M.P.S.I., Colles, Semaine 14, Sujet 1. EX 1. EX 2. EX 3. M.P.S.I.

E F f ∈ L(E, F ) (ei)i=1,...,n E
f(f(ei))i=1,...,n F
f1f2
f1: ] −1; 1[ →R
x7→ 1
x−1
f2: ] −1; 1[ →R
x7→ 1
x+1
f1f2
f:x7→ 2
x2−1f1f2
F={(x, y, z)∈R3x+y+z= 0 2x−y+z= 0}
FR3
F
E F f ∈ L(E, F ) (ei)i=1,...,n E
f(f(ei))i=1,...,n F
k1 (αk)k=1,...,n ]0; 1[
(fk)k=1,...,n
∀k= 1, . . . , n ∀x∈]0,1[ fk(x) = 1x=αk,
0
(gk)k=1,...,n k= 1, . . . , n x ∈]0; 1[ gk(x) = 1
1−αkx
R4u= (1,1,1,1) v= (1,2,−1,3)
E(u1, . . . , up)E p ∈N\{1}v E
(u1, . . . , up, v)v(u1, . . . , up)
F={(x, y, z, t)∈R3x+y+z+t= 0}
FR3
F
k1 (αk)k=1,...,n 0< α1<· · · < αn<1x∈]0,1[ k= 1, . . . , n fk(x) = xαk
(fk)k=1,...,n
E F f ∈ L(E, F ) (ei)i=1,...,n E
f(f(ei))i=1,...,n F
f1f2
f1: ] −1; 1[ →R
x7→ 1
x−1
f2: ] −1; 1[ →R
x7→ 1
x+1
f1f2
f:x7→ 2
x2−1f1f2
F={(x, y, z)∈R3x+y+z= 0 2x−y+z= 0}
FR3
F
E F f ∈ L(E, F ) (ei)i=1,...,n E
f(f(ei))i=1,...,n F
k1 (αk)k=1,...,n ]0; 1[
(fk)k=1,...,n
∀k= 1, . . . , n ∀x∈]0,1[ fk(x) = 1x=αk,
0
(gk)k=1,...,n k= 1, . . . , n x ∈]0; 1[ gk(x) = 1
1−αkx
R4u= (1,1,1,1) v= (1,2,−1,3)
E(u1, . . . , up)E p ∈N\{1}v E
(u1, . . . , up, v)v(u1, . . . , up)
F={(x, y, z, t)∈R3x+y+z+t= 0}
FR3
F
k1 (αk)k=1,...,n 0< α1<· · · < αn<1x∈]0,1[ k= 1, . . . , n fk(x) = xαk
(fk)k=1,...,n