Linear Algebra - Sebastian Pancratz

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d d(e1, . . . , en) = 1 i = j d(v1, . . . , vj, . . . , vi, . . . , vn) = −d(v1, . . . , vi, . . . , vj, . . . , vn). 0 = d(v1, . . . , vi + vj, . . . , vi + vj, . . . , vn) = 0 + d(v1, . . . , vi, . . . , vj, . . . , vn) + d(v1, . . . , vj, . . . , vi, . . . , vn) + 0 σ ∈ Sn d F n v1, . . . , vn ∈ F n d d(v σ(1), . . . , v σ(n) = ε(σ)d(v1, . . . , vn) d(e σ(1), . . . , e σ(n) = ε(σ)d(e1, . . . , en) = ε(σ) d F n A = (aij) = (A (1) , . . . , A (n) ) ∈ Mn(F) d(A (1) , . . . , A (n) ) = det Ad(e1, . . . , en) d(A (1) , . . . , A (n) ) = d( j1 aj11ej1 , A(2) , . . . , A (n) ) = aj11d(ej1 , A(2) , . . . , A (n) ) j1 = aj11aj22d(ej1 , ej2 , A(3) , . . . , A (n) ) j1,j2 = · · · = j1,...,jn aj11 · · · ajnnd(ej1 , . . . , ejn) = aσ(1)1 · · · aσ(n)nε(σ)d(e1, . . . , en) σ∈Sn = (det A)d(e1, . . . , en). d : F n × · · · × F n → F d(A (1) , . . . , A (n) ) = det A A = (A (1) , . . . , A (n) ) d n j=1 a σ(j)j det A A (k) = A (l) k = l det A = 0 τ = (kl) Sn det A = ε(σ) aσ(j)j σ∈Sn j
σ στ ε(σ) = 1 ε(στ) = −1 det A = ⎛ ⎝ aσ(j)j − ⎞ a ⎠ στ(j)j = 0 σ det A = σ∈Sn ε(σ) n det A T = det A j a σ(1)1 · · · a σ(k)k · · · a σ(l)l · · · a σ(n)n − a σ(1)1 · · · a σ(l)k · · · a σ(k)l · · · a σ(n)n = 0. j=1 δ σ(j)j = ε(ι) · 1 = 1 σ ∈ Sn n j=1 a σ(j)j = n j=1 a jσ(j) σ Sn σ −1 ε(σ −1 ) = ε(σ) det A = ε(σ) σ∈Sn = ε(σ) σ∈Sn = ε(σ) σ∈Sn = det A T j n aσ(j)j j=1 n ajσ−1 (j) j=1 n ajσ(j) det I A aij = 0 i > j det A = a11 · · · ann det A = σ∈Sn j=1 ε(σ)a σ(1)1 · · · a σ(n)n. σ(i) ≤ i i = 1, . . . , n σ(1) = 1 σ(2) = 2 σ(n) = n σ = ι det A = a11 · · · ann E n × n A det(AE) = det A det E = det(EA). A det A det Tij = −1 det Mi,λ = λ det Ci,j,λ = 1 det A −1 λ 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 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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