Problem Set 1 Solutions Math 109


 Easter Thomas
 2 years ago
 Views:
Transcription
1 Problem Set 1 Solutions Math 109 Exercise 1.6 Show that a regular tetrahedron has a total of twentyfour symmetries if reflections and products of reflections are allowed. Identify a symmetry which is not a rotation and not a reflection. Check that this symmetry is a product of three reflections. Solution. How many ways are there to arrange the lels a, b, c, and d at the four vertices of the tetrahedron? There are 4 options for where to place a, and once it s placed there are 3 options for where to place b, then 2 options for c, and then d goes on whichever vertex is left. Thus there are ways to lel the vertices of the tetrahedron. Any symmetry sends vertices to vertices, and in fact is determined by where it sends the vertices. So if we start the tetrahedron with our favorite leling of the vertices, and apply products of reflections, then the result will give us one of the 24 leling options. Thus there are at most 24 distinct reflections and products of reflections. Given any two vertices, there is a reflection which swaps those two vertices and leaves the other two fixed  for example, in Figure 1.4 of Armstrong, to switch vertex 2 and 3, you reflect through the plane which goes through the center point of the line segment 23 and is normal to it. These swapping reflections are called a transpositions, and they allow us to get to any leling, a.k.a permutation, of the vertices by working one vertex at a time. Start with the vertices in a preferred leling, and take any of the 24 possible permutations as the goal. Do the following steps: 1. If vertex a is in the correct position, go to the next step. If not, apply the transposition that swaps a into its correct position. 2. If vertex b is in the correct position, go to the next step. If not, swap b into its correct position (because of step 1, a is not in b s correct position, so this leaves a fixed). 3. If vertex c is in the correct position, go to the next step. If not, swap c into its correct position (neither a nor b are in c s correct position, so this leaves them fixed). 4. a, b, and c are in the correct position, so d must be too. We re done! This gives a method for reaching any permutation of the vertices by a sequence of reflections, so there are 24 symmetries if products of reflections are allowed. Suppose the tetrahedron starts with its lels positioned as (1,2,3,4), and a reflection swaping 1 and 4 is performed to get (4,2,3,1), another to get (4,3,2,1), and a final to get (3,4,2,1). The symmetry (1,2,3,4) (3,4,2,1) can t be just a reflection, since all of the vertices have moved. For the same reason, it can t be a rotation which fixes one vertex. 1
2 Problem Set 8 2 of 8 Finally, it can t be a rotation which swaps a with b and c with d, for example because 1 goes to 4 s place, but 4 does not go to 1 s place. Exercise 2.3 Which of the following collections of 2 2 matrices with real entries form groups under matrix multiplication? (i) The set S 1 of matrices of the form bc for which ac b 2. (ii) The set S 2 of matrices of the form ca for which a2 bc. (iii) The set S 3 of matrices of the form for which ac 0. (iv) The set S 4 of matrices which have nonzero determinant and whose entries are integers. Solution. (i) and are matrices in S 1, but their product is not. Therefore S 1 is not closed under matrix multiplication and thus is not a group with this operation. (ii) and are matrices in S 2, but their product is not. S 2 is not closed under matrix multiplication and thus is not a group with this operation. (iii) closure: For any and a b in S3, the product a b aa +bc S c 3. Therefore S 3 is closed under matrix multiplication. associativity: Matrix multiplication is always associative, but it is easier to check the formula for matrices in our set S 3 than for general matrices: ( a ) b a b aa +bc a b c aa a aa b + c +bc c c c a a a b +b c c ( a b a b ) identity: is an element of S 3, and we know cd cd cd for any matrix cd, so there is an identity element in S3. inverses: For all in S3, ac 0 (and so a 0 c), so the matrix 1/a b/(ac) is in S 0 1/c 3 and is an inverse for because 1/a b/(ac) /c 01. 1/a b/(ac) 0 1/c
