Math 403A assignment 7. Due Friday, March 8, 2013.
Chapter 12.
Gausss lemma.
A nonconstant polynomial f (x) Q[x] is primitive if it has integer coefficients, the g.c.d. of
its coefficients is 1, and its leading coefficient is positive.
Gauss. The product
10 23
PERFECT SETS
Theorem Let P be a nonempty perfect set in R k . Then P is uncountable.
Proof :
Since P has limit points, P must be infinite.
Suppose that P is countable and let P = cfw_x1 , x 2 ,., x n ,. .
Let V1 = N r (x1 ) be a neighborhood of x1
10 28
Definition
Given a sequence cfw_ pn in a metric space X , consider a sequence cfw_nk
of positive integers, such that n1 < n2 < n3 < L .
Then the sequence cfw_ pni is called
a subsequence of cfw_ pn . If cfw_ pni converges, its limit is called
10 30
Recall
1. A sequence cfw_ pn in a metric space X is said to a Cauchy sequence if for every
> 0 there is an integer N such that d ( pn , pm ) < if n N and m N .
2. If cfw_ pn is a sequence in a compact metric space X, then there exists a
subseque
11 4
Theorem Let cfw_sn be a sequence of real numbers with lim sup sn R . Then
= lim sup sn if and only if
E .
> 0 , there is an integer N such that sn < + , n N .
(a)
(b)
Proof :
() We shall prove by contradiction that there is only one number that
11 6
Theorem Let cfw_sn be a sequence of real numbers with lim sup sn R . Then
= lim sup sn if and only if
E .
> 0 , there is an integer N such that sn < + , n N .
(a)
(b)
Theorem Let cfw_sn be a sequence of real numbers. = lim sup sn if and only if
11 11
Theorem (Root Test) Given an , put = lim sup n | an | .
n
Then
(a) if < 1 ,
a
(b)
if > 1 ,
converges;
n
n =1
a
diverges;
n
n =1
(c) if = 1 , the test gives no information.
Outline of the Proof :
(a) Suppose that < 1 , then r such that < r < 1 .
He
11 13
Recall
Theorem (Dirichlets test) Suppose
(a)
a
the partial sums An of
(b)
n
form a bounded sequence;
b0 b1 b2 L;
(c)
n =0
lim bn = 0 .
n
Then
a b
n =0
converges.
nn
In fact, similar proof gives the following theorem ( Rudin p.79, exercise # 8):
Th
11 20
LIMITS OF FUNCTIONS
Definition
Let X and Y be metric spaces; suppose E X , f : E Y , and p is a
limit point of E. We say lim f ( x) exists, if q Y with the following property:
x p
Given any > 0 there exists a > 0 such that
dY ( f ( x), q ) < , when
11 25
Theorem Let X and Y be metric spaces, f : E Y is continuous on X if and only
if f 1 (V ) is open in X for every open set V in Y.
Proof :
() Suppose that f 1 (V ) is open in X for every open set V in Y.
Given p X , > 0 , we need to find a > 0 such t
11 27
Definition
Let f be a mapping of a metric space X into a metric space Y.
We
say that f is uniformly continuous on X if for every > 0 there exists > 0 such
that
dY ( f ( p), f (q ) <
for all p and q in X which d X ( p, q ) < .
Question :
1. Whats t
12 2
Chapter 5 DIFFERENTIATION
THE DERIVATIVE OF A REAL FUNCTION
Definition Let f be defined (and real-valued) on [a, b] .
say f is differentiable at x provided that
For any x [a, b] , we
f (t ) f ( x)
exists,
(*)
t x
tx
which is called the derivative of
12 4
Let f : (a, b) R and c (a, b) .
If f is differentiable at c, then : (a, b) R which is continuous at
Recall
c such that
f ( x) = f (c) + f '(c)( x c) + ( x c) ( x) ,
for all x (a, b) , with (c) = 0 .
Consequently, we have
Theorem Let f : (a, b) R and
12 11
Recall
Darbouxs Theorem Suppose f
interval I , and suppose [a, b] I
f (a ) and f (b) , then c (a, b)
That is, the image of the derivative
is a real differentiable function on an open
with f (a ) f (b) . If is a number between
such that f (c) = .
f
12 16
Theorem (The Lagrange Remainder Theorem) Suppose f is a real function on
[a, b] , n is a positive integer, f ( n 1) is continuous on [a, b] , f ( n ) ( x) exists for
every x (a, b) . Let , be distinct points of [a, b] , and define
f ( k ) ( )
( x )
12 18
Recall
Let be a monotonically increasing function on [a, b] . Corresponding to each
partition P = cfw_a = x0 , x1 ,., xn = b of [a, b] , we write
i = ( xi ) ( xi 1 ).
