R3 p, q, r 4p+ 5q + 6r = R3 .

f, g E (f) (f) g f g

f, g E

∀A,B ∈ n(K), (AB) = (BA)

∀A,B ∈ n(K), (AB) = (A) (B)

E K A,B,C E.

A+(B ∩ (A+ C)

)= A+

(C ∩ (A+B)

).

(fa : [0 ; +∞[ −→ R, x �−→ 1

x+ a

)a∈ ]0 ;+∞[(

fa : R −→ R, x �−→ (x− a))a∈R

.

A ∈ 3,2(R) B ∈ 2,3(R) AB = C, C

C =

⎛⎝1 0 00 0 00 0 0

⎞⎠ ,

⎛⎝1 1 11 1 10 0 0

⎞⎠ ,

⎛⎝1 1 11 1 01 0 0

⎞⎠ ?

n ∈ N∗, X = {x1, ..., xn} n F = KX

i ∈ {1, ..., n}, i : F −→ K, f �−→ f(xi), xi.( i)1�i�n F ∗.

n ∈ N∗ (A,B) ∈ (n(C)

)2AB −BA = n

n, p ∈ N∗ M =

(A B0 C

)A ∈ n(K), B ∈ n,p(K), C ∈ p(K).

M A C

A C M−1

(fa : R −→ R, x �−→ |x− a|3/2)

a∈RRR.

E K[ ].

E

E

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) A2 = A, B2 = B, C2 = C.

M = A+√2B +

√3C M2 = 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 ).

M M =

(RS

),

(M) � (R) + (S).

M M =

(A BC D

),

A D

(M) � (A) + (B) + (C) + (D).

m, n, p ∈ N∗ m � n p � n, M =

(A BC 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 = (aij)ij ∈ n,p(K).

||A||� =1�j�p

( n∑i=1

|aij |), ||A||c =

1�i�n

( p∑j=1

|aij |),

X = (xj)1�j�p ∈ p,1(K) ||X||1 =

p∑j=1

|xj |, ||X||∞ =1�j�p

|xj |.

||A||� =X∈ p,1(K)−{0}

||AX||1||X||1 , ||A||c =

X∈ p,1(K)−{0}

||AX||∞||X||∞ .

E K F,G E E.

G ={f ∈ L(E) ; (f) = F (f) = G

}.

G ◦.E n = (E), p = (F ),

B1 = (e1, ..., ep) F, B2 = (ep+1, ..., en) G, B = (e1, ..., en)E.

θ : f �−→ B(f) (G, ◦)(H, ·) H =

{(M 00 0

)∈ n(K) ; M ∈ p(K)

}.

n(K) n(K)

n ∈ N− {0, 1}. n(K) n(K).

n, p ∈ N∗, B ∈ n,p(K), C ∈ p(K).

(n B0 C

)= n+ (C).

n, p ∈ N∗, R ∈ n,p(K), S ∈ p,n(K).

p+ ( n +RS) = n+ ( p + SR).

n, p ∈ N∗, A ∈ n(K), B ∈ p(K).

(A 00 B

)= (A) + (B).

n ∈ N∗, A,B ∈ n(K).

(A 00 A

) (B 00 B

)A B

n, p ∈ N∗, A,B ∈ n(K), U, V ∈ p(K). A B(A 00 U

) (B 00 V

)U 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.

M =

(A BC D

), A ∈ n(K), B ∈ n,p(K), C ∈ p,n(K), D ∈ p(K).

M D − CA−1BM−1

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 .

n, p ∈ N∗ M =

(A BC D

)A ∈ n(K) B ∈ n,p(K) C ∈ p,n(K) D ∈ p(K)

(M) = n ⇐⇒ D = CA−1B.

K E K p ∈ N∗, F1, ..., Fp Ep⋃

i=1

Fi = E. i ∈ {1, ..., p} Fi = E.

GL(E)

E K e = E , G GL(E)

n = (G) p =1

n

∑g∈G

g.

∀h ∈ G, p ◦ h = p.

p E.⋂g∈G

(g − e) = (p).

