Linear Algebra - Sebastian Pancratz

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