exos maths 2016

R

R2
E={(2,1)}
F={(1,2t)|t∈R}
G={(2t, −t)|t∈R}
H={(t, t)|t > 0}
I={(t2, t)|t∈R}
J={(s, s +t)|s, t ∈R}
R3
E={(2t, 0, t)|t∈R}
F={(0,2t, s)|s, t ∈R}
G={(t+s, 2t−3s, s)|s, t ∈R}
H={(2t2,0, t)|t∈R}
I={(t, st, s)|s, t ∈R}
J={(s+t2,−s, t2)|s, t ∈R}
R3
E={(x, y, z)∈R3|x+ 3z= 0,2y−z= 0, x −2y+ 3z= 0}
F={(x, y, z)∈R3|2y=x, x2+z= 0}
G={(x, y, z)∈R3|x2=y2, x +y+z= 0, z = 0}
E={(x, y, z)∈R3|x−2y+z= 0 x+y−2z= 0 5x−4y−z= 0}
I={(x, y, z)∈R3|2x+y−z= 0 −4x−y+z= 0}
J={(x, y, z)∈R3|2x−4y+ 2z= 0 3x−6y+ 3z= 0}
K={(x, y, z, t)∈R4|2x+y−2z−t= 0 x+y−z+t= 0 −4x−
y+ 4z+ 5t= 0}
L={(x, y, z, t)∈R4|x+y+ 3z−2t= 0 x+ 3y+ 4z−3t= 0 2x+
4y+ 8z−3t= 0}
M={(x, y, z, t)∈R4|x−y−3z+2t= 0 2x+y−3z+t= 0 y+z−t=
0}
R2
L1={(x, y)∈R2|xy = 1}
L2={(x, y)∈R2|x+ 3y= 2}
L3={(x, y)∈R2|3x+ 2y= 0}R2
E
E1={(x1, x2, x3, x4, x5)∈E=R5|x1−x2+x3+x4+x5= 0}
E2={(x1, x2, x3)∈E=R3|x2
1+x2 2=x2 3}
E3={(x1, x2, x3, x4)∈E=R4|x1−x2+x3+x4= 1}
F={(x, y, z)∈R3=E|x+y−z= 0}
G={(a−b, a +b, a −3b)|a, b ∈R}.
F G R3
F∩G
R2
L1={(x, y)∈R2|x≤y}
L2={(x, y)∈R2|xy = 0}
L3={(x, y)∈R2|x=y}
L4={(x, y)∈R2| |x|=|y|}
L5={(x, y)∈R2|x2=y2}
L6={(x, y)∈R2|x= 2y}R2
n∈N
n Mn(R)
ER R
E
F1
F20
F3
F4f(1) = 0
F5
F6f(0) = 1
F7
F8f(1) = 2f(√2)

E=R[X]R
R[X]
E1={P∈R[X]|deg P=n}n
E2={P∈R[X]|deg P≤n}n
E3={P∈R[X]|P(1) = P0(1) = 0}
E={(x, y)∈R2|x=y}F={(x, y)∈R2|x= 0}
E F E ∪F
R4
F={(r+ 2t, r +s+t, r + 2s, t −s)|r, s, t ∈R},
G={(x, y, z, t)|x+y−2z−t= 0}.
F G R4
F∩G
R4
F={(a, b, c, d)∈R4|b−2c+d= 0},
G={(a, b, c, d)∈R4|a=d b = 2c}.
F G R4
F∩G
E U V E
W={(x, y)∈E×E|x∈U, y ∈V}W
E×E
S={x+y|x∈U, y ∈V}S
E×E
D={(x, x)∈E×E|x∈E}D
E×E
3
R5
x1+ 2x2+ 3x3+ 4x4+x5= 1
x1−2x2+x3+ 2x4−x5=−1
2x1−x2+x3+x4+x5= 1.
S
SR5
v0= (0,0,0,0,1)
S=v0+E={v0+v|v∈E}E
R5
E
E
E={(x, y, z)∈R3|x+y+z= 0}E
R3
V={(t2,−t2,0) |t∈R}V
E
ERF E
F E
E1={(x, y)∈R2|x y y ≤0}
E2={(x, y, z)∈R3|2x−y−z= 2}
E3={(x, y, z)∈R3|2x+y−z= 0}
E4={(2a−b, a + 3b)|a b }
E5{fR R f(1) = f(−1) = 0}
F G E
H H =F∩G={u∈E|u∈F u ∈G}
H
H E
E U V
E H =U+V={u+v|u∈U, v ∈V}
U∪V⊂U+V
U+V E
F E U ∪V⊂F
U+V⊂F
ERH E

