Math 518 review for Exam 2 Spring 2002
1. Let T∈ L(V), ma positive integer, v∈Vsuch that Tm−1(v)6= 0 but Tm(v) = 0.
Prove that {v, T (v), T 2(v), ..., T m−1(v)}is linearly independent.
2. Let Vbe a finite dimensional vector space and T∈ L(V, V ). Suppose that range(T) =
range(T2). Prove that V= range(T)⊕null(T).
3. Let Vbe a finite dimensional vector space, T∈ L(V), and v∈V. Let pdenote the
monic polynomial of smallest degree such that p(T)(v) = 0. Prove that pdivides the minimal
polynomial of T.
4. Let Vbe a finite dimensional vector space, T, S ∈ L(V). show that ST is nilpotent if and
only if T S is nilpotent.
5. Prove that similar matrices have the same characteristic polynomial and the same minimal
polynomial.
6. Let Vbe a finite dimensional vector space, T∈ L(V) with Tm= 0 for some positive integer
M. Show that the only eigenvalue of Tis zero.
7. Let Vbe a finite dimensional vector space, T∈ L(V) such that T2=I. Show that
V=V+⊕V−, where V+={v∈V|T(v) = v}and V−={v∈V|T(v) = −v}.
8. Let Vbe a finite dimensional vector space, T∈ L(V). If T2=Twe say Tis a projection.
If T2=Iwe say Tis an involution. Find the minimal polynomials of all projections and all
involutions.
9. Let Pn(C) denote the vector space of polynomials in xwith coefficients in Cand degree
≤n.
(a) Define D:Pn(C)→Pn(C) as D(p(x)) = p0(x). Find the minimal polynomial of D.
(b) Define S:Pn(C)→Pn(C) as S(p(x)) = p(x+ 1). Find the minimal polynomial of S.
10. Let Vbe a finite dimensional vector space of n×nmatrices and let T∈ L(V) be defined
as T(A) = At(Tmaps any n×nmatrix to its transpose). Show that the only eigenvalues are
±1 and describe the corresponding eigenspaces.
11. Let T∈ L(V) have characteristic polynomial p(x) = (x−1)5(x+ 4)4x2and minimal
polynomial m(x)=(x−1)2(x+ 4)3x. Find all possible non-similar Jordan canonical forms of
T.
12. Classify up to similarity all complex 3 ×3 matrices Awith A3=I.
13. For each of the following matrices A, find etA.
(a) "0 1
−1 0 #(b)
010
000
001
(c)
0 2 −1
1 2 −2
0 2 −1
(d)
5−6−6
−1 4 2
3−6−4
14. Solve the following systems of differential equations:
(a)y0
1(t) = 2y1(t)−y2(t)
y0
2(t) = y1(t)
y1(0) = 0
y2(0) = 1 .
(b)y00(t) = 2y0−y y(0) = 0 y0(0) = 1.
15. Previous homework assignments.