It is clear that i 0 . For any real function f which is bounded on [a, b] we
put
12 23
Let be a monotonically increasing function on [a, b] and f be a bounded
function on [a, b] . Let P = cfw_a = x0 , x1 ,., xn = b be a partition of [a, b] , we have
n
U ( P, f , ) L( P, f , ) = ( M i mi ) i
i =1
where M i = sup cfw_ f ( x) | xi 1 x x
12 25
Recall
Let be a monotonically increasing function on [a, b] and f be a bounded
function on [a, b] . f R ( ) ,if and only if for all > 0 , a partition P of
[a, b] such that
U ( P, f , ) L ( P, f , ) < .
In this case, with the same P we have
b
a
b
b
10 21
Theorem Let E R k , then the following are equivalent :
(a) E is closed and bounded.
(b) E is compact.
(c) Every infinite subset of E has a limit point in E.
Remark
In general metric spaces :
(b)
(c)
(a)
(b)
ex. 16
(b)
ex. 23, 24, 26
(c)
/
(a)
Chap
10 14
Let X be a metric space, E X .
Recall
A point p X is a limit point of E if for every neighborhood N of p there
exists a point q p such that q N .
E is closed if every limit point of E is a point of E.
Remark
Let p be a limit point of E , then for e
Math 403A Assignment 8. Due Friday, March 15, 2013.
Chapter 15.
1.(1.1) Suppose that R is an integral domain that contains a field F such that R is a finitedimensional F -vector space. Show that R is a field.
Solution. The ideal way to solve this problem
Math 403 Assignment 5. Due Friday, Feb 22, 2013.
1(6.2) (a). Is Z/(6) isomorphic to the product ring Z/(2) Z/(3)?
Solution. Take the natural homomorphism Z Z/(2) Z/(3) that take each integer n to
(n+(2), n+(3). The kernel equals (2)(3) = (6). Hence, there
Math 403A Assignment 6. Due March 1, 2013.
1.(8.1) Which principal ideals in Z[x] are maximal ideals?
Solution. No principal ideal (f (x) is maximal. If f (x) is an integer n 6= 1, then (n, x)
is a bigger ideal that is not the whole ring. If f (x) has pos
Math 403 Assignment 1. Due Jan. 2013.
Chapter 11.
1. (1.2) Show that, for n 6= 0, cos(2/n) is algebraic over Q.
Solution. Since e2i/n and e2i/n are each algebraic (each satisfies the equation xn 1 = 0),
half their sum is algebraic too, i.e., cos(2/n) = (1
Math 403 Assignment 4. Due Monday, Feb 11, 2013
1(4.2) What does the correspondence theorem tell us about the ideals of Z[x] that contain
the element x2 + 1?
Solution. The ideals of Z[x] that contain the element x2 + 1 are the ideals of Z[x] that
contain
Math 403 Assignment 2. Due Wednesday, January 23, 2013
Chapter 3.
1(3.2) Which of the following subsets is a subspace of the vector space F nn of nn matrices
with entries in a field F (matrices combined by addition)?
(a). Symmetric matrices A = At , i.e.,
Math 403 assignment 3. Due Friday. Feb 1, 2013.
Ch 11, section 3.
1. (3.2) Show that every nonzero ideal of the ring Z[i] of Gaussian integers contains a nonzero
integer.
Solution. Let m + ni be a nonzero element of the ideal. Then the element (m ni)(m +
Math 113 Homework 1
Solutions
due June 27, 2011
1. Determine the elements of the cyclic group generated by the matrix
1 1
A=
1 0
explicitly.
Solution: The characteristic polynomial of A is x2 x+1, so A2 A+I =
0. Thus A2 = A I, A3 = A(A2 ) = A(A I) = A2 A
Math 113 Homework 7
Solutions
due August 7, 2011
1. Let R be an integral domain containing a field F as subring and which is
finite-dimensional when viewed as a vector space over F . Prove that R is
a field.
Solution: Let x R. Observe that the positive in
Math 113 Homework 3
Solutions
due July 11, 2011
1. Let G and G0 be two groups whose orders have no common factor. Prove
that the only homomorphism : G G0 is the trivial one (x) = 1 for
all x.
Solution: Let : G G0 . We know that |G| = |ker()| |im()|, so
in