Linear Algebra - Sebastian Pancratz

u ∈ U
A ′ = Q −1 AP
[α(u)]C = A[u]B = AP [u]B ′
= Q[α(u)]C ′
m × n A, A ′ ∈ Mm,n(F)
Q ∈ Mm,m(F) P ∈ Mn,n(F) A ′ = QAP
Mm,n(F)
U
V m n
U V F dim U = n dim V = m
α : U →
V
r
B U C V
[α]B,C = Ir 0
0 0
m × n Ir 0
0 0
r
ur+1, . . . , un N(α) B =
{u1, . . . , ur, ur+1, . . . , un} U α(u1), . . . , α(r) Im(α)
C = {α(u1), . . . , α(ur), vr+1, . . . , vm} V
[α]B,C = Ir 0
0 0
A ∈ Mm,n(F) α : Fn → Fm , x ↦→ Ax
Fn Fm
α A A
Ir 0
0 0 r
A ∈ Mm,n(F) A rank(A)
A F n
A
α : U → V B U C V
A = [α]B,C rank(α) = rank(A)
θ : Im(α) → colsp(A), α(u) ↦→ [α(u)]C
A, A ′ ∈ Mm,n(F)
rank(A) = rank(A ′ )
A ′ = Q −1 AP Q, P α : F n → F m , x ↦→ Ax
α B, C A B ′
P C ′ Q ′ [α]B ′ ,C ′ = Q−1 AP
P C B B ′ C C ′
rank(A) = rank(α) = rank(A ′ )
A A ′
Ir 0 Ir ′ 0
r r ′
0 0 0 0
rank(A) = r rank(A ′ ) = r ′ rank(A) = rank(A ′ ) r = r ′
A A ′