9.3. CYCLIC SETS 241
Proof: Claim: If x ∈ kerϕ (A) , and |βx| denotes the length of βx, then |βx| = d thedegree of the irreducible polynomial ϕ(λ) and so
βx ={x,Ax,A2x, · · · , Ad−1x
}also span (βx) is A invariant, A (span (βx)) ⊆ span (βx).
Proof of the claim: Let m = |βx| . That is, there exists monic η (λ) of degree m andη (A)x = 0 with m is as small as possible for this to happen. Then from the usual process ofdivision of polynomials, there exist l (λ) , r (λ) such that r (λ) = 0 or else has smaller degreethan that of η (λ) such that
ϕ (λ) = η (λ) l (λ) + r (λ)
If deg (r (λ)) < deg (η (λ)) , then the equation implies 0 = ϕ (A)x = r (A)x and so m wasincorrectly chosen. Hence r (λ) = 0 and so if l (λ) ̸= 1, then η (λ) divides ϕ (λ) contraryto the assumption that ϕ (λ) is irreducible. Hence l (λ) = 1 and η (λ) = ϕ (λ) . The claimabout span (βx) is obvious because A
dx ∈ span (βx). This shows the claim.Suppose now x ∈ U \W where U ⊆ ker (ϕ (A)). Consider
{v1, · · · , vs, βx} .
Is this set of vectors independent? Suppose
s∑i=1
aivi +
d∑j=1
djAj−1x = 0.
If z ≡∑d
j=1 djAj−1x, then z ∈W ∩ span
(x,Ax, · · · , Ad−1x
). Then also for each m ≤ d−1,
Amz ∈W ∩ span(x,Ax, · · · , Ad−1x
)because W, span
(x,Ax, · · · , Ad−1x
)are A invariant. Therefore,
span(z,Az, · · · , Ad−1z
)⊆ W ∩ span
(x,Ax, · · · , Ad−1x
)⊆ span
(x,Ax, · · · , Ad−1x
)(9.3)
Suppose z ̸= 0. Then from the Lemma 9.3.2 above,{z,Az, · · · , Ad−1z
}must be linearly
independent. Therefore,
d = dim(span
(z,Az, · · · , Ad−1z
))≤ dim
(W ∩ span
(x,Ax, · · · , Ad−1x
))≤ dim
(span
(x,Ax, · · · , Ad−1x
))= d
ThusW ∩ span
(x,Ax, · · · , Ad−1x
)= span
(x,Ax, · · · , Ad−1x
)which would require x ∈ W but this is assumed not to take place. Hence z = 0 and sothe linear independence of the {v1, · · · , vs} implies each ai = 0. Then the linear indepen-dence of
{x,Ax, · · · , Ad−1x
}, which follows from Lemma 9.3.2, shows each dj = 0. Thus{
v1, · · · , vs, x, Ax, · · · , Ad−1x}is linearly independent as claimed.
Let x ∈ U \W ⊆ ker (ϕ (A)) . Then it was just shown that {v1, · · · , vs, βx} is linearlyindependent. Let W1 be given by
y ∈ span (v1, · · · , vs, βx) ≡W1