Linear Algebra - Sebastian Pancratz

u, v ∈ V
ψ(u, v) = [u] T B[ψ]B[v]B
[ψ]B ′ = P T [ψ]BP [ψ]B ′
= (P [u]B ′)T [ψ]B(P [v]B ′)
= [u] T B ′P T [ψ]BP [v]B ′,
A, B B = P T AP
P
Mn(R)
ψ ψ(u, v) = ψ(v, u) u, v ∈ V
A = [ψ]B A = A T
P T AP = D P A = A T
V Q : V → R
V Q(λv) = λ 2 Q(v) λ ∈ R, v ∈ V
ψ V Q(v) + Q(w) + 2ψ(v, w) = Q(v + w) v, w ∈ V
ψ Q(v) = ψ(v, v)
(Q(v + w) − Q(v) − Q(w))
Q ψ(v, w) = 1
2
⎛
Ip
⎝ −Iq
⎞
0
⎠ .
0 0
dim V = n
ψ(u, v) = 0 u, v ∈ V [ψ]B = 0
B e ∈ V ψ(e, e) = 0
2ψ(u, v) = ψ(u + v, u + v) − ψ(u, u) − ψ(v, v) 0 u, v ∈ V
W = {v ∈ V : ψ(e, v) = 0} V = 〈e〉 ⊕ W v ∈ V v = λe + (v − λe)
〈e〉 ∩ W = {0}
λ ∈ R λ v − λe ∈ W λ = ψ(e,v)
ψ(e,e)
ψ(e, λe) = 0 λ = 0 ψ ′ ψ W
e2, . . . , en W ψ ′
ψ ′
⎛
⎜
⎝
d2
0 dn
⎞
0
⎟
⎠ .