Linear Algebra - Sebastian Pancratz

B = {u1, . . . , un} C = {v1, . . . , vm} U, V
ε = εB : U → F n , u ↦→ [u]B φ = φC : V → F m , v ↦→ [v]C
εB
U
⏐
F n
α
−−−−→ V
⏐
⏐
φC
A·
−−−−→ F m
A [α(u)]C = A · [u]B
u ∈ U A ′ · [u]B = [α(u)]C u ∈ U
u1, . . . , un [uk]B = ek A ′ · ek k A ′
1 ≤ k ≤ n
α : U → V dim U = n dim V = m L(U, V )
Mm,n(F)
B = {u1, . . . , un} C = {v1, . . . , vm} U V
θ : L(U, V ) → Mm,n(F), α ↦→ [α]B,C
U α −→ V β −→ W B, C, D U, V, W
[β ◦ α]B,D = [β]C,D · [α]B,C
A = [α]B,C B = [β]C,D
[β ◦ α]B,D = BA
β ◦ α(uk) = β
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U V F U B = {u1, . . . , un}
B ′ = {u ′ 1 , . . . , u′ 2 } V C = {v1, . . . , vm} C ′ = {v ′ 1 , . . . , v′ m}
P = (pij) B B ′ u ′
j = i pijui
P = [u ′ 1 ]B . . . [u ′
n]B = [ιU]B ′ B.
[u]B = P [u]B ′ u ∈ U u′ j [u′ j ]B ′ = ej
P P −1 B ′ B
Q C C ′
α : U → V B, B ′ , C, C ′
A = [α]B,C A ′ = [α]B ′ ,C ′ A′ = Q −1 AP