Résumé de la semaine du 9 Novembre 2015 - ANMC

L(V, W )K
K
ker(f)fIm (f).
V, W K f :V→W
ker(f)V Ker(f) = {0}f
Im (f)WIm (f) = W f
f:V→W f−1:W→V
V, W K f :V→W
{v1, . . . , vn}V{f(v1), . . . , f(vn)}
W
{v1, . . . , vn}V f
{f(v1), . . . , f(vn)}W
V, W K
f:V→Wdim(V) = dim(W)
A∈Mm×n(K)fA:Kn→Km
fA(x) = Ax rang (fA) = rang (A)
V, W K dim V <
∞f:V→W n = dim(V) = rang (f) + dim ker(f)
V, W K dim V=
dim W f :V→W
f
f
f

V, W K dim V= dim W < ∞
BV={v1, . . . , vn}BW={w1, . . . , wn}V W
f:V→W f(vi) = wii= 1, . . . , n
V K B ={v1, . . . , vn}
V[·] : V→Kn[vi]B=ei
i= 1, . . . , n {e1, . . . , en}Kn
f:Kn→KmK
A∈Mm×n(K)f(x) = Ax
f BVBW
[f]BV,BW
V, W K BV, BW
V W n =dimV m =dimW
ψ:L(V, W )→Mm×n(K)f7→ [f]BV,BW
V, W K dim V=ndim W=m
dim(L(V, W )) = m·n
U, V, W K
BU, BV, BWg:U→V f :V→W
[fg]BU,BW= [f]BV,BW·[g]BU,BV
V, W K BVBW
dim V= dim W f ∈L(U, W )f[f−1]BW,BV=
([f]BV,BW)−1