118 CHAPTER 5. LINEAR TRANSFORMATIONS

32. ↑In the situation of the above problem, show that there exist finitely many vectors inU, {x1, · · · ,xm} such that

{w1, · · · ,ws,β x1

, · · · ,β xm

}is a basis for U +W . This last

vector space is defined as the set of all y+w where y ∈U and w ∈W .

33. ↑ In the situation of the above where φ (λ ) is irreducible. Let U be defined as

U = φ (A)(ker(φ (A)m))

Explain why U ⊆ ker(

φ (A)m−1)

. Suppose you have a linearly independent set in U

which is{

β x1, · · · ,β xr

}. Here the notation means

β x ≡{

x,Ax, · · · ,Am−1x}

where these vectors are independent but Amx is in the span of these. Such exists anytime you have x ∈ ker(g(A)) for g(λ ) a polynomial. Letting φ (A)yi = xi, explainwhy

{β y1

, · · · ,β yr

}is also linearly independent. This is like the theorem presented

earlier that the inverse image of a linearly independent set is linearly independent butit is more complicated here because instead of single vectors, we are considering setsβ x.