Linear Algebra - Sebastian Pancratz

α α −1 : W → V
α −1
α −1 (w1 + w2) = α −1 (α(v1) + α(v2))
= α −1 (α(v1 + v2))
= v1 + v2
= α −1 (w1) + α −1 (w2)
α −1 (λw) = α −1 (λα(v))
= α −1 (α(λv))
= λv
= λα −1 (w)
V F n V F n
v1, . . . , vn α : V → F n , n
i=1 λivi ↦→ (λ1, . . . , λn) T
V W F
v1, . . . , vn w1, . . . , wn V W
α : V → W, λivi ↦→ λiwi
V W B
V α : V → W α(B) W
• w ∈ W w = α(v) v ∈ V v = n
i=1 λivi v1, . . . , vn ∈ B
λi ∈ F w = α( λivi) = λiα(vi)
• λ1α(v1)+· · ·+λnα(vn) = 0 α(λ1v1+· · ·+λnvn) = 0 λ1v1+· · ·+λnvn = 0
α λi = 0 i B
α : V → W ker(α) = {v ∈ V : α(v) =
0} = N(α) α Im(α) = {w ∈ W : w = α(v) v ∈ V }
N(α) ≤ V Im(α) ≤ W α N(α) = {0}
Im(α) = W n(α) = dim N(α) α
rank(α) = dim Im(α) α
V W F dimF V
α : V → W dim V = rank(α) + n(α)
v1, . . . , vk N(α) v1, . . . , vk, vk+1, . . . , vn
V α(vk+1), . . . , α(vn) Im(α)
• w ∈ Im(α) w = α(v) v ∈ V v = n i=1 λivi λi ∈ F
w = α(v) = n i=1 λiα(vi) = n k+1 λiα(vi) α(vi) = 0 i = 1, . . . , k