Linear Algebra - Sebastian Pancratz

e1, . . . , ek W e1, . . . , ek, ek+1, . . . , en
V ek+1, . . . , en W ⊥
v ∈ V v = k j=1 λjej + n j=k+1 λjej V = W + W ⊥ W ∩ W ⊥ = {0}
〈v, v〉 = 0 v = 0
V W ≤ V
π = πW V W π 2 = π W = Im π W ⊥ = ker π
e1, . . . , ek W
e1, . . . , ek, ek+1, . . . , en V v = n j=1 λjej πw(v) =
k j=1 λjej πw(v) = k j=1 〈ej, v〉ej
ιV = πW + π W ⊥ πW π W ⊥ = 0
W ≤ V v ∈ V πW (v) W v
w0 = πW (v) d(w0, v) ≤ d(w, v) w ∈ W
V α ∈ End(V )
α ∗ V α
α ∗ : V → V 〈αv, w〉 = 〈v, α ∗ w〉
v, w ∈ V B V [α ∗ ]B = [α] T
B
B = {e1, . . . , en} V A = [α]B = (aij)
α ∗ [α ∗ ]B = ĀT C = ĀT 1 ≤ i, j ≤ n
〈αei, ej〉 = 〈
=
n
k=1
akiek, ej〉
n
aki〈ek, ¯ ej〉
k=1
= ¯
aji
= cij
n
= ckj〈ei, ek〉
k=1
= 〈ei,
n
k=1
= 〈ei, α ∗ ej〉
ckjek〉
〈αv, w〉 = 〈v, α ∗ w〉 v, w ∈ V α ∗
[α ∗ ]B = ĀT
(α + β) ∗ = α ∗ + β ∗ (λα) ∗ = ¯ λα ∗
α ∗∗ = α ι ∗ = ι (αβ) ∗ = β ∗ α ∗
〈α ∗ v, w〉 = 〈v, α ∗∗ w〉
〈α ∗ v, w〉 = 〈w, α ∗ v〉