Linear Algebra - Sebastian Pancratz

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hj(λi) = δij 1 ≤ i, j ≤ k h(t) = k j=1 hj(t) = 1 h(t) − 1 k λ1, . . . , λk k v ∈ V v = ι(v) = h(α)(v) = k hj(α)(v) = vj = hj(α)(v) (α − λjι)vj = 0 p(α) = 0 vj α λj v ∈ V hj(α) A ∈ Mn(F) P −1AP P p(A) = 0 p ∈ F[t] P P −1 ⎛ d1 ⎜ AP = D = ⎝ ⎞ 0 ⎟ ⎠ 0 dn AP = P D AP (j) = djP (j) j P A dj α1, α2 ∈ End(V ) α1α2 = α2α1 B V [α1]B [α2]B V = V1⊕· · ·⊕Vk Vj α1 α1(vj) = λjvj vj ∈ Vj α2(Vj) ⊂ Vj v ∈ Vj α1(α2(v)) = α2(α1(v)) = α2(λjv) = λjα2(v) α2(v) ∈ Vj α2| Vj Bj Vj α2 α1 B α1 α2 p(t) ∈ F[t] j=1 p(t) = ant n + · · · + a1t + a0 ai ∈ F 0 ≤ i ≤ n p(t), q(t) ∈ F[t] F[t] m ≤ n p(t) = ant n + · · · + a1t + a0 q(t) = bmt m + · · · + b1t + b0 (p + q)(t) = ant n + · · · + (am + bm)t m + · · · + (a1 + b1)t + (a0 + b0) k j=1 (pq)(t) = anbmt n+m + · · · + (a1b0 + a0b1)t + a0b0 vj
deg p p l al = 0 −∞ p 0 deg pq = deg p + deg q F[t] a, b ∈ F[t] b = 0 q, r ∈ F[t] a = bq + r deg r < deg b r = 0 a = antn + · · · + a0 b = bmtm + · · · + b0 bm = 0 n ≥ m q = 0 r = a a a ′ = a − an bm tn−mb deg a ′ < deg a a ′ = bq ′ + r q ′ , r deg r < deg b q = an bm tn−mq ′ F[t] p ∈ F[t] λ ∈ F p(λ) = 0 q ∈ F[t] p(t) = (λ − t)q(t) λ p e (λ − t) e p (λ − t) e+1 n n p1, p2 n n p, q ∈ F[t] α ∈ End(V ) p(α)q(α) = q(α)p(α) α(v) = λv p ∈ F[t] p(α)(v) = p(λ)v V F dim V = n α ∈ End(V ) p n 2 p(α) = 0 dim End(V ) = n 2 a n 2, . . . , a1, a0 ∈ F an2α n2 + an2−1α n2−1 + · · · + a1α + a0ι = 0 n2 +1 p(t) = an2tn2 +· · ·+a1t+a0 α ∈ End(V ) mα α mα(α) = 0 α ∈ End(V ) p ∈ F[t] p(α) = 0 mα p F[t] p = mαq + r q, r ∈ F[t] deg r < deg mα r = 0 p(α) = 0 = mα(α) r(α) = 0 r = 0 deg mα V α ∈ End(V ) χα(α) = 0 A ∈ Mn(F) χA(A) = 0 A ∈ Mn(F) (−1) n χA(t) = t n + an−1t n−1 + · · · + a1t + a0 = det(tI − A) B B · adj B = (det B)I adj(tI − A) n (tI − A)(Bn−1t n−1 + · · · + B1t + B0) = (tI − A) adj(tI − A) = (t n + an−1t n−1 + · · · + a1t + a0)I
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: α ∈ End(V ) U V α(U) ≤ U
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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