Linear Algebra - Sebastian Pancratz

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m = mj Cl = J(nl, λ) aj aj = mj i=1 ni gj = mj ej = n1 n1 ≥ n1 ≥ · · · nm > 0 n1 + n2 + · · · + nm = 0 C λ1, . . . , λk V Wj = N(α − λjι) aj V = k j=1 Wj B = k j=1 Bj Bj Wj ⎛ B1 ⎜ [α]B = ⎝ ⎞ 0 ⎟ ⎠ . 0 Bk pj(t) = (λj −t) −aj k r=1 (λr −t) ar qj k Wj = Im(hj(α)) j=1 pjqj = 1 α(Wj) ⊂ Wj Wj α| ∈ End(Wj) Wj λ = λj V = Wj n = aj (α − λι) n = 0 α − λι V ⎛ λ ⎜ ⎜1 ⎜ ⎝ ⎞ 0 ⎟ ⎠ 0 1 λ . λ v1, . . . , vm α−λι v1 ↦→ v2 ↦→ . . . ↦→ vm ↦→ 0 n = 3 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎝ λ1 λ2 λ3 ⎠ ⎝ λ1 λ2 1 ⎠ (λ1 − t)(λ2 − t)(λ3 − t) (λ1 − t)(λ2 − t) 2 (λ1 − t)(λ2 − t)(λ3 − t) (λ1 − t)(λ2 − t) 2 ⎛ ⎞ λ ⎛ ⎞ λ 1 ⎝ λ ⎠ ⎝ λ ⎠ λ λ (λ − t) 3 (λ − t) 3 λ − t (λ − t) 2 λ2 ⎝ λ1 λ2 λ2 ⎠ (λ1 − t)(λ2 − t) 2 (λ1 − t)(λ2 − t) ⎛ ⎞ λ 1 ⎝ λ 1⎠ λ (λ − t) 3 (λ − t) 3
n = 4 (λ−t) 4 4 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ λ 1 λ 1 λ ⎟ 1⎠ λ ⎜ ⎝ (λ − t) 4 ⎛ λ 1 ⎜ ⎝ λ (λ − t) 2 λ λ ⎞ ⎟ ⎠ ⎜ ⎝ λ 1 (λ − t) 3 ⎛ λ ⎜ λ ⎝ λ − t λ 1 λ n((α − λι) r ) r ⎛ 2 0 0 ⎞ 0 ⎜ A = ⎜3 ⎝0 2 0 0 2 −2 ⎟ 0 ⎠ 0 0 2 2 χA(t) = (2 − t) 4 ⎛ 0 0 0 ⎞ 0 ⎜ A − 2I = ⎜3 ⎝0 0 0 0 0 −2 ⎟ 0 ⎠ 0 0 2 0 λ λ λ ⎟ ⎠ ⎞ ⎟ ⎠ ⎜ ⎝ λ 1 λ (λ − t) 2 ⎟ λ 1⎠ λ (A − 2I) 2 ⎛ 0 0 0 ⎞ 0 ⎜ = ⎜0 ⎝0 0 0 −4 0 0 ⎟ 0⎠ 0 0 0 0 A mA(t) = (2 − t) 3 n(A − 2I) = 2 A ⎛ 2 1 0 ⎞ 0 ⎜ JNF = ⎜0 ⎝0 2 0 1 2 0 ⎟ 0⎠ 0 0 0 2 . v3 ∈ ker(A − 2I) 2 v3 = (0, 0, 1, 0) T A − 2I ⎛ ⎞ 0 ⎜ v3 = ⎜0 ⎟ ⎝1⎠ 0 ↦→ v2 ⎛ ⎞ 0 ⎜ = ⎜0 ⎟ ⎝0⎠ 2 ↦→ v1 ⎛ ⎞ 0 ⎜ = ⎜−4 ⎟ ⎝ 0 ⎠ ↦→ 0. 0 v4 v4 = (2, 0, 0, 3) T
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 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: α p(α) = 0 mα
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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