# Math 225A, Differential Topology: Homework 3

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Math 225A, Differential Topology: Homework 3 Ian Coley October 17, 2013 Problem Suppose that y is a regular value of f : X Y, where X is compact and dim X = dim Y. Show that f 1 (y) is a finite set {x 1,..., x N }. Prove there exists a neighbourhood U of y in Y such that f 1 (U) is a disjoint union V 1 V N, where V i is an open neighbourhood of x i and f maps each V i diffeomorphically onto U. Since dim X = dim Y, dim f 1 (y) = 0, i.e. it is locally diffeomorphic to a point. Hence f 1 (y) is diffeomorphic to a set of discrete points, and if this set is infinite, we violate compactness (namely there exists an open cover which has no finite sub cover). Therefore the preimage of y is a set finite set of points, as required. Now since f is a local diffeomorphism at each x 1,..., x N (taking the terminology of the problem), there is some neighbourhood W i of x i and U i of y such that f : U i V i is a diffeomorphism. Since f 1 (y) is discrete, we may assume that W i are pairwise disjoint since X is Hausdorff. Let U = U i and let W i = W i f 1 (U ). Then it is clear that f : W i U is a diffeomorphism, since it is a restriction of the above diffeomorphism. Consider X \ W i. This is compact, since W i is open, and hence Z = f(x \ W i ) is compact as well. It clearly does not contain y since the preimage of y was contained in W i. Finally, if we let U = U \ Z and V i = W i f 1 (U), we have that U is a neighbourhood of y, V i is a neighbourhood of x i, f 1 (U) = V i is a disjoint union, and f : V i U is a diffeomorphism for all i. This completes the proof. Problem Verify that the tangent space to O(n) at the identity matrix I is the vector space of skew symmetric n n matrices. Recall that O(n) is constructed as the preimage of I S(n) under the map f : M(n) S(n) where A AA T. We can construct the tangent space to O(n) at I by looking at the derivative of f. In general, the tangent space of x f 1 (y) is equal to ker(df x ). In particular, we would like to find ker(df I ), where Df is given in 4 to be Df A (B) = BA T + AB T. 1

2 Suppose that B ker Df I. This is true if and only if 0 = B + B T B = B T. Therefore the tangent space of O(n) at I is exactly the skew-symmetric matrices, as claimed. Problem (a) Prove that SL(n) is a submanifold of M(n) and thus is a Lie group. (b) Check that the tangent space to SL(n) at the identity matrix consists of all matrices with trace equal to zero. (a) Let det : M(n) R be the determinant map on matrices. We claim that this is a homogeneous polynomial in the entries of M(n). It is certainly a polynomial in its entries, since if A = (a i,j ), det A = σ S n sgn(σ) n a i,σ(i). Additionally, we have det(ta) = t n det A, so it is homogeneous. By 1.4.6, the preimage of any a 0 under a homogeneous polynomial is a submanifold of M(n). In particular, SL(n) = det 1 (1), so it is a submanifold of M(n). Thus SL(n) is a Lie group. (b) As in , the tangent space at I is equal to the ker D det I. We explore this derivative via elementary matrices. Let B ij = (b kl ) be the n n matrixes with b ij = 1 and 0 everywhere else. Then i=1 det(i + sb) det I D det I (B ij ) = lim. If b i j is on the diagonal (i.e. i=j), then det(i + sb) = 1 + s, so D det I (B ij ) = lim s s 1 s If b i j is off the diagonal, then det(i + sb) = 1, so D det I (B ij ) = 0. Therefore if a general (b i j) = B M(n) is any element, since D det is linear, we have = 1. D det I (B) = i,j b ij D det I (B ij ) = i b ii = tr B. Therefore ker D det I is exactly those matrices with trace zero, and we are done. Problem Exhibit a smooth map f : R R whose set of critical values is dense. 2

