COR TD 2

fi
f1:R2→R2f1(x, y) = (2x+y, x −y)
f2:R3→R3f2(x, y, z)=(xy, x, y)
f3:R3→R3f3(x, y, z) = (2x+y+z, y −z, x +y)
f4:R2→R4f4(x, y) = (y, 0, x −7y, x +y)
f5:R3[X]→R3f5(P) = P(−1), P (0), P (1)
f1(x, y)∈R2(x0, y0)∈R2
f1(x, y)+(x0, y0)=f1x+x0, y +y0
=2(x+x0)+(y+y0),(x+x0)−(y+y0)
=2x+y+ 2x0+y0, x −y+x0−y0
=2x+y, x −y+2x0+y0, x0−y0
=f1(x, y) + f1(x0, y0)
(x, y)∈R2λ∈R
f1λ·(x, y)=f1λx, λy=2λx+λy, λx−λy=λ·2x+y, x−y=λ·f1(x, y).
f2f2(1,1,0) + f2(1,1,0)
f2(2,2,0)
f3(x, y, z) (x0, y0, z0)
f3(x, y, z)+(x0, y0, z0)=f3(x, y, z)+f3(x0, y0, z0)
(x, y, z)λ f3λ·(x, y, z)=λ·f3(x, y, z)
f4(x, y) (x0, y0)
f4(x, y)+(x0, y0)=f4(x, y) + f4(x0, y0)
(x, y)λ f4λ·(x, y)=λ·f4(x, y)
f5P, P 0∈R3[X]
f5P+P0=(P+P0)(−1),(P+P0)(0),(P+P0)(1)
=P(−1) + P0(−1), P (0) + P0(0), P (1) + P0(1)
=P(−1), P (0), P (1)+P0(−1), P 0(0), P 0(1)
=f5(P) + f5(P0)

P∈R3[X]λ∈R
f5λ·P=(λP )(−1),(λP )(0),(λP )(1)
=λ×P(−1), λ ×P(0), λ ×P(1)
=λ·P(−1), P (0), P (1)
=λ·f5(P)
E E1E2
E f :E1×E2→E
f(x1, x2) = x1+x2
f
f
f
Im f={f(x1, x2)|x1∈E1, x2∈E2}={x1+x2|x1∈E1, x2∈E2}=E1+E2.
ker f={(x1, x2)|f(x1, x2)=0}={(x1, x2)|x1+x2= 0}
(x1, x2)∈ker f
x1∈E1x2∈E2x1=−x2x1∈E2x1∈E1∩E2
x∈E1∩E2(x, −x)∈ker f
ker f={(x, −x)|x∈E1∩E2}.
x7→ (x, −x) ker f
E1∩E2
dim ker f+ dim Im f= dim(E1×E2).
ker f E1∩E2
dim(E1∩E2) + dim(E1+E2) = dim(E1×E2).
dim(E1×E2) = dim E1+ dim E2
dim(E1+E2) = dim E1+ dim E2−dim(E1∩E2).

E n f
E
ker f= Im f
f2= 0 n= 2 ·(f)
⇒ker f= Im f x ∈E f(x)∈
Im f f(x)∈ker f f(f(x)) = 0 f2= 0
dim ker f+ (f) = ndim ker f=
dim Im f=f2 (f) = n
⇒f2= 0 Im f⊂ker f y ∈Im f x
y=f(x)f(y) = f2(x) = 0 2 (f) = n
dim ker f= (f) dim ker f= dim Im f
Im fker f
ker f= Im f
ker fi
Im fifi
f1:R2→R2f1(x, y) = (2x+y, x −y)
f2:R3→R3f2(x, y, z) = (2x+y+z, y −z, x +y)
f3:R2→R4f3(x, y)=(y, 0, x −7y, x +y)
f4:R3[X]→R3f4(P) = P(−1), P (0), P (1)
dim ker f+ dim Im f= dim E f :E→F
f1
(x, y)∈ker f1⇐⇒ f1(x, y) = (0,0) ⇐⇒ (2x+y, x −y) = (0,0)
⇐⇒ (2x+y= 0
x−y= 0 ⇐⇒ (x, y) = (0,0)
ker f1={(0,0)}f1

(X, Y )f1(x, y)
f1(x, y) = (X, Y )⇐⇒ (2x+y, x −y) = (X, Y )
⇐⇒ (2x+y=X
x−y=Y⇐⇒ (x=X+Y
3
y=X−2Y
3
⇐⇒ (x, y) = X+Y
3,X−2Y 3
(X, Y )∈R2
(x, y)=(X+Y
3,X−2Y 3)f1(x, y)=(X, Y )
Im f1=R2f1
f1
f2
f2
(x, y, z)∈ker f2⇐⇒ f1(x, y, z) = (0,0,0)
⇐⇒ (2x+y+z, y −z, x +y) = (0,0,0)
⇐⇒ 
2x+y+z= 0
y−z= 0
x+y= 0
⇐⇒ · · ·
⇐⇒ (x, y, z) = (0,0,0)
ker f2={(0,0,0)}f2
ker f2={(0,0,0)}
dim ker f2= 0 f2:
R3→R3dim ker f2+dim Im f2= dim R3dim Im f2=
3 Im f23
R33 Im f2=R3f2
f2
f3:R2→R4

(x, y)∈ker f3⇐⇒ f3(x, y) = (0,0,0,0)
⇐⇒ (y, 0, x −7y, x +y) = (0,0,0,0)
⇐⇒ 
y= 0
0=0
x−7y= 0
x+y= 0
⇐⇒ · · ·
⇐⇒ (x, y) = (0,0)
ker f3={(0,0)}f3
f3:R2→R4dim ker f3+
dim Im f3= dim R2dim Im f3= 2 Im f3
2R3f3
Im f3
Im f3
v1=f(1,0) = (0,0,1,1) v2
v1v2=f(0,1) =
(1,0,−7,1) v1, v2∈Im f
dim Im f3= 2 {v1, v2}
Im f3
Im f3={v1, v2}=λ(0,0,1,1) + µ(1,0,−7,1) |λ, µ ∈
R
f4:R3[X]→R34
f4
P≤3
P(X) = aX3+bX2+cX +d P (0) = d P (1) =