Linear Algebra - Sebastian Pancratz

U, V F ψ : U × V → F
u ∈ U ψ(u, v) v
v ∈ V ψ(u, v) u U = V
V
F = R V = R n ψ(x, y) = n
i=1 xiyi = x T y
V = F n A ∈ Mn(F) ψ(u, v) = u T Av
V F dim V = n B = {v1, . . . , vn}
V ψ V B A =
(ψ(vi, vj)) = [ψ]B
ψ V B V ψ(u, v) =
[u] T B [ψ]B[v]B u, v ∈ V
B = {v1, . . . , vn} u = n
i=1 aivi v = n
i=1 bivi
ψ(u, v) = ψ( aivi, bjvj)
=
aibjψ(vi, vj)
i,j
⎛
b1
bn
⎞
= ⎜
a1 · · ·
⎟
an [ψ]B ⎝ ⎠
= [u] T B[ψ]B[v]B.
[ψ]B u, v ∈ V ψ(u, v) =
[u] T B A[v]B u, v ∈ V u = vi v = vj Aij = ψ(vi, vj)
B = {v1, . . . , vn}, B ′ = {v ′ 1 , . . . , v′ n} V P
B B ′
v ′ j =
n
i=1
pijvi
[ψ]B ′ = P T [ψ]BP
[v]B = P [v]B ′