3 We take the hint. Let r i, i N be an enumeration of the rationals. We construct a smooth function f i supported on (i, i + 1) which has a critical value at r i. Fortunately, in problem , we constructed a smooth bump function h on R such that { 0 x a h(x) = 1 x b and smooth interpolation on (a, b). Fix a small 1/2 > ε > 0. Then we define f i as follows: 0 x i r i h(x) i < x < i + ε f i (x) = i + ε x i + 1 ε r i r i (1 h(x)) i + 1 ε < x < i x i + 1 This function exhibits hinted at behaviour: it smoothly interpolates from 0 to r i, remains constant at r i for a nontrivial interval, then smoothly interpolates to r i. f i itself is smooth by construction. Further, f i has critical values at r i (since the derivative is zero there) and 0. Now let f = r i Q f i be a function on all of R. This sum makes sense because f i are supported on disjoint intervals of R. Further, f has a critical value whenever any f i does, which occurs at every r i Q. Therefore the set of critical values of f is dense in R since Q R is dense. Problem Prove that the sphere S k is simply connected if k > 1. Let f : S 1 S k be a map. We would like to show that this is homotopic to a constant. Taking the hint, Sard s theorem allows us to choose p S k \ f(s 1 ) when k > 1. Recall that we showed in that S k \ {p} is diffeomorphic to R k for k 1. Further, R k is contractible, so it is simply connected. Therefore f(s 1 ) is diffeomorphic to a loop in R k, which is homotopic to a constant. Therefore this homotopy can be pulled back along the diffeomorphism so f(s 1 ) is simply connected in S k \{p}, hence in S k as well. This completes the proof. Problem When dim X < dim Y, the image of any smooth map f : X Y has measure zero in Y. Prove this assuming that if A has measure zero in R l and g : R l R l is smooth, then g(a) also has measure zero. Taking the hint, we reduce to the case where f : U R k R l, where k < l. This is sufficient because X is locally diffeomorphic to R k, and µ(f(x)) i N µ(f(u i)) for an 3

4 open cover U i of X. Hence µ(x) = 0 if µ(f(u i )) = 0 for every open subset of R k, which is to say every open subset of X. We construct the function F : U R l k R l where F (x, t) = f(x). This function is smooth since f is smooth. If U has measure zero in R k, so does U R l k R l. Therefore F (U R l k ) = f(u) has measure zero, so we are done. Problem Show that T (R k ) = R k R k. By definition, T (R k ) = {(x, v) R k R k : v T x (R k )}. Hence we need only show that, for every x R k, T x (R k ) = R k ). But since R k is locally diffeomorphic to R k everywhere, T x (R k ) is diffeomorphic to R k as well. Hence T (R k ) = R k R k and we are done. Problem Show that the tangent bundle to S 1 is diffeomorphic to the cylinder S 1 R. View S 1 as the unit circle in R 2. Then T (S 1 ) is a submanifold of R 4, where T (S 1 ) = {(x, v) S 1 R 2 : v T x (S 1 )} = {(x, v) S 1 R 2 : x, v = 0} The second formulation is exactly suggesting that v is orthogonal to x in R 2, hence in its tangent space. Let x = (x 1, x 2 ) S 1 and v = (v 1, v 2 ) T x (S 1 ). Then since x satisfies x 2 1 +x 2 2 = 1 and x 1 v 1 +x 2 v 2 = 0, v must be a multiple of (x 2, x 1 ). Further, if v = t(x 2, x 1 ), this t R unique determines v, and all such t are possible. Therefore we define a map S 1 R T (S 1 ) R 4 by (x, t) (x, t(x 2, x 1 )). This map is certainly smooth and a homeomorphism, hence it is a diffeomorphism and we are done. Problem Let S(X) be the set of points (x, v) T (X) with v = 1. Prove that S(X) is a (2k 1)- dimensional submanifold of T (X) Consider the map f : T (X) R by f(x, v) = v 2, which is smooth. We claim that 1 is a regular value of f. Assuming this, f 1 (1) is precisely S(X). Further, since dim T (X) = 2k, dim S(X) = 2k dim R = 2k 1, and we are done. We now show 1 is a regular value. We need only show that Df (x,v) is not zero for all (x, v) with v 2 = 1. We have f((x, v) + s(y, w)) f(x, v) Df (x,v) (y, w) = lim v + sw 2 v 2 = lim sw 2 + 2s Re v, w = lim = 2 Re v, w, which is not identically zero since v 0. Hence 1 is a regular value, and we are done. 4