( ⋂g∈G

(g − e))=

1

n

∑g∈G

(g).

(fa)a∈[0 ;+∞[

(fa)a∈R

(f−1, f0, f1)

A,B(A,B)

i ∈ {1, ..., n} i ∈ F ∗.

( i)1�i�n

j ∈ {1, ..., n} fj : xi �−→ δij .

M =

(X YZ T

)

a ∈ R, fa C2

R− {a} C2 R.

n = (E).

(P1, ..., Pn+1)(P1) � ... � (Pn+1),(Q1, ..., Qn+1) Qn+1 = Pn+1

∀i ∈ {1, ..., n}, (Qi) < (Pn+1),

(P1, ..., Pn)(P1) < ... < (Pn)

(S1, ..., Sn) Sn = Pn

∀i ∈ {1, ..., n}, (Si) = (Pn).

A ∈ n(K),ϕA : n(K) −→ K, X �−→ (AX)

n(K)∗.

θ

(α, β, γ) ∈ Z3 α + β√2 + γ

√3 = 0,

α = β = γ = 0.

n,p,r.

(M) = n.

�∀X ∈ p,1(K), ||AX||1 � ||A||� ||X||1.

� j j

||A||� =n∑

i=1

|aij |.

�∀X ∈ p,1(K), ||AX||∞ � ||A||c ||X||∞.

� X =

⎛⎜⎝ε1

εp

⎞⎟⎠ ,

εj =

⎧⎪⎨⎪⎩

|ai0j |ai0j

ai0j �= 0

1 ai0j = 0,

i0 ||A||c =

p∑j=1

|ai0j |.

G ◦,G

GL(E).

F G,

� f ∈ G f

B(M 00 0

), M ∈ p(K).

�

A =

(M 00 0

)H, M ∈ p(K),

f E, A BG.

θ ϕθ

(G, ◦) (H, ·).

H n(K).H ∩ n(K) = ∅.

H

(n B0 C

)(n −B0 p

)=

(n 00 C

).

n+RS p+SRn+ p

....

(u, v) ∈ (L(E))2

u ◦ f ◦ v = p, p FG.

....

N =

(X YZ T

), MN = n+p.

(n 0

−CA−1p

)(A BC D

)(n −A−1B0 p

)

=

(A B0 D − CA−1B

).

E K

m,n,a p,q,b a = (A), b = (B)a � b

r = (A) < nMr ∈ r+1(K) r

Nr =

(Mr 00 0

)∈ n(K).

(n 0

CA−1 − p

)(A BC D

)(n −A−1B0 p

)

=

(A 00 CA−1B −D

).

p.

F1, ..., Fp+1 E

p+1⋃i=1

Fi = E, Fp+1 �= E,

p⋃i=1

Fi �= E,

x, y ∈ E x /∈ Fp+1 y /∈p⋃

i=1

Fi,

yx

h ∈ Gg �−→ g ◦ h G∑

g∈G

g ◦ h =∑g∈G

g.

p2

x ∈⋂g∈G

(g − e),

p(x) = x.

x ∈ (p) g(x) = (g ◦ p)(x),g ◦ p = p.

n = 2 n � 3

(E1 ∩ E2) + (E1 ∩ E3) ⊂ E1 ∩ (E2 + E3).

E1 + (E2 ∩ E3) ⊂ (E1 + E2) ∩ (E1 + E3).

E1 E2 E E1 ∩ E2 = {0}0 f E f = g1+g2

g1 : x �−→ f(x) + f(−x)

2, g2 : x �−→ f(x)− f(−x)

2, (g1, g2) ∈ E1 × E2

(p, q, r)

3 = ( R3) = (4p+5q+6r) = 4 (p)+5 (q)+6 (r) = 4 (p)+5 (q)+6 (r),

(p), (q), (r) N

E = R2 f, g

A =

(0 10 0

), B =

(1 00 0

)

n � 2 A = B = n

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∗, a1, ..., an ∈ ]0 ; +∞[

λ1, ..., λn ∈ R

n∑k=1

λkfak = 0.

