Cours d`algèbre linéaire

Rn
RnR
Rn
Rn
Rn
Rn

Mn,p

Rn
n≥1Rn~u = (x1, . . . , xn)
Rnn xii
~u ~ 0n= (0,...,0) n
RnR
~u = (x1, . . . , xn)~v = (y1, . . . , yn)Rnα
~u +~v = (x1+y1, . . . , xn+yn) et α.~u = (α.x1, . . . , α.xn)
Rn×Rn→Rn
(~u, ~v)7→ ~u +~v
R×Rn→Rn
(α, ~u)7→ α.~u
(Rn,+, .)R~u ~v ~w
Rnα β
[EV1] (~u +~v) + ~w =~u + (~v +~w)
[EV2]~u +~v =~v +~u
[EV3]~u +~ 0p=~ 0p+~u =~u
[EV4]~u + (−1).~u =~ 0p
[EV5]α.(~u +~v) = α.~u +α.~v
[EV6] (α+β).~u =α.~u +β.~u
[EV7] (αβ).~u =α.(β.~u)
[EV8] 1.~u =~u
p≥1F= (~u1, . . . , ~up)
RnF
p
X
k=1
αk.~uk=α1.~u1+· · · +αp.~up,avec (α1, . . . , αp)∈Rp