5 Problem Show that if X is a compact k-dimensional manifold, then there exists a map X R 2k 1 that is an immersion except at finitely many points of X. We know that an immersion f : X R 2k exists by the Whitney immersion theorem. Also, F : T (X) R 2k is a smooth map where F (x, v) = Df x (v). Let a be a regular value of F. Then F 1 (a) has dimension dim T (X) dim R 2k = 0, and by the same logic as it must be a finite set of discrete points. Let π f : X R 2k 1, where π : R 2k R 2k 1 is the orthogonal projection onto the subspace perpendicular to a. We claim that π f is an immersion except on the set f 1 (a) = {x X : F (x, v) = a for some v}. Let y / f 1 (a). Then D(π f) y = Dπ f(y) Df y. Df y is injective by assumption, so we need only show Dπ f(y) is injective. Suppose that Dπ f(y) (v) = 0. Then 0 = lim s 0 π(f(y) + sv) π(f(y)) s = π(v) = 0, which means that v span(a). If v 0, then there exists a constant t such that tv = a. Let w be such that Df y (w) = v (so that D(π f) y (w) = 0). Then Df y (tw) = a, which is impossible. Hence we must have v = 0, so Dπ f(y) is injective, and we are done. Problem Use partition-of-unity techniques to prove the following: suppose that the derivative of f : X Y is an isomorphism whenever x lies in the submanifold Z X, and assume that f maps Z diffeomorphically onto f(z). Prove that f maps a neighbourhood of Z diffeomorphically onto a neighbourhood of f(z). Let U i be a locally finite collection of open subsets of Y covering f(z). Because the derivative of f is an isomorphism, we may find inverses g i : U i X i. Let W = {y U i : g i (y) = g j (y) whenever y U i U j }, where the first U i is arbitrary. This set is the set of y that behave nicely in the intersection of the open cover. Hence we may glue together the g i to some inverse g : W X because elements lying in more than one U i have a well defined image. We claim that W contains an open neighbourhood of f(z). If so, this is an inverse for f from a neighbourhood of f(z) to a neighbourhood of Z, which would prove the result. Let x Z. Then f maps some neighbourhood V of x diffeomorphically to a neighbourhood U of f(x) = y f(z). Since y U i for finitely many U i, say i = {1,..., n}, there is another neighbourhood U = U U 1... U n. Then U W, and V = f 1 (U ) is a neighbourhood of x. Then the neighbourhood of Z given by x Z V x is mapped diffeomorphically onto a neighbourhood of f(z) inside W, namely y f(z) U y. 5

### Math 312 Homework 1 Solutions

Math 31 Homework 1 Solutions Last modified: July 15, 01 This homework is due on Thursday, July 1th, 01 at 1:10pm Please turn it in during class, or in my mailbox in the main math office (next to 4W1) Please

### MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.

MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar

### NOTES ON LINEAR TRANSFORMATIONS

NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all

### ( ) which must be a vector

MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are

### Linear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)

MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of

### some algebra prelim solutions

some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

### 8.1 Examples, definitions, and basic properties

8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A k-form ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)-form σ Ω k 1 (M) such that dσ = ω.

