Linear Algebra - Sebastian Pancratz

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A A det A = 0 A A det A det A = 0 A 0 A A 0 det A = 0 A, B ∈ Mn(F) det(AB) = det(A) det(B) A dA : (B (1) , . . . , B (n) ) ↦→ det(AB) B = (B (1) , . . . , B (n) ) F n det(AB) = dA(AB (1) , . . . , AB (n) ) dA det(AB) = dA(B (1) , . . . , B (n) ) = det BdA(e1, . . . , en) = det(B) det(A), ⎛ det(AB) = det ⎝ bj11A (j1) , . . . , bjnnA (jn) ⎞ ⎠ σ∈Sn j1 = ⎛ ⎞ n ⎝ b ⎠ σ(j)j det(A σ(1) , . . . , A σ(n) ) σ∈Sn j=1 jn = ⎛ ⎞ n ⎝ b ⎠ σ(j)j ε(σ) det A j=1 = det(A) det(B). B AB det B = 0 = det(AB) B B = E1 · · · Ek det(AB) = det(AE1 · · · Ek) = det A det E1 · · · det Ek = det A det B. A det A −1 = (det A) −1 A AA −1 = I (det A)(det A −1 ) = det I = 1 det A −1 = (det A) −1 n × n
det(P −1 AP ) = det P −1 det A det P = (det A)(det P )(det P ) −1 = det A α : V → V det α = det[α]B B V det : End(V ) → F det ι = 1 det α ◦ β = det α det β det α = 0 α α det α −1 = (det α) −1 GL(V ) V GLn(F) n × n F GL(V ) GLn(F) det : GLn(F) → F A ∈ Mm(F) B ∈ Mk(F) C ∈ Mm,k(F) det A C 0 B = det A det B B, C dB,C : A ↦→ det A C 0 B Fm dB,C(A) = det A det I C 0 B C B ↦→ det I C 0 B Fk det I C 0 B = det B I C 0 I det I C 0 I = 1 I C 0 I det A C 0 B = det A det B X = A C 0 B A C det = 0 B m+n ε(σ) xσ(j)j σ∈§m+n j=1 x σ(j)j = 0 j ≤ m σ(j) > m σ j ∈ [1, m] σ(j) ∈ [1, m] x σ(j)j = a σ1(j)j σ1 ∈ Sm σ [1, m] j ∈ [m + 1, m + k] σ(j) ∈ [m + 1, m + k] l = j − m x σ(j)j = b σ2(l)l σ2(l) = σ(m + l) − m ε(σ) = ε(σ1)ε(σ2) σ det ⎛ A C 0 B = ⎝ ⎞ ⎛ m ε(σ1) a ⎠ ⎝ σ1(j)j ⎞ k ε(σ2) a ⎠ σ2(j)j σ1∈Sm = det A det B j=1 σ2∈Sk A = (aij) n × n A ij (n − 1) × (n − 1) A i j l=1
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 and 14: • n i=k+1 λiα(vi) = 0 α( n i
Page 15 and 16: B = {u1, . . . , un} C = {v1, . . .
Page 17: A ∈ Mm,n(F) rowrk(A) = dim rowsp
Page 20 and 21: d d(e1, . . . , en) = 1
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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