3 Problem Set 8 3 of 8 This shows that S 3 is a group under matrix multiplication. (iv) The matrix has nonzero determinant and integer entries, but we can see that it does not have an inverse with integer entries. Indeed, if cd 10 01, then we must have a 1/2 c, but there is no such matrix in S 4. Therefore S 4 does not have multiplicative inverses and cannot be a group. Exercise 2.5 A function from the plane to itself which preserves the distance between any two points is called an isometry. Prove that an isometry must be a bijection and check that the collection of all isometries of the plane forms a group under composition of functions. Solution. Suppose that f : R 2 R 2 is an isometry. We denote the distance between points a and b as d(a, b). f is injective: Let a, b R 2. If f(a) f(b), then 0 d(f(a),f(b)) d(a, b), so a b. f is surjective: Suppose p R 2 is a point in the codomain of f. We wish to show that there is some point in R 2 which is mapped to p by f. Choose two distinct points a, b R 2 in the domain of f, and let f(a) a, f(b) b. Looking in the codomain, let d 1 d(a,p ) and d 2 d(b,p ). Now draw circles of radius d 1 around a and d 2 around b in the domain, and draw corresponding circles of radius d 1 around a and d 2 around b in the codomain. These circles are exactly the points of distance d 1 from a, d 2 from b, etc., so in particular p lies in the intersection of the circles in the codomain. Notice that f must map the intersection of the circles in the domain to the intersection of the circles in the codomain. Indeed, if x is an intersection point of the two circles in the domain, then d(a, x) d 1 and d(b, x) d 2, and since f is an isometry, this means d(a,f(x)) d 1 and d(b,f(x)) d 2. But this last statement just says that f(x) must lie in the intersection of the two circles in the codomain. Using the fact that f is an isometry and the triangle inequality, we see that d(a, b) d(a,b ) d(a,p )+d(p, b )d 1 + d 2. If this inequality is an equality, then the circles in the codomain intersect in exactly one point, as do the circles in the domain (this happens when p lies on the line segment between a and b ). Calling p the point in the intersection of circles in the domain, we see that f(p) must be the intersection point in the codomain, i.e. f(p) p and we are done.
4 Problem Set 8 4 of 8 Now suppose that d(a, b) d(a,b ) <d 1 + d 2. In this case the circles in the domain intersect at exactly two points, as do the circles in the codomain. Lel the intersection points of the domain x and y; one of the intersection points in the codomain is p, so lel the other one q. If f(x) p, we are done. If not then f(x) q, and since f is injective, f(y) q so we must have f(y) p. This completes the proof of surjectivity. isometries form a group: closure: If f, g are isometries of R 2, f g : R 2 R 2, and for all a, b R 2, d(a, b) d(g(a),g(b)) d(f(g(a)),f(g(b))), so f g is an isometry. associativity: Function composition is always associative when it s defined: (f g) h(x) f(g(h(x))) f (g h). identity: The identity function on R 2 is certainly distancepreserving. inverses: Since we have proved that any isometry f is a bijection, we know that the function f 1 exists, and we have d(f 1 (a),f 1 (b)) d(ff 1 (a), ff 1 (b)) d(a, b). Therefore f 1 is an isometry. This shows that isometries of the plane form a group. Exercise 2.6 Show that the collection of all rotations of the plane out a fixed point P forms a group under composition of functions. Is the same true of the set of all reflections of lines which pass through P? What happens if we take all the rotations and all the reflections? Solution. Without loss of generality, we may suppose that P is the origin. Any rotation of the plane can be specified by the angle θ through which it rotates the positive xaxis, and it s clear that the composition of rotations θ, θ is the rotation θ + θ. As ove, function composition is always associative, and the identity function on R 2 ( rotation by 0 ) is an identity element. The inverse of rotation by θ is just rotation by θ. Therefore rotations of the plane out a fixed point form a group. The set of reflections through lines through P does not form a group under composition of functions. To see this intuitively, draw a triangle in the plane and lel the vertices a, b, c in a clockwise order. You can see that any reflection will reverse the orientation of the leling, so that when you read off a, b, c, it will now be in counterclockwise order. When you look at the image of the triangle after applying two reflections along different lines, the orientation of the leling will be clockwise again. Thus the composition of these two reflections can t be equal to a single reflection. Reflections are not closed under