H H E
F G R3
F={(x, y, z)∈R3|x+y+z= 0},
G={(x, y, z)∈R3|x−y−z= 0}.
F G R3
F∩G={(0, a, −a)|a∈R}
F∪GR3
HR3F∪G⊂H
u= (1,1,−2) v= (0,1,−1) w= (1,0,1) H
λ µ ν λu +µv +νw H
x y z ∈Rλ µ ν
(x, y, z) = λu +µv +νw.
H=R3
u vi
E
E=R2u= (1,2) v1= (1,−2) v2= (2,3)
E=R3u= (2,5,3) v1= (1,3,2) v2= (1,−1,4)
E=R3u= (3,1, m)v1= (1,3,2) v2= (1,−1,4)
m
R3
x1= (1,0,1) x2= (1,2,2)
x1= (1,0,0) x2= (1,1,0) x3= (1,1,1)
x1= (1,2,1) x2= (2,1,−1) x3= (1,−1,−2)
x1= (1,−1,1) x2= (2,−1,3) x3= (−1,1,−1)
u= (1,2,3) v= (1,−4,6)
u= (1,2,−1) v= (1,0,1) w= (−1,2,−3)
u= (1,2,1,0) v= (−1,1,1,1) w= (2,−1,0,1) t= (2,2,2,2)
3Pi
P1(X)=1 P2(X)=2X+ 3X2P3(X)=3X−X2+X3
3R
f1(x) = e−xf2(x)=1 f3(x) = ex
3Qi
Q1(X) = X−1Q2(X) = X2−X+ 1 Q3(X) = 2X2
A1=1 1
1 0 A2=0 1 1 0 A3=1 0 1 1 A4=1 1 1 1 
B1=1−1
1 0 B2=1 0 0 1 B3=2−1 1 1 
R3u= (1,−1,4) v= (1,2,3)
w= (1,−1,2) (u, v, w)R3
u0= (1,0,−1) v0= (0,1,1) w0= (1,−1,−2)
R3
u= (2,1,1) v= (1,3,1) w= (−2,1,3)
u= (1,0,3) v= (0,1,2) w= (2,−3,0)
u= (2,1,1) v= (1,2,1) w= (1,1,2)
E={(x, y, z)∈R3|x+y+z= 0}E
R3E
R3u= (−1,1,1) v= (1,−1,1)
(u, v)R3
R4u= (1,1,1,1) v= (−1,1,1,1)
(u, v)R4
A1={(x, y, z)∈R3|x+ 2y+ 2z= 0},
A2={(x, y, z)∈R3|y+z= 0 x= 0},
A3={(a+b, b −a, a +b), a, b ∈R},
A4={(a, 2a, a), a ∈R}.

M1=1 0
0 0 , M2=0 1 1 0 , M3=0 0 0 1 , M4=0−1 0 0 .
(M1, M2, M3, M4)M2(R)
E1={M=a b
c d a, b, c, d ∈R|ad −bc = 0}
E2={M=a b
c d a, b, c, d ∈R|a+d= 0}
E3={M=a0
0aa∈R}
E4={M=0b
c d b, c, d ∈R}
E3+E4
B1B2R3
B1= ((1,1,1),(1,−1,1),(1,1,−1)),
B2= ((1,1,0),(1,0,1),(0,1,1)).
B1B2R3
X(x, y, z)B=
((1,0,0),(0,1,0),(0,0,1)) R3X
B1x y z
X B2x y z
u1= (0,1,1) u2= (1,0,1) u3= (1,1,0)
R3
u= (1,1,1)
u1= (−1,2,3) u2= (1,−2,3) u3= (1,2,−3)
R3u= (1,1,1)
v1= (0,1,1,1) v2= (1,0,1,1) v3= (1,1,0,1)
v4= (1,1,1,0) R4
u= (1,1,1,1) v= (1,0,0,0)
R4
E1= Vect(u1, u2, u3, u4)u1= (1,2,−3,4) u2= (2,4,−5,9)
u3= (−2,−1,3,1) u4= (3,0,−1,−4)
E2={(x1, x2, x3, x4)∈R4|x1= 2x2−x3, x4=x1+x2+x3}
e1= (1,0,0) e2= (0,1,0) e3= (0,0,1)
3 (e1, e2, e3)R3v1= 2e1+e2+e3
v2=e1+ 2e2+e3v3=e1+e2+e3
v1v2v3
(v1, v2, v3)R3
(v1, v2, v3)w= 2e1+e3
(u1, u2, u3, u4, u5)R5
u1= (1,0,2,1,2), u2= (1,2,−1,−1,0), u3= (2,1,0,0,1),
u4= (0,1,1,0,1), u5= (3,4,0,−1,2).
(u1, u2, u3, u4, u5)
F= Vect(u1, u2, u3, u4, u5)F
(u1, u2, u3, u4, u5)R5
u1= (1,0,0,1,2), u2= (1,0,1,1,0), u3= (2,−1,0,4,3),
u4= (0,−1,−1,2,1), u5= (2,1,2,0,1).
(u1, u2, u3, u4, u5)
F= Vect(u1, u2, u3, u4, u5)F