Linear Algebra - Sebastian Pancratz

A ∈ Mn(F) tr A = n
i=1 aii tr : Mn(F) → F
tr(AB) = tr(BA)
tr(AB) =
i j aijbji =
j i bjiaij = tr(BA)
tr(P −1 AP ) = tr(AP P −1 ) = tr(A)
α ∈ End(V ) tr(α) = tr[α]B B V
Sn {1, . . . , n}
(σ ◦ τ)(j) = σ(τ(j)) σ
+1
ε(σ) =
−1
ε : Sn → {+1, −1}
A ∈ Mn(F)
det A =
σ∈Sn
ε(σ)a σ(1)1 · · · a σ(n)n.
A (i) i A A = (A (1) , . . . , A (n) )
A n F n {e1, . . . , en}
F n
d : F n × · · · × F n → F F n
d(v1, . . . , λivi, . . . , vn) = λd(v1, . . . , vi, . . . , vn)
d(v1, . . . , vi + v ′ i, . . . , vn) = d(v1, . . . , vi, . . . , vn) + d(v1, . . . , v ′ i, . . . , vn)
i = j vi = vj d(v1, . . . , vn) = 0