Math 125A, Fall 2009 Solutions for Homework 4
(Prepared by Matt Low)
1. Regardless of the argument of the cosine function, we have |cos u| 1, and so:
x0
lim x cos 3x2 + esin x - ln|x|
lim |x| = 0
x0
and consequently:
x0
lim x cos 3x2 + esin x - ln|x| = 0
Math 125A, Fall 2009 Homework 3
Due Date: October 16, 2009
Problem 1: Let f be a uniformly continuous function. Prove that if (xn ), (yn ) are sequences such that |xn - yn | 0, then |f (xn ) - f (yn )| 0 also. Problem 2: Determine if the function is unifo
Math 125A, Fall 2009 Solutions for Homework 2
(Prepared by Matt Low)
1. Let f (x) = x - cos x. By direct computation, we see that f (0) < 0 and f ( ) > 0. Hence, by the 2 intermediate value theorem (Theorem 18.2), there exists a number c (0, ) such that f
Math 125A, Fall 2009 Homework 8 ONLY TURN IN PROBLEMS 14 Problem 1: Let S be a subset of a metric space (X, d).
Due Date: November 30, 2009
(a) If S is the union of a collection of open balls, then prove that S is open. (b) If S is open, then prove that S
MAT067
University of California, Davis
Winter 2007
Solutions to Homework Set 4
1. Define the map T : R2 R2 by T (x, y) = (x + y, x). (a) Show that T is linear. (b) Show that T is surjective. (c) Find dim nullT . (d) Find the matrix for T with res
MAT067
University of California, Davis
Winter 2007
Homework Set 7: More Exercises on Eigenvalues
Directions: Please work on all of the following exercises. Submit your solutions to Problems 3(d) and 5(b) as your Calculational Problems and Problems
MAT067
University of California, Davis
Winter 2007
Homework Set 3: Exercises on Linear Spans and Bases
Directions: Please work on all exercises! Hand in Problems 1 and 2 as your "Calculational Homework" and Problems 5 and 7 as your "Proof-Writing
MAT067
University of California, Davis
Winter 2007
Homework Set 1: Exercises on Complex Numbers
Directions: You are assigned the Calculational Problems 1(a, b, c), 2(b), 3(a, b), 4(b, c), 5(a, b), and the Proof-Writing Problems 8 and 11. Please su
MAT067
University of California, Davis
Winter 2007
Solutions to Homework Set 1
1. Express the following complex numbers in the form x + yi for x, y R: (a) (2 + 3i) + (4 + i) Solution: By direct computation, (2 + 3i) + (4 + i) = (2 + 4) + (3 + 1)i
MAT067
University of California, Davis
Winter 2007
Solutions to Homework Set 8
As usual, we are using F to denote either R or C. We also use , to denote an arbitrary inner product and to denote its associated norm. 1. Let (e1 , e2 , e3 ) be the
Name:
Student ID:
MAT 67: Homework 6
Due before class, Friday, November 15th, 2013
1. (5 points.) Given the following vectors in R2 :
1 = (1, 1),
1 = (1, 0),
2 = (2, 1),
2 = (0, 1),
3 = (3, 2),
3 = (1, 1),
determine whether there is a linear transformatio
Mathematics 32A
Ciprian Preda
Final Exam
Try to do all the problems, and be as explicit as required in your answers. Show your
work! Incomplete reasoning will lose points. There is plenty of working space, and
2 blank pages at the end. The number of poin
Name:
Student ID:
MAT 67: Homework 7
Due before class, Friday, November 22nd, 2013
1. (10 points.) Let V and W be nite-dimensional vector spaces over F. Given
T L(V, W ), show that there is a subspace U of V such that the following are
true:
U null(T ) =
32A Stovall
Midterm 2
Name:
November 9
Section: Tu/Th
Duncan/Melissa
I certify that the work appearing on this exam is completely my
own:
Signature:
There are 5 problems and a total of 8 pages. Please make sure that
you have all pages.
Please show your
M IDTERM 1
May 20, 2002
Instructions.
Please show your work. You will receive little or no credit for an answer
not accompanied by appropriate explanations, even if the answer is correct.
If you have a question about a particular problem, please raise you
Name:
Student ID:
MAT 67: Homework 3
Due before class, Friday, October 18th, 2013
1. (5 points.) Let V be the set of all pairs (x, y ) of real numbers and suppose vector
addition and scalar multiplication are dened in the following way:
(x1 , y1 ) + (x2 ,
Math32a
R. Kozhan
Midterm2 Summary
Midterm2 will be focused on the sections listed below, and will not explicitly test the knowledge
of the material included for Midterm1. However the student is assumed to know it and be able to
use it when needed.
The ma
Math32a
R. Kozhan
Final Summary
Material for the Final includes everything that we covered in the course.
Namely:
Chapter 12;
Chapter 13 (except Section 13.4);
Sections 10.110.2.
Basics of the conic sections (Section 10.5 without foci and directrices)
Math 125A, Fall 2009 Solutions for Homework 8
(Prepared by Matt Low)
1. (a) Suppose S = Bi where each Bi is an open ball. For each x in S, there exists a ball Bi such that x Bi . This ball Bi is in the form of Bri (xi ) of radius ri > 0 and center xi X. L
Solution to Mat125A Midterm Exam
Instructor: Qinglan Xia Date: Monday, Nov. 2, 2009
1. Suppose f : A B and g : B C are uniformly continuous functions.
Prove that the composition function h = g f : A C is also uniformly continuous.
Answer: For any > 0, sin
Math 125A, Fall 2009 Homework 7
Due Date: November 20, 2009
Problem 1: Let f be a function satisfying f (x) = 0 for all x R. Prove that f has the form f (x) = ax + b for some constants a, b. (Integration is not an acceptable answer since we have not yet c
Winter 2011
Math 67
Linear Algebra
Homework 4
Problems 3.3, 3.4, 3.5, 3.7 and 3.9 from Axler.
Solutions.
3.3 Let U V be a subspace and S : U W be a linear map. Pick a
basis (u1 , . . . , u ) of U and extend it to a basis of V ,
(u1 , . . . , u , v +1 , .