Algèbre linéaire en dimension finie. 1 Espaces vectoriels, premières

E
E×E→E
(u, v)7→ u+v
R×E→E
(λ, u)7→ λu
0
E E
0E
(x1,· · · , xn)E n
(λ1,· · · , λn)λ1x1+· · · +λnxn= 0
(x1,· · · , xn)E
λ1x1+· · · +λnxn= 0 =⇒λ1=· · · =λn= 0
(x1,· · · , xn)E E
xix E n
(λ1,· · · , λn)λ1x1+· · · +λnxn=x
E
E
E n E
n n
n

E F
f:E→F f(λu) = λf(u)
f(u+v) = f(u) + f(v)λ u v E L(E, F )
E F L(E) = L(E, E)
E E
f Ker(f) = f−1({0}) = {x∈E|f(x)=0}
E f f f
E Im(f) = {f(x), x ∈E} ⊂ F
f(u) = f(v)u=v
{0}
F Im(f) = F
E F
E F f−1:F→E
f◦f−1F f−1◦f E
E F f ⇐⇒ f⇐⇒ f
ERn(e1,· · · , en)E
L(E)E
EMn(R)n
f(e1,· · · , en)
Mate1,··· ,en(f) (λij )i,j∈[1,n]λij
∀j∈[1, n], f(ej) =
n
X
j=1
λij ei.
j f(ej)ei
f g E E
f◦g f g
Mate1,··· ,en(f◦g) = Mate1,··· ,en(f)Mate1,··· ,en(g).
f
E
D:Rn[X]→Rn[X]
P(X)7→ P0(X)
(1, X, · · · , Xn)
(e1,· · · , en) (ε1,· · · , εn)E Mate1,··· ,en(f)
Matε1,··· ,εn(f)P
Mate1,··· ,en(f) = P Matε1,··· ,εn(f)P−1

A= (aij )i,j∈[1,n]X=
x1
xn
Y=
y1
yn
AX =Y⇐⇒ 
a11x1+· · · +a1nxn=y1,
an1x1+· · · +annxn=yn.
aij
yixi
A−1n AA−1=A−1A=In
AX =Y Y X
AX =Y Y X
AX = 0 X= 0
A0
A
A X AX =Y
AX =Y A =
100
010
000
Y=
1
1
1
Y=
1
1
0

det : Mn(R)→R
det A= 0 ⇐⇒ A
•det(AB) = det A·det B= det(BA)
•A
det A=Qn
i=1 aii
•Aij n−1i j
A i
det A=
n
X
j=1
(−1)i+jaij det Aij ,
j
det A=
n
X
i=1
(−1)i+jaij det Aij .
det(A+B)6= det A+ det B
•A B P
A=P BP −1det A= det B A
•2 3
det a b
c d =ad −bc,
det 
a b c
d e f
g h i
=aei +bfg +cdh −gec −dbi −hfa.
221
010
101

A
λ∈RA X
AX =λX λ
λ
λRn
λ
A A
A A
A
n
n n
λ A det(A−λIn) = 0
A χA
χA(x) = det(A−xIn)
A A
101
121
003
tr(A) = Pn
i=1 aii
tr(AB) = tr(BA)
401
021
003
,
101
020
101
,
111
222
333
,
1 0 0
1 1 2
0 0 3
,0 1
−2 0 ,0 1 2 0 ,1−1 1 3 .