Unformatted text preview: 11/22/2001 THU 19:40 FAX 6434330 MOFFITT LIBRARY 001 George M. Bergman Spring 1995, Math 110, Section 1 7 April, 1995
85 Evans Hall Second Midterm Exam 10:1011:00 1. (Read all three parts of this question before answering.) (a) (6 points) ASSuming one knows how to deﬁne the determinant of a square matrix,
deﬁne the determinant of a linear operator T: V ——> V for V a ﬁnite—dimensional vector space. (b) (6 points) What result must be proved to show that the determinant of T, as you
deﬁned it in (a), is welldeﬁned? (c) (8 points) Give the proof of the result referred to in (b). You may assume general
results about the arithmetic of determinants, and about matrices of linear transformations. 2. (20 points) Below, I give a slightly reworded version of Gerschgorin’s Disk Theorem,
and a brief proof. At four spots in the proof, a statement is given in bold type. You are
to give a brief justiﬁcation of each of these statements below. (The assertions are preceded
by marks [A] to [D]. Give each justiﬁcation after the corresponding mark at the bottom of
the page.) If you cannot justify some step, you can still give justiﬁcations for later steps assuming that step. Gerschgorin’s Disk Theorem. Let AeMnxn(C). For i = 1,...,n. let Ci denote
the closed disk in the complex plane centered at Aﬁ, and having radius equal to
Ejii IAij. Then each eigenvalue of A lies in one of the disks Ci. Proof. Let A be an eigenvalue of A. Then [A] there exist complex numbers
x1,...,xn, not all zero, such that for all i, Zj Aij xj = Axi. Let k be chosen so that
xk has largest absolute value among the xi. Observe that [B] Axk — Akk xk = Ejgﬁk AkJxj. Hence [luck — Akkxk = {Ejgék Akj le S thk Akj Ixj. [C] This last term is S kal.
Hence Put —A x S E— A  x , or, factorin out Jr on both sides,
k lck lc Jik k} k g k
2L — A x S (2 A  ) x . But [D] x > 0. Hence we can divide this ine uality
kk k ji‘k k] k k q
by x , getting A — A S E— A  , which sa s that A lies in the disk C of
k kk j¢k k] Y k radius 21: k IAkJI about the point Akk, as claimed. El 11/22/2001 THU 19:40 FAX 6434330 MOFFITT LIBRARY 002 3. Let T be a linear operator on a vector space V. (a) (6 points) Deﬁne what is meant by a T—invariant subspace of V. (b) (10 points) Suppose V is ﬁnite—dimensional,of dimension n, and m is an integer S n. Show that if V has an ordered basis ,6 such that [Th5 has the form , where A is me, then V has an mdimensional T—invariant subspace. (This is really an “if and only if” result, but to save time I am just asking you to prove this direction.) 4. Let A be an n><n matrix over a ﬁeld F.
(a) (6 points) Deﬁne what is meant by the “characteristic polynomial of A”. (b) (6 points) The CayleyHamilton Theorem says that A “satisﬁes” its characteristic
polynomial. Say precisely what this means. (If you use the concept of “substituting” a matrix into a polynomial, say what you mean by this.) (c) (10 points) Show that the subspace of Mnxn (F) spanned by I, A, A2, A3,... has dimension S n. 5. Suppose U and V are ﬁnite—dimensional inner product spaces over F (the ﬁeld of real or complex numbers), and let T: U —> V be a linear map. (a) (15 points) Prove that there exists a map 7””: V —> U such that for all 105 U, yEV
one has <T(x), y>V = (x, T*(y)>U, where <, >U and <, >V denote the inner
products of U and V respectively. (For the sake of time, I do not ask you to prove that
T* is also linear.) This is a modiﬁed version of a result in the text; you may use any results actually proved in the text in proving it. (b) (7 points) Pr0ve that the map T* constructed in (a) also satisﬁes the equation <36, T(y)>v = <T*(x), y>U for all xEV, yEU. ...
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