Linear Algebra - Sebastian Pancratz

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vi λik = 0 1 ≤ i ≤ m 1 ≤ k ≤ n εij α ∈ L(U, V ) α(uk) = i aikvi (aijεij(uk)) = aikvi = α(uk) i,j 1 ≤ k ≤ n U m × n F A = (aij) m n aij ∈ F 1 ≤ i ≤ m 1 ≤ n ≤ n Mm,n(F) m × n F Mm,n(F) dimF Mm,n(F) = mn (aij) + (bij) = (aij + bij) λ(aij) = (λaij) 1 ≤ i ≤ m 1 ≤ j ≤ n eij = 1 Eij = i ei ′ j ′ = 0 (i′ , j ′ ) = (i, j). U V F α : U → V B = {u1, . . . , un} C = {v1, . . . , vm} U V A = (aij) α(uj) = i aijvi u ∈ U u = i λiui [u]B = (λ1, . . . , λn) T A = [α(u1)]C · · · [α(un)]C , A = [α]B,C u ∈ U [α(u)]C = [α]B,C[u]B u ∈ U u = j λjuj [u]B = (λ1, . . . , λn) T α(u) = λjα(uj) [α(u)]C = A · [u]B j = j λj = i j i aijvi aijλjvi = (A · [u]B)ivi i
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 = j j ajk = i j ajkvj i bijwi bijajkwi = (BA)ikwi i 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
Page 5 and 6: F R C F • F +
Page 7 and 8: V = C C 1, i v1, . .
Page 9 and 10: U ∪ W U ⊂ W W ⊂ U
Page 11 and 12: V W F α : V → W v, v1
Page 13: • n i=k+1 λiα(vi) = 0 α( n i
Page 17: A ∈ Mm,n(F) rowrk(A) = dim rowsp
Page 20 and 21: d d(e1, . . . , en) = 1
Page 22 and 23: A A det A = 0 A A
Page 24 and 25: j det A = n i=1 (−1)i+j aij det(
Page 27 and 28: V F F R C α : V → V
Page 29 and 30: α ∈ End(V ) U V α(U) ≤ U
Page 31 and 32: deg p p l al = 0 −∞
Page 33 and 34: α p(α) = 0 mα
Page 35: n = 4 (λ−t) 4 4 4 = 3 +
Page 38 and 39: λ ∈ F θ ∈ V ∗ = α ∗ (θ
Page 41 and 42: U, V F ψ : U × V → F
Page 43 and 44: B0 = {e1, e2, . . . , en} e1 = e
Page 45 and 46: C e1, . . . , en ψ
Page 47: ψ : U × V → F ψL : U → V
Page 50 and 51: R n C n V = C[0, 1] 〈(, f
Page 52 and 53: = 〈αw, v〉 = 〈v, αw〉 〈v
Page 54: A n × n V n B

Linear Algebra - Sebastian Pancratz

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