lI.:l"LiiI<.'i'J *-.f- 0 -"df-Ii ~.f±~: ~ HUt~ 1>til! # 1 11

lI.:l"LiiI<.'i'J *-.f- 0 -"df-Ii ~.f±~: ~ HUt~ ^{1>til! # 1 11}

I

~i.;t*3Jt: fi:$.~"*~~.(f.ffl}j$.~~J:.fi:$.ji'lQ}JJl!§:.

^{0 1}10 1.05.08

Advanced Calculus 1. Bolzano-Weierstrass Theorem

(a) (15) Show the IRⁿ version of this theorem.

(b) (10) Show that the assumption" ]Rn " is essential.

2. Norms

(a) (15) Show that all norms in ]RT/. are equivalent.

(b) (10) Show that the assumption ,: ]Rn " is essentiaL 3. Compactness

(25) Let

S

be a subset of a metric space. Then

S

is compact if and ouly if every sequence ill S contains a convergent subsequence in S.

4. Series

(a) (15) If the series L~=l an converges absolutely, then every rear­

rangement of it also converges to the same value.

(b) (10) Show that the assumption of absolute convergence is essentiaL

Work out all of the following problems with details.

116

^Pts

1

^{1. Let}

^V

^and

^W

be finite dimensional vector spaces over the field

F.

Let

L : V

--+

W

be a linear map. Prove that the dimension of the kernel of

L

plus the dimension of the image of

L

is equal to the dimension of

V.

2. In each of the following cases decide whether there is a linear map T : R.2 --+ R.3 such that the following holds:

(a)

^T

([~])

^{= [}

~2J

^T

([:J)

⁼

[H ^and

^T

^(UJ)

^{= [ ; }}

~)

^T

([!])

⁼

[~2] ^and

^T

([m

⁼

[~}

In case such a linear map T exists, determine its matrix with respect to the standard basis ofR2 and with respect to the standard basis oflR? If no such a linear map T exists, explain why it is so.

120

^Pts

1

^{3. Let F} be a field and V the vector space FZ. Let T : V --+ V be a linear operator. A vector

a

E V is said to be a ryclic vector for T if{Tia

I

^{i =} 0, 1,2 ... } spans V.

(a) Prove that any nonzero vector of V which is not a eigenvector for T is a cyclic vector for T.

(b) Prove that either

T

has a cyclic vector or

T

is a scalar multiple of the identity operator.

116

^Pts

1

^{4. Let S be} a subspace of a finite dimensional inner product space V over either R or C. Prove that each coset in

V I

S contains exactly one vector that is orthogonal to S.

116

^Pts

I

5. Let}'t/ be an

n x n

with real entries,

n

2: 1. Suppose that M is unita.ry, upper triangular, and has positive entries on the main diagonal. Prove that M is the identity matrix

116

^Pts

I

6. A square matrix N over a field is said to be

nilpotent

if N ^k = 1 for some k ::: O. Let Nl and N

z

^{be 3}

x

3 nilpotent matrices over the field F. Prove that Nl and N

z

are similar if and only if they have the same minimal polynomial.

Total number of points: 100

II lL,o;)(. >II :k'" - 0 - '" ~ ^Ii ~:~: ^'If ~ ^it ~tl::Oli!"tI:~ ^{# 1 :A}

I

}.t..;t*~: ff~~~:lf!.il.:-{f.PtT" ~$~J:..ff~ji#a},J,tlit

^{0 1101.05.08}

May 8,2012 PhD Entrance Exam

Real Analysis

1. (8 points) Compute the limit

. 1

00 ^ncosx

lim 1 2;c3/2 dx.

n-+oo 0

+

n 2. (12 points) For

I

E V (0,00), 1:::; p :::; 00, define

00

(TJ)

(y)

=

1

^(x

^{+ y)2 e-(~+Y) I}

^{(x) dx for}

^y

^{E (0,00).}

Show that

TI

E V (0,00) and

IITlllp:::; 211/11p·

3. (10 points) Suppose p, is a positive measure on X and

I :

X -+ (0,00) satisfies Ixldp, = 1. Prove, for every E C X with 0

<

p,(E)

<

00, that

L

^{(log J)}

^dp,:::;

^p,

^(E)

^log

^{p,! E)}

and, when 0 <p

<

1,

L IPdp, :::;

p, (E)l-P.

4. (10 points) Suppose E

~

JR is measurable with

lEI '

A

>

0, where Ais a finite number.

Show that for any 0

<

t

<

A, there exists a subset

A

of

E

such that

A

is measurable and

IAI = t.

That is, the Lebesgue measure

H

on JR satisfies the Intermediate Value Theorem.

5. (10 points) Let

IA:

^and

I

be (Lebesgue) measurable on a measurable set E C JRn,

lEI <

00. Then

1 If,., - II

^{dx -+ 0 as k -+} 00.

110

^-+

I

in measure iff

E 1

+ Ilk - II

101. 05. 08 Algebra Exam May 2012

Show ALL work for full credit.

(1) (10pts) Let p

<

q be primes with q

"I-

p (mod p).

(a) Show that every group of order pq is cyclic.

(b) Let G be a group and H

<

Z(G), where Z(G) denotes the center of G.

Suppose

G j H

is cyclic. Prove that

G

is abelian.

(2) (10pts) Let

R

be a commutative ring with identity. Let

N

=

{r

E

R : rn

=

o

for some n

> o}.

(a) Prove N is an ideal.

(b) Suppose N is a maximal ideal of R. Prove that N is the unique maximal ideal of

R.

(3) (lOpts) Let R be a commutative ring with unity. Suppose the following dia­

gram R-modules commutes

0

---+ A ---+ B _---+

C

---+

0

tf tg th

0

---+ A' ---+ B' _---+

C'

---+

⁰ and the rows are exact.

(a) Prove that if

f

and h are surjective, then 9 is surjective.

(b) Prove that if

f

and h are injective, then 9 is injective.

(4) (lOpts) Let K be the splitting field of :f3 ~ 2 over Q. Compute the Galois group of

Kover Q

and all the intermediate fields.

(5) (lOpts) Let IF be a finite field.

(a) Prove that

IlFl

^{= pr where p, r E} Z+ with p a prime.

(b) Let p E Z be a prime and IFp the finite field of p elements. Let IF be an extension field of

lFp-

Prove that the Galois group Gal(lF jFp) is cyclic. (Hint:

Consider the Frobenius isomorphism x ^I---t x^{P )}

1

L---~-~---