∀x ∈ [0 ; +∞[,

n∑k=1

λk

x+ ak= 0.

[0 ; +∞[R,

∀x ∈ R− {a1, . . . , an},n∑

k=1

λk

x+ ak= 0.

x −ak∀k ∈ {1, ..., n}, λk = 0.

(fa)a∈ ]0 ;+∞[

a ∈ R

∀x ∈ R, fa(x) = (x− a) = a x− a x,

fa

(fa)a∈R,2

x ∈ R

(f−1 + f1)(x) = (x+ 1) + (x− 1)

= 2 1 x = (2 1)f0(x),

f−1 − 2 1 f0 + f1 = 0,

(fa)a∈R

A =

⎛⎝1 00 00 0

⎞⎠ , B =

(1 0 00 0 0

)

A =1

2

⎛⎝1 11 10 0

⎞⎠ , B =

(1 1 11 1 1

)

(A,B)

3 = (C) = (AB) � (A) � 2,

(A,B)

i ∈ {1, ..., n}, i ∈ F ∗i

F K i

∀α ∈ K, ∀f, g ∈ F, i(αf + g) = (αf + g)(xi)

= αf(xi) + g(xi) = α i(f) + i(g).

(α1, ..., αn) ∈ Knn∑

i=1

αi i = 0.

j ∈ {1, ..., n}

fj : X −→ K, xi �−→{1 i = j

0 i �= j.

0 =n∑

i=1

αifj(xi) = αj .

( 1, ..., n) F ∗.

X n F = KX

n F ∗n

( 1, ..., n) nF ∗ F ∗.

(A,B) ∈ ( n(C))2

AB −BA = n

(AB −BA) = ( n) = n.

(AB −BA) = (AB)− (BA) = 0,

(A,B) ∈ ( n(C))2

AB −BA = n.

(M) =

(A B0 C

)= (A) (C)

(M) �= 0 ⇐⇒ ((A) �= 0 (C) �= 0

)M A C

A C M

M−1 M

M−1 =

(X YZ T

).

MM−1 = n+p ⇐⇒(A B0 C

)(X YZ T

)=

(n 00

)

⇐⇒

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

AX +BZ = n

AY +BT = 0

CZ = 0

CT = p

⇐⇒C

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

Z = 0

T = C−1

AX = n

AY = −BC−1

⇐⇒A

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

Z = 0

T = C−1

X = A−1

Y = −A−1BC−1.

M−1 =

(A−1 −A−1BC−1

0 C−1

).

n ∈ N∗ a1, ..., an ∈ R

λ1, ..., λn ∈ R

n∑k=1

λkfak = 0.

i ∈ {1, ..., n}. λi �= 0.

fai = − 1

λi

∑1�k�n, k �=i

λkfak .

a ∈ R, fa C2

R− {a} C2 R.

fai C2

ai,fai C2

ai,

∀i ∈ {1, ..., n}, λi = 0.

(fa)a∈R

n = (E).

� n = 1.

� n.

E K[ ] n+ 1. EB = (P1, ..., Pn+1). B,

∀i ∈ {1, ..., n+ 1}, (Pi) � (Pn+1).

C = (Q1, ..., Qn+1)Qn+1 = Pn+1 i ∈ {1, ..., n}

Qi =

⎧⎨⎩ Pi (Pi) < (Pn+1)

Pi − αiPn+1 (Pi) = (Pn+1),

αi (Pi − αiPn+1) < (Pn+1).

αi

Pi Pn+1

Q1, ..., Qn+1

P1, ..., Pn+1.

Pn+1 = Qn+1

i ∈ {1, ..., n} Pi = Qi Pi = Qi + αiQn+1,P1, ..., Pn+1

Q1, ..., Qn+1.

(C) = (B) = E.

(E) = n+ 1 C E n+ 1C E.

F = (Q1, ..., Qn),n R[ ]. F

F = (R1, ..., Rn)

G = (R1, ..., Rn, Pn+1).

E = F ⊕ Pn+1K[ ] F F,G E.

