Linear Algebra - Sebastian Pancratz

A ∈ Mm,n(F) rowrk(A) = dim rowsp(A) = rank(A T )
A ∈ Mm,n(F) rowrk(A) = rank(A)
A ∈ Mm,n(F) r = rank(A) A
Ir 0
0 0
m×n
Ir 0
0 0 = QAP Q P
m×n
P T A T Q T
Ir 0
=
0 0
n×m
AT
Ir 0
0 0 n×m rank
Ir 0
0 0 = r rowrk(A) =
n×m
rank(AT ) = rank
Ir 0
0 0 = r = rank(A)
n×m
m × n
F
i j
i λ i λ ∈ F \ {0}
λ i j i = j λ ∈ F
In
Tij Mi,λ Ci,j,λ A
m × n
j 1 ij i1 ≤ i2 ≤ · · ·
ij k k < j 0
A
A n × n
In A −1
A ↦→ AE1E2 · · · Ek = I
In ↦→ InE1E2 · · · Ek = A −1 .
A n × n A
A −1 = E1 · · · Ek
A = E −1
k · · · E −1
1
n × n A A ′ A ′ = P −1 AP
P