Linear Algebra - Sebastian Pancratz

F R C F
• F + 0
• F \ {0} · 1
• a(b + c) = ab + ac a, b, c ∈ F
V F
V +
+ : V → V (v1, v2) ↦→ v1 + v2 ∈ V
v1, v2, v3 ∈ V (v1 + v2) + v3 = v1 + (v2 + v3)
v1, v2 ∈ V v1 + v2 = v2 + v1
0 ∈ V v + 0 = v v ∈ V
v ∈ V −v ∈ V −v + v = 0
· V
· : F × V → V (λ, v) ↦→ λv
λ ∈ F, v1, v2 ∈ V λ(v1 + v2) = λv1 + λv2
λ1, λ2 ∈ F, v ∈ V (λ1 + λ2)v = λ1v + λ2v
λ1, λ2 ∈ F, v ∈ V (λ1λ2)v = λ1(λ2v)
v ∈ V 1v = v
V F v ∈ V, λ ∈ F
0 · v = 0 λ · 0 = 0
−v = (−1) · v
λv = 0 λ = 0 v = 0
0 · v = (0 + 0) · v = 0 · v + 0 · v 0 = 0 · v
−v + v = 0 = 0v = (−1 + 1)v = (−1)v + 1v = (−1)v + v −v = (−1)v