∀i ∈ {1, ..., n}, Ri ∈ (Q1, ..., Qn)

(Q1, ..., Qn) < (Pn+1)∀i ∈ {1, ..., n}, (Ri) < (Pn+1).

G E

n.

n = (E). E

E B = (P1, ..., Pn)

(P1) < ... < (Pn).

i ∈ {1, ..., n} Si =

⎧⎨⎩Pi + Pn i < n

Pn i = n.

∀i ∈ {1, ..., n}, (Si) = (Pn).

S1, ..., Sn

P1, ..., Pn.

∀i ∈ {1, ..., n}, Pi =

⎧⎨⎩Si − Sn i < n

Sn i = n,

P1, ..., Pn S1, ..., Sn.

(E) = n C = (S1, ..., Sn) nE, C E.

E

A ∈ n(K).

ϕA : n(K) −→ K, X �−→ (AX)

∀α ∈ K, ∀X,Y ∈ n(K),

ϕA(αX + Y ) =(A(αX + Y )

)= (αAX +AY )

= α (AX) + (AY ) = αϕA(X) + ϕA(Y ).

ϕA ∈ n(K)∗.

θ : n(K) −→ n(K)∗

∀A ∈ n(K), ∀X ∈ n(K), θ(A)(X) = (AX).

∀A ∈ n(K), θ(A) = ϕA.

� 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 = (aij)ij . (i, j) ∈ {1, ..., n}.

0 = A ij) =

⎛⎜⎝

a1i

(0) (0)ani

⎞⎟⎠ = aji,

i Aj

A = 0.

(θ) = {0}, θ

� θ : n(K) −→ n(K)∗

n(K) n(K)∗θ K

A,B,C,M

(M) = (A+√2B +

√3C)

= (A) +√2 (B) +

√3 (C),(

(A)− (M)︸ ︷︷ ︸α

)+ (B)︸ ︷︷ ︸

β

√2 + (C)︸ ︷︷ ︸

γ

√3 = 0.

(α, β, γ) ∈ Z3 α+ β√2 + γ

√3 = 0.

(α, β, γ) = (0, 0, 0).

γ√3

α2 + 2β2 + 2αβ√2 = 3γ2,

αβ �= 0√2 =

3γ2 − α2 − 2β2

2αβ∈ Q,

√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 0r,p−r

0n−r,r 0n−r,p−r

).

n,p,r =

(r

0n−r,r

)(r 0r,p−r

),

A

A = P

(r

0n−r,r

)︸ ︷︷ ︸

U

(r 0r,p−r

)Q︸ ︷︷ ︸

V

,

U ∈ n,r(K), V ∈ r,p(K).

U1, ..., Up U V1, ..., Vq

V,

(U1, ..., Up, V1, ..., Vq)

= (U1, ..., Up) + (V1, ..., Vq),

(U1, ..., Up, V1, ..., Vq)

� (U1, ..., Up) + (V1, ..., Vq),

(M) � (U) + (V ).

(M) =

(RS

)=( (R

S

))=

(R S

)� ( R) + ( S) = (R) + (S).

(M) =

(A BC D

)�

(AC

)+

(BD

)�(

(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 =

⎛⎜⎝x1

xp

⎞⎟⎠ ∈ p,1(K)

||AX||1 =

n∑i=1

∣∣∣ p∑j=1

aijxj

∣∣∣�

n∑i=1

p∑j=1

|aij | |xj | =p∑

j=1

( n∑i=1

|aij |)|xj |

�p∑

j=1

||A||� |xj | = ||A||�p∑

j=1

|xj | = ||A||� ||X||1.

∀X ∈ p,1(K)− {0}, ||AX||1||X||1

� ||A||�.

� ||A||� =1�j�p

( n∑i=1

|aij |),

j ∈ {1, ..., p} ||A||� =

n∑i=1

|aij |.

X = j ,j

1.

||X||1 = 1 AX =

⎛⎜⎝a1j

anj

⎞⎟⎠ ,

||AX||1 =

p∑j=1

|aij | = ||A||�,

||AX||1||X1||

= ||A||�.

||A||�

X∈ p,1(K)−{0}||AX||1||X||1

= ||A||�.