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### Finite dimensional topological vector spaces

Chapter 3 Finite dimensional topological vector spaces 3.1 Finite dimensional Hausdorff t.v.s. Let X be a vector space over the field K of real or complex numbers. We know from linear algebra that the

### MATH 110 Spring 2015 Homework 6 Solutions

MATH 110 Spring 2015 Homework 6 Solutions Section 2.6 2.6.4 Let α denote the standard basis for V = R 3. Let α = {e 1, e 2, e 3 } denote the dual basis of α for V. We would first like to show that β =

### SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

### Mathematical Physics, Lecture 9

Mathematical Physics, Lecture 9 Hoshang Heydari Fysikum April 25, 2012 Hoshang Heydari (Fysikum) Mathematical Physics, Lecture 9 April 25, 2012 1 / 42 Table of contents 1 Differentiable manifolds 2 Differential

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS

TOPOLOGY: THE JOURNEY INTO SEPARATION AXIOMS VIPUL NAIK Abstract. In this journey, we are going to explore the so called separation axioms in greater detail. We shall try to understand how these axioms

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### 5. Linear algebra I: dimension

5. Linear algebra I: dimension 5.1 Some simple results 5.2 Bases and dimension 5.3 Homomorphisms and dimension 1. Some simple results Several observations should be made. Once stated explicitly, the proofs

### Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,

### Sec 4.1 Vector Spaces and Subspaces

Sec 4. Vector Spaces and Subspaces Motivation Let S be the set of all solutions to the differential equation y + y =. Let T be the set of all 2 3 matrices with real entries. These two sets share many common

### Name: Section Registered In:

Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

### The Tangent Bundle. Jimmie Lawson Department of Mathematics Louisiana State University. Spring, 2006

The Tangent Bundle Jimmie Lawson Department of Mathematics Louisiana State University Spring, 2006 1 The Tangent Bundle on R n The tangent bundle gives a manifold structure to the set of tangent vectors

### Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =

Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and

### Matrix Representations of Linear Transformations and Changes of Coordinates

Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under

### 4.6 Null Space, Column Space, Row Space

NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

### 1 Local Brouwer degree

1 Local Brouwer degree Let D R n be an open set and f : S R n be continuous, D S and c R n. Suppose that the set f 1 (c) D is compact. (1) Then the local Brouwer degree of f at c in the set D is defined.

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### Notes on metric spaces

Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

### The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit

The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),

### LINEAR ALGEBRA. September 23, 2010

LINEAR ALGEBRA September 3, 00 Contents 0. LU-decomposition.................................... 0. Inverses and Transposes................................. 0.3 Column Spaces and NullSpaces.............................

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### Polynomial Invariants

Polynomial Invariants Dylan Wilson October 9, 2014 (1) Today we will be interested in the following Question 1.1. What are all the possible polynomials in two variables f(x, y) such that f(x, y) = f(y,

### 12.5 Equations of Lines and Planes

Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### RANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis

RANDOM INTERVAL HOMEOMORPHISMS MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis This is a joint work with Lluís Alsedà Motivation: A talk by Yulij Ilyashenko. Two interval maps, applied

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

### 1 Introduction to Matrices

1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

### 8 Square matrices continued: Determinants

8 Square matrices continued: Determinants 8. Introduction Determinants give us important information about square matrices, and, as we ll soon see, are essential for the computation of eigenvalues. You

### COBORDISM IN ALGEBRA AND TOPOLOGY

COBORDISM IN ALGEBRA AND TOPOLOGY ANDREW RANICKI (Edinburgh) http://www.maths.ed.ac.uk/ aar Dedicated to Robert Switzer and Desmond Sheiham Göttingen, 13th May, 2005 1 Cobordism There is a cobordism equivalence

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### T ( a i x i ) = a i T (x i ).

Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)

### Orthogonal Projections

Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.

Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### NOV - 30211/II. 1. Let f(z) = sin z, z C. Then f(z) : 3. Let the sequence {a n } be given. (A) is bounded in the complex plane

Mathematical Sciences Paper II Time Allowed : 75 Minutes] [Maximum Marks : 100 Note : This Paper contains Fifty (50) multiple choice questions. Each question carries Two () marks. Attempt All questions.