5 Problem Set 8 5 of 8 function composition, so they don t form a group. We will see this more rigorously below. Notice that we can already say that the collection of reflections, rotations, and any of their compositions forms a group under composition of functions. Associativity and existence of an identity are as ove. Because the composition of a reflection with itself is the identity, and rotations have inverses, any composition of reflections and rotations has an inverse  just write the composition backwards and put inverses everywhere. For example, it s easy to see (r 1 r 2 s 1 r 3 s 2 s 3 ) 1 s 1 3 s 1 2 r 1 3 s 1 1 r 1 2 r 1 1. However we can actually show that any composition of rotations and reflections fixing the origin can be written as a single reflection or a single rotation (contrast with Exercise 1.6!). This shows that the collection of reflections and rotations is a group  the statement and any of their compositions ove was unnecessary. Recall from linear algebra that rotation counterclockwise by θ is a linear map, and its matrix with respect to the standard basis of R 2 is cos θ sin θ R θ, sin θ cos θ and that the matrix corresponding to reflection across a line making an angle of φ with the positive xaxis is cos 2φ sin 2φ S φ. sin 2φ cos 2φ (If you didn t remember these, you could produce them yourself: recall that the first column should be the image of the vector 0, 1 and the second is the image of 1, 0.) Using the trigonometric identities cos(s + t) cos s cos t sin s sin t and sin(s + t) sin s cos t + cos s sin t liberally, we compute the matrix for the following compositions by matrix multiplication: R θ S φ S φ R θ S φ S ψ cos(θ +2φ) sin(θ +2φ) sin(θ +2φ) cos(θ +2φ) cos(2φ θ) sin(2φ θ) sin(2φ θ) cos(2φ θ) cos(2φ 2ψ) sin(2φ 2ψ) sin(2φ 2ψ) cos(2φ 2ψ) S θ+2φ S 2φ θ R 2φ 2ψ In particular we can see explicitly that a composition of reflections is not a reflection.
6 Problem Set 8 6 of 8 Exercise 2.7 Let x and y be elements of a group G. Prove that G contains elements w and z which satisfy wx y and xz y, and show that these elements are unique. Solution. wx y if and only if wxx 1 yx 1, i.e. if and only if w we yx 1. Likewise xz y if and only if x 1 xz x 1 y, if and only if z x 1 y. Exercise 3.1 Show that each of the following collections of numbers forms a group under addition. (i) The even integers. (ii) All real numbers of the form a + b 2 where a, b Z. (iii) All real numbers of the form a + b 2 where a, b Q. (iv) All real numbers of the form a + bi where a, b Z. Solution. (i) The sum of two even integers 2n and 2m is 2(n + m), which is also even, so the set of even integers is closed under addition. Addition of integers is associative, and 0 is an even integer and is the additive identity element. The inverse of 2n is 2n, which is also an even integer. Therefore the even integers form a group under addition. (ii) For a, b, a,b Z, (a + b 2) + (a + b 2) (a + a ) + (b + b ) 2, and since (a + a ), (b + b ) Z, the given set is closed under addition. Addition of real numbers is always associative (though here we can get away with using only associativity for Z). The identity element is clearly , and the inverse of a + b 2 is a +( b) 2, which is in our set. This shows that the set is a group under addition. (iii) The proof of (iii) is the same as that of (ii) after replacing every occurrence of Z with Q. (iv) The proof of (iv) is the same as (ii) after replacing 2 with i. Exercise 3.8 Show that if a subset of {1, 2,..., 21} contains an even number, or contains the number 11, then it cannot form a group under multiplication modulo 22. Solution. We will show that even numbers and the number 11 cannot have multiplicative inverses modulo 22, which will show that a set containing an even number or 11 cannot be a group under multiplication mod 22. Let 2m be an even integer, and suppose that there were some integer a such that (2m)a 1 mod 22. This means there would be some integer b such that 2ma 1+(22)b. However, we could then write 2(ma + (11)b) 1, and this implies that 2 divides 1, a contradiction. Thus 2m cannot have a multiplicative inverse mod 22.
