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ER4E= (e1, e2, e3, e4)E f
E
A= Mat
E(f) = 
1−1 2 −2
001−1
1−1 1 0
1−1 1 0
λ∈Rdet(f−λIdE)
A A2A−I4(A−I4)2
A= (a1, a2, a3, a4)E
Mat
A(f) = 
0100
0000
0011
0001
E
det(f−λIdE) = λ2(λ−1)2
rg A= 3,rg A2= 2,rg(A−I4)=3,rg(A−I4)2= 2
dim(ker f) = 1,dim(ker f2) = 2,dim(ker(f−IdE)) = 1,dim(ker(f−IdE)2)=2
a1ker f a3ker(f−IdE)
f(a1) = 0Ef(a3) = a3
xker f2ker f f(x)∈
ker f= Vect a1λ6= 0 f(x) = a1a2=1
λx
f(a2) = a1
yker(f−IdE)2ker(f−IdE)
f(y)−y∈ker(f−IdE) = Vect(a3)µ6= 0
f(y)−y=µa3a4=1
µy f(a4) = a4+a3
(a1, a2, a3, a4)
λ1λ2λ3λ4λ1a1+λ2a2+λ3a3+λ4a4= 0E
f
λ1a2+λ3a3+ (λ3+λ4)a4= 0E
λ3a3+ (2λ3+λ4)a4= 0E)⇒λ1a2−λ3a4= 0E
f λ3a4= 0Eλ3= 0 λ1= 0
λ4= 0 λ2= 0
a1
x−y+ 2z−2t= 0
z−t= 0
x−y+z= 0
x−y+z= 0
⇔
x−y+z= 0
z−2t= 0
z−t= 0
⇔
x−y= 0
z= 0
t= 0
a1=e1+e2

a2
x−y+ 2z−2t= 1
z−t= 1
x−y+z= 0
x−y+z= 0
⇔
x−y+z= 0
z−2t= 1
z−t= 1
⇔
x−y=−1
z= 1
t= 0
a2=e2+e3
a3
−y+ 2z−2t= 0
−y+z−t= 0
x−y= 0
x−y+z−t= 0
⇔
x−y= 0
−y+ 2z−2t= 0
−y+z−t= 0
z−t= 0
⇔
x−y= 0
−y+z−t= 0
z−t= 0
a3=e3+e4
a4
−y+ 2z−2t= 0
−y+z−t= 0
x−y= 1
x−y+z−t= 1
⇔
x−y= 1
−y+ 2z−2t= 0
−y+z−t= 0
z−t= 0
⇔
x−y= 1
−y+z−t= 0
z−t= 0
a4=e1