### The Topology of Fiber Bundles Lecture Notes. Ralph L. Cohen Dept. of Mathematics Stanford University

The Topology of Fiber Bundles Lecture Notes Ralph L. Cohen Dept. of Mathematics Stanford University Contents Introduction v Chapter 1. Locally Trival Fibrations 1 1. Definitions and examples 1 1.1. Vector

### 9 More on differentiation

Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......

### Lecture Notes: Matrix Inverse. 1 Inverse Definition

Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### The cover SU(2) SO(3) and related topics

The cover SU(2) SO(3) and related topics Iordan Ganev December 2011 Abstract The subgroup U of unit quaternions is isomorphic to SU(2) and is a double cover of SO(3). This allows a simple computation of

### Finite dimensional C -algebras

Finite dimensional C -algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for self-adjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n

### Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

### Recursion Theory in Set Theory

Contemporary Mathematics Recursion Theory in Set Theory Theodore A. Slaman 1. Introduction Our goal is to convince the reader that recursion theoretic knowledge and experience can be successfully applied

### MATH 551 - APPLIED MATRIX THEORY

MATH 55 - APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points

### 2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair

2.1 J.A.Beachy 1 2.1 Functions from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 21. The Vertical Line Test from calculus says that a curve in the xy-plane

### PROJECTIVE GEOMETRY. b3 course 2003. Nigel Hitchin

PROJECTIVE GEOMETRY b3 course 2003 Nigel Hitchin hitchin@maths.ox.ac.uk 1 1 Introduction This is a course on projective geometry. Probably your idea of geometry in the past has been based on triangles

### 1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]

1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not

### Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### 4.5 Linear Dependence and Linear Independence

4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then

### FIBER PRODUCTS AND ZARISKI SHEAVES

FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also

### Practice with Proofs

Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

### Linear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:

Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),

### Notes on Symmetric Matrices

CPSC 536N: Randomized Algorithms 2011-12 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.

### SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS

SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS Ivan Kolář Abstract. Let F be a fiber product preserving bundle functor on the category FM m of the proper base order r. We deduce that the r-th

### Solution to Homework 2

Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if

### ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction

ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity

### A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE

A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE DANIEL A. RAMRAS In these notes we present a construction of the universal cover of a path connected, locally path connected, and semi-locally simply

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### MAT 200, Midterm Exam Solution. a. (5 points) Compute the determinant of the matrix A =

MAT 200, Midterm Exam Solution. (0 points total) a. (5 points) Compute the determinant of the matrix 2 2 0 A = 0 3 0 3 0 Answer: det A = 3. The most efficient way is to develop the determinant along the

### 1 Sets and Set Notation.

LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### 4. MATRICES Matrices

4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

### Eigenvalues and eigenvectors of a matrix

Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a non-zero column vector V such that AV = λv then λ is called an eigenvalue of A and V is

### Introduction to Topology

Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................

### Comments on Quotient Spaces and Quotient Maps

22M:132 Fall 07 J. Simon Comments on Quotient Spaces and Quotient Maps There are many situations in topology where we build a topological space by starting with some (often simpler) space[s] and doing

### Notes from February 11

Notes from February 11 Math 130 Course web site: www.courses.fas.harvard.edu/5811 Two lemmas Before proving the theorem which was stated at the end of class on February 8, we begin with two lemmas. The

### Section 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj

Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### Orthogonal Diagonalization of Symmetric Matrices

MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

### 17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function

17. Inner product spaces Definition 17.1. Let V be a real vector space. An inner product on V is a function, : V V R, which is symmetric, that is u, v = v, u. bilinear, that is linear (in both factors):

### NOTES on LINEAR ALGEBRA 1

School of Economics, Management and Statistics University of Bologna Academic Year 205/6 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura

### by the matrix A results in a vector which is a reflection of the given

Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that

### We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

### (January 14, 2009) End k (V ) End k (V/W )

(January 14, 29) [16.1] Let p be the smallest prime dividing the order of a finite group G. Show that a subgroup H of G of index p is necessarily normal. Let G act on cosets gh of H by left multiplication.

### Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

### 2 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

### Chapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors

Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col