7 Problem Set 8 7 of 8 Very similarly, if there were an integer a such that 11a 1 mod 22, then there would be an integer b such that 11a b, which implies 11(a +2b) 1, and hence that 11 divides 1, a contradiction. (More generally, this method will show that any number sharing a divisor with n does not have a multiplicative inverse mod n.) Exercise 4.5 An element x of a group satisfies x 2 e precisely when x x 1. Use this observation to show that a group of even order must contain an odd number of elements of order 2. Solution. Let G be a group of even order. Define S 1 to be the set of elements in G of order 1, S 2 to be the set of elements of order 2, and S to be the set of elements of order higher than 2. These subsets partition G as a set, meaning that their union is G and the intersection of any two is empty, and so G S 1 + S 2 + S. Our goal is to show that S 2 is odd, and we know that G is even. There is exactly one element of G of order 1, namely the identity e, so S 1 1 is odd. Thus it will suffice to show that S is even. If S 0 we are done. Otherwise, lel the elements of S as {s 1,s 2,..., s n }. Where does the inverse of s 1 lie? It s easy to show that an element and its inverse have the same order (do this), so the inverse of s 1 lies in S. Furthermore, since the order of s 1 is higher than 2, we know that s 2 1 e, and hence that s 1 1 s 1. Therefore s 1 1 is an element in the set {s 2,..., s n }, and after releling the elements if necessary, we may assume that s 2 s 1 1. If n 2, we are done. Otherwise, there is some s 3, and we may ask where its inverse is. Again it must be in S, and it is not equal to s 1,s 2 or s 3, so there must be some s 4. Continue this process until reaching s n ; if n is odd we reach a contradiction, for s n will have no inverse. Thus S must be even, as desired. Additional Problems Exercise 1 Show that if a in a group G, then b e. Solution. a implies a 1 a 1 a implies eb e implies b e. Exercise 2 Show that the equation ax b has a unique solution in G. Solution. ax b in G if and only if a 1 ax a 1 b, i.e. if and only if x a 1 b. This shows that a 1 b satisfies the equation (a solution exists), and that if some element x of G
8 Problem Set 8 8 of 8 satisfies the equation, then x a 1 b (the solution is unique). Exercise 3 Let G be a group with multiplicative notation. Define the opposite group G op to be the same underlying set as G, but endowed with the operation a b ba. Prove that G op is in fact a group. Solution. binary operation: Since G is a group, ba G is an element of the set G op, so the star operation is well defined in G o p. associativity: For all a, b, c G op, a (b c )a (cb) (cb)a c(ba) by associativity in G (ba) c (a b ) c. identity: e G is also an identity for G op, since e a ae a ea a e for all a G. inverses: The inverse of an element a in G is also the inverse of a in G op, since a a 1 a 1 a e aa 1 a 1 a. This shows that G op is a group.
Geometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More information1 Symmetries of regular polyhedra
1230, notes 5 1 Symmetries of regular polyhedra Symmetry groups Recall: Group axioms: Suppose that (G, ) is a group and a, b, c are elements of G. Then (i) a b G (ii) (a b) c = a (b c) (iii) There is an
More informationBaltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions
Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.
More information3. 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,
More informationGROUPS ACTING ON A SET
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for
More information1. Determine all real numbers a, b, c, d that satisfy the following system of equations.
altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c
More information1 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
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More information2.1 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 xyplane
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More informationChapter 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
More informationMath 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
More informationMAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
More informationVector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.
1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called
More informationMATH 304 Linear Algebra Lecture 24: Scalar product.
MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent
More informationFoundations of Geometry 1: Points, Lines, Segments, Angles
Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More information5.1 Commutative rings; Integral Domains
5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationChapter 7. Permutation Groups
Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationTHE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
More informationLecture 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,
More informationElements of Abstract Group Theory
Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for
More informationINCIDENCEBETWEENNESS GEOMETRY
INCIDENCEBETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More informationI. 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
More informationwww.pioneermathematics.com
Problems and Solutions: INMO2012 1. Let ABCD be a quadrilateral inscribed in a circle. Suppose AB = 2+ 2 and AB subtends 135 at the centre of the circle. Find the maximum possible area of ABCD. Solution:
More informationTOPIC 3: CONTINUITY OF FUNCTIONS
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More informationUNIFORMLY DISCONTINUOUS GROUPS OF ISOMETRIES OF THE PLANE
UNIFORMLY DISCONTINUOUS GROUPS OF ISOMETRIES OF THE PLANE NINA LEUNG Abstract. This paper discusses 2dimensional locally Euclidean geometries and how these geometries can describe musical chords. Contents
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationBisections and Reflections: A Geometric Investigation
Bisections and Reflections: A Geometric Investigation Carrie Carden & Jessie Penley Berry College Mount Berry, GA 30149 Email: ccarden@berry.edu, jpenley@berry.edu Abstract In this paper we explore a geometric
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
More information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
More informationMATH PROBLEMS, WITH SOLUTIONS
MATH PROBLEMS, WITH SOLUTIONS OVIDIU MUNTEANU These are free online notes that I wrote to assist students that wish to test their math skills with some problems that go beyond the usual curriculum. These
More informationL12. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014
L12. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 Unfortunately, no one can be told what the Matrix is. You have to see it for yourself.  Morpheus Primary concepts:
More informationThe Inverse of a Matrix
The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationSolutions to Practice Problems
Higher Geometry Final Exam Tues Dec 11, 57:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles
More informationThe Use of Dynamic Geometry Software in the Teaching and Learning of Geometry through Transformations
The Use of Dynamic Geometry Software in the Teaching and Learning of Geometry through Transformations Dynamic geometry technology should be used to maximize student learning in geometry. Such technology
More information1 Review of complex numbers
1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely
More informationGRE Prep: Precalculus
GRE Prep: Precalculus Franklin H.J. Kenter 1 Introduction These are the notes for the Precalculus section for the GRE Prep session held at UCSD in August 2011. These notes are in no way intended to teach
More information2.1 Increasing, Decreasing, and Piecewise Functions; Applications
2.1 Increasing, Decreasing, and Piecewise Functions; Applications Graph functions, looking for intervals on which the function is increasing, decreasing, or constant, and estimate relative maxima and minima.
More informationPOWER 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
More informationWe know a formula for and some properties of the determinant. Now we see how the determinant can be used.
Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we
More informationSection 10.7 Parametric Equations
299 Section 10.7 Parametric Equations Objective 1: Defining and Graphing Parametric Equations. Recall when we defined the x (rcos(θ), rsin(θ)) and ycoordinates on a circle of radius r as a function of
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationPythagorean Triples. Chapter 2. a 2 + b 2 = c 2
Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the
More informationAbstract Algebra Cheat Sheet
Abstract Algebra Cheat Sheet 16 December 2002 By Brendan Kidwell, based on Dr. Ward Heilman s notes for his Abstract Algebra class. Notes: Where applicable, page numbers are listed in parentheses at the
More informationLecture 6. Inverse of Matrix
Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that
More informationsome 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
More informationComputing the Symmetry Groups of the Platonic Solids With the Help of Maple
Computing the Symmetry Groups of the Platonic Solids With the Help of Maple Patrick J. Morandi Department of Mathematical Sciences New Mexico State University Las Cruces NM 88003 USA pmorandi@nmsu.edu
More informationMATH 131 SOLUTION SET, WEEK 12
MATH 131 SOLUTION SET, WEEK 12 ARPON RAKSIT AND ALEKSANDAR MAKELOV 1. Normalisers We first claim H N G (H). Let h H. Since H is a subgroup, for all k H we have hkh 1 H and h 1 kh H. Since h(h 1 kh)h 1
More informationMath 497C Sep 9, Curves and Surfaces Fall 2004, PSU
Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 15 sometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics We say
More informationMatrix 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
More informationWeek 5: Binary Relations
1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all
More informationGeometric Transformations
CS3 INTRODUCTION TO COMPUTER GRAPHICS Geometric Transformations D and 3D CS3 INTRODUCTION TO COMPUTER GRAPHICS Grading Plan to be out Wednesdas one week after the due date CS3 INTRODUCTION TO COMPUTER
More informationLecture Notes 1: Matrix Algebra Part B: Determinants and Inverses
University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,
More informationWe call this set an ndimensional 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
More informationIntroduction to Modern Algebra
Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write
More informationGeometry in a Nutshell
Geometry in a Nutshell Henry Liu, 26 November 2007 This short handout is a list of some of the very basic ideas and results in pure geometry. Draw your own diagrams with a pencil, ruler and compass where
More informationSolution 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
More informationAdditional Topics in Math
Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
More informationIntroduction to Matrices for Engineers
Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 08 4 01 1 0 11
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More information4. An isosceles triangle has two sides of length 10 and one of length 12. What is its area?
