Linear Algebra - Sebastian Pancratz

V F V ∗ = L(V, F)
V V ∗ V
V F B = {e1, . . . , en} V
B ∗ = {ε1, . . . , εn} V ∗ B εj(ek) = δjk
n j=1 λjεj = 0 k = 1, . . . , n λk = ( n j=1 λjεj)(ek) = 0
B∗ ε ∈ V ∗ ε = n j=1 ε(ej)εj B∗ V ∗ ε = n
j=1 ajεj v = n
j=1 xjej
ε(v) =
n
ajxj = ⎛ ⎞
x1
⎜
a1 . . .
⎟
an ⎝ ⎠ .
j=1
F n n
U ≤ V U ◦ = {ε ∈ V ∗ : ε(u) = 0 ∀u ∈ U} U ◦ U
V ∗
U ≤ V U ◦ ≤ V ∗
U ≤ V dim U + dim U ◦ = dim V
U ≤ V e1, . . . , ek U B = {e1, . . . , ek, . . . , en}
V U ◦ = 〈εk+1, . . . , εn〉 ε1, . . . , εn V ∗ B
i > k εi(ej) = 0 j ≤ k εi ∈ U ◦ ε ∈ U ◦ ε = n
j=1 λjεj
j ≤ k λj = ε(ej) = 0 ε ∈ 〈εk+1, . . . , εn〉
U, V F U α −→ V
V ∗ α∗
−→ U ∗ α∗ (ε) = ε ◦ α ε ∈ V ∗ α
ε ◦ α : U → F α ∗ ∈ U ∗ θ1, θ2 ∈ V ∗
α ∗ (θ1 + θ2) = (θ1 + θ2) ◦ α
= θ1 ◦ α + θ2 ◦ α
xn