� X =

⎛⎜⎝x1

xp

⎞⎟⎠ ∈ p,1(K)

||AX||∞ =1�i�n

∣∣∣ n∑j=1

aijxj

∣∣∣�

1�i�n

p∑j=1

|aij | |xj | �1�i�n

( p∑j=1

|aij | ||X||∞)

=(

1�i�n

p∑j=1

|aij |)||X||∞ = ||A||c ||X||∞.

∀X ∈ p,1(K)− {0}, ||AX||∞||X||∞

� ||A||c.

� ||A||c =1�i�n

( p∑j=1

|aij |),

i0 ∈ {1, ..., n} ||A||c =

p∑j=1

|ai0j |.

X =

⎛⎜⎝ε1

εp

⎞⎟⎠ ∈ p,1(K)

j ∈ {1, ..., p} εj =

⎧⎪⎪⎨⎪⎪⎩

|ai0j |ai0j

ai0j �= 0

1 ai0j = 0.

||X||∞ = 1, X 1X �= 0.

||AX||∞ =1�i�n

( p∑j=1

|aijεj |)�

p∑j=1

|ai0jεj |.

j ∈ {1, ..., p} |ai0jεj | = |ai0j |,ai0j �= 0, ai0j = 0.

||AX||∞ �p∑

j=1

|ai0j | = ||A||c.

X ∈ p,1(K)||AX||∞||X||∞

� ||A||c.

X∈ p,1(K)−{0}||AX||∞||X||∞

= ||A||c.

◦ G.f1, f2 ∈ G.

� ∗ (f2 ◦ f1) ⊂ (f2) = F.

∗ z ∈ F. z ∈ F = (f2), y ∈ Ez = f2(y). E = F⊕G, u ∈ F, v ∈ Gy = u+ v.

z = f2(y) = f2(u+ v) = f2(u) + f2(v).

u ∈ F = (f1), x ∈ E u = f1(x),v ∈ G = (f2), f2(v) = 0.

z = f2(f1(x)

)= f2 ◦ f1(x) ∈ (f2 ◦ f1).

F ⊂ (f2 ◦ f1).

(f2 ◦ f1) = F.

� ∗ (f2 ◦ f1) ⊃ (f1) = G.

∗ x ∈ (f2 ◦ f1); f2(f1(x)

)= 0,

f1(x) ∈ (f1) ∩ (f2) = F ∩ G = {0},x ∈ (f1) = G.

(f2 ◦ f1) ⊂ G.

(f2 ◦ f1) = G.

f2 ◦ f1 ∈ 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.

◦

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 ∈ Ef(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,

f B(M 00 0

), M

f ′ f F.

(M) =

(M 00 0

)= (f) = (F ) = p.

M ∈ p(K),

(M 00 0

)∈ H.

θ : G −→ H, f �−→ B(f).

� ϕA H, f E

B(f) = A

A =

(M 00 0

)= B(f)

M ∈ p(K),

(f) = F (f) = G,

f ∈ G.� θ ϕ

�

∀f1, f2 ∈ G, θ(f2)θ(f1) =

(M2 00 0

)(M1 00 0

)

=

(M2M1 0

0 0

)= θ(f2 ◦ f1).

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∗ Nk = 0⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩( n − αN)

( k−1∑p=0

(αN)p)= n − αkNk = n

( k−1∑p=0

(αN)p)( n − αN) = n − αkNk = n,

n − αN ∈ n(K).

M ∈ H ∩ n(K),

H

n(K)

N1 =

⎛⎜⎜⎜⎜⎜⎝0 . . . . . . 0

(0)

0 (0)1 0 . . . 0

⎞⎟⎟⎟⎟⎟⎠ , N2 =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 . . . 0

0 (0)

0

(0) 10 . . . . . . . . . 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

N1 N2

N1 ∈ H N2 ∈ H, HN1 +N2 ∈ H.

N1 +N2 =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 . . . 0

0 (0)

0

0 (0) 11 0 . . . . . . 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

n(K) n(K).