1 1 2 + 1 3 + 1 5 = 2 The sum of three numbers is 17 The first is 2 times the second The third is 5 more than the second What is the value of the largest of the three numbers? 3 A chemist has 100 cc of
More informationAlgebra 2 Chapter 1 Vocabulary. identity  A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity  A statement that equates two equivalent expressions. verbal model A word equation that represents a reallife problem. algebraic expression  An expression with variables.
More informationPreCalculus Review Problems Solutions
MATH 1110 (Lecture 00) August 0, 01 1 Algebra and Geometry PreCalculus Review Problems Solutions Problem 1. Give equations for the following lines in both pointslope and slopeintercept form. (a) The
More information1. 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 MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationLINEAR 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,
More informationa 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
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
More informationMaximizing Angle Counts for n Points in a Plane
Maximizing Angle Counts for n Points in a Plane By Brian Heisler A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE OF STETSON UNIVERSITY IN PARTIAL FULFILLMENT OF
More informationMODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.
MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on
More information5.1 Midsegment Theorem and Coordinate Proof
5.1 Midsegment Theorem and Coordinate Proof Obj.: Use properties of midsegments and write coordinate proofs. Key Vocabulary Midsegment of a triangle  A midsegment of a triangle is a segment that connects
More informationSection 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,
More informationSolutions 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
More informationarxiv:1404.6042v1 [math.dg] 24 Apr 2014
Angle Bisectors of a Triangle in Lorentzian Plane arxiv:1404.604v1 [math.dg] 4 Apr 014 Joseph Cho August 5, 013 Abstract In Lorentzian geometry, limited definition of angles restricts the use of angle
More informationPractice 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
More informationTriangles. (SAS), or all three sides (SSS), the following area formulas are useful.
Triangles Some of the following information is well known, but other bits are less known but useful, either in and of themselves (as theorems or formulas you might want to remember) or for the useful techniques
More informationSelected practice exam solutions (part 5, item 2) (MAT 360)
Selected practice exam solutions (part 5, item ) (MAT 360) Harder 8,91,9,94(smaller should be replaced by greater )95,103,109,140,160,(178,179,180,181 this is really one problem),188,193,194,195 8. On
More informationLecture 16 : Relations and Functions DRAFT
CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationMATRIX 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 +
More informationOn the generation of elliptic curves with 16 rational torsion points by Pythagorean triples
On the generation of elliptic curves with 16 rational torsion points by Pythagorean triples Brian Hilley Boston College MT695 Honors Seminar March 3, 2006 1 Introduction 1.1 Mazur s Theorem Let C be a
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationSection 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
More information. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9
Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a
More informationTriangle Congruence and Similarity A CommonCoreCompatible Approach
Triangle Congruence and Similarity A CommonCoreCompatible Approach The Common Core State Standards for Mathematics (CCSSM) include a fundamental change in the geometry program in grades 8 to 10: geometric
More informationMath 319 Problem Set #3 Solution 21 February 2002
Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod
More informationMathematical Procedures
CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,
More informationYou know from calculus that functions play a fundamental role in mathematics.
CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.
More informationWhat is inversive geometry?
What is inversive geometry? Andrew Krieger July 18, 2013 Throughout, Greek letters (,,...) denote geometric objects like circles or lines; small Roman letters (a, b,... ) denote distances, and large Roman
More informationOrthogonal 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
More information