# MCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 15 January. Elimination Methods

Save this PDF as:

Size: px
Start display at page:

Download "MCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 15 January. Elimination Methods"

## Transcription

1 Elimination Methods In this lecture we define projective coordinates, give another application, explain the Sylvester resultant method, illustrate how to cascade resultants to compute discriminant, and then state the main theorem of elimination theory. Because of the algebraic nature of the methods, we assume the coefficients to be exact (in Z or Q) and all calculations (with Maple or Sage) are performed with exact arithmetic and/or with the manipulation of symbolic coefficients. 1 Projective Coordinates Consider the problem of intersecting two circles in the plane. Without loss of generality, we may choose our coordinate system so that the x-axis passes through the centers of the circles and let the first circle be centered at the origin and have radius one. Let then the second circle have radius r and denote the coordinates of its center by (c, ). The corresponding system is then x x = f(x 1, x 2 ) = (1) (x 1 c) 2 + x 2 2 r 2 =. By elimination subtracting from the second equation the first we see that there can be at most two isolated roots. Just like two parallel lines meet at infinity, the system has solutions at infinity. We can compute these solutions by transforming the system into projective coordinates. The map ψ [ ψ : C n P n : (x 1, x 2,..., x n ) [x : x 1 : x 2 : : x n ] = 1 : x 1 : x 2 : : x ] n. (2) x x x defines an embedding of C n into projective space P n, mapping points into equivalence classes. While most common, note that ψ is just a projective transformation, we will encounter other ones. The equivalence relation is defined by [x : x 1 : : x n ] [y : y 1 : : y n ] λ : x i = λy i, i =, 1,..., n. (3) From the embedding in (2) we see that as x, that then the coordinates x i, for i = 1, 2,..., n. So a point with x = lies at infinity. To find the solution at infinity of the system (1), we embed it into projective space replacing first x 1 by x 1 /x, x 2 by x 2 /x and then multiplying by x 2, to obtain x x2 2 x2 = x 2 1 cx 1x + x 2 2 r2 x 2 =. (4) Notice how every monomial in the system has now the same degree. We say that the system is now homogeneous. To find the solutions at infinity, we set x = and solve x x 2 2 =. Since coordinates in P 2 are only determined up to a factor, we may set x 1 = 1 and find then x 2 = ± 1. Thus we find [ : 1 : ± 1] as the two solutions at infinity of the problem of intersecting two circles. Except for the case when there are infinitely many solutions, the theorem of Bézout in projective space states that there are always as many solutions as the product of the degrees. The importance of projective space to practical applications lies in the scaling of the coordinates. Computing solutions with huge coordinates leads to an ill-conditioned numerical problem. Embedding the problem into projective space provides a numerically more favorable representation of the solutions with huge coordinates. Obviously, the coefficients of the polynomials also naturally belong to a projective space, because the solution set does not change as we multiply the polynomials with a nonzero coefficient. Jan Verschelde UIC, Dept of Math, Stat & CS Lecture 2, page 1

2 2 Molecular Configurations The picture at the left below represents a robot arm with 6 revolute joints. The middle picture is a configuration of six atoms. Placing these two pictures on top of each other suggests that the bonds between the atoms correspond to the joints of the robot arm while the links of the robot arm correspond to the atoms. Note that the configurations are spatial. To determine the configuration, we label the atoms clockwise as p 1, p 2,..., p 6. Consider then the three triangles T 1, T 2, and T 3, spanned respectively by the respective triplets (p 1, p 2, p 6 ), (p 2, p 3, p 4 ), and (p 4, p 5, p 6 ). Denote by θ i, for i = 1, 2, 3, the angle between T i and the base triangle spanned by (p 2, p 4, p 6 ). Fixing the distances between the atoms and angles θ i determines the configuration. To design a configuration, we must solve the system α 11 + α 12 cos(θ 2 ) + α 13 cos(θ 3 ) + α 14 cos(θ 2 )cos(θ 3 ) + α 15 sin(θ 2 )sin(θ 3 ) = α 21 + α 22 cos(θ 3 ) + α 23 cos(θ 1 ) + α 24 cos(θ 3 )cos(θ 1 ) + α 25 sin(θ 3 )sin(θ 1 ) = α 31 + α 32 cos(θ 1 ) + α 33 cos(θ 2 ) + α 34 cos(θ 1 )cos(θ 2 ) + α 35 sin(θ 1 )sin(θ 2 ) = cos 2 (θ 1 ) + sin 2 (θ 1 ) 1 = cos 2 (θ 2 ) + sin 2 (θ 2 ) 1 = cos 2 (θ 3 ) + sin 2 (θ 3 ) 1 = where the α ij are input coefficients. Instead of solving directly for the cosines and sines of the angles, we can obtain a smaller system transforming to half angles: letting t i = tan(θ i /2), for i = 1, 2, 3, implies cos(θ i ) = (1 t 2 i )/(1 + t2 i ) and sin(θ i ) = 2t i /(1 + t 2 i ). After clearing denominators, we obtain the system β 11 + β 12 t β 13t β 14t 2 t 3 + β 15 t 2 2 t2 3 = β 21 + β 22 t β 23t β 24t 3 t 1 + β 25 t 2 3 t2 1 = (6) β 31 + β 32 t β 33t β 34t 1 t 2 + β 35 t 2 1 t2 2 = where the β ij are the input coefficients. Note that the second system has fewer equations. However the equations of (6) are of degree four. While the decrease in dimension is often seen as favorable for symbolic elimination methods, numerically, dealing with polynomials of higher degree may often be challenging. This application is described in greater detail in [3], along with elimination methods to solve such systems. (5) Jan Verschelde UIC, Dept of Math, Stat & CS Lecture 2, page 2

3 3 Resultants The following Lemma is taken from [1]: Lemma 3.1 Two polynomials f and g have a common factor there exist two nonzero polynomials A and B such that Af + Bg =, with deg(a) < deg(g) and deg(b) < deg(f). Proof. Let f = f 1 h and g = g 1 h, then A = g 1 and B = f 1 : Af + Bg = g 1 f 1 h + f 1 g 1 h =. Suppose f and g have no common factor. By the Euclidean algorithm we then have that GCD(f, g) = 1 = Ãf + Bg. Assume B. Then B = 1B = (Ãf + Bg)B. Using Bg = Af we obtain B = (ÃB BA)f. As B, it means that deg(b) deg(f) which gives a contradiction. Thus f and g must have a common factor. The importance of this lemma is that via linear algebra we will get a criterion on the coefficients of the polynomials f and g to decide whether they have a common factor. Consider for example f(x) = a + a 1 x + a 2 x 2 +a 3 x 3 and g(x) = b +b 1 x+b 2 x 2. If f and g have a common factor, then there must exist two nonzero polynomials A(x) = c +c 1 x and B(x) = d +d 1 x+d 2 x 2 such that A(x)f(x)+B(x)g(x) =. Executing the polynomial multiplication and requiring that all coefficients with x are zero leads to a homogeneous system in the coefficients of A and B: x 4 : a 3 c 1 + b 2 d 2 = x 3 : a 2 c 1 + a 3 c + b 1 d 2 + b 2 d 1 = x 2 : a 1 c 1 + a 2 c + b d 2 + b 1 d 1 + b 2 d = (7) x 1 : a c 1 + a 1 c b d 1 + b 1 d = x : a c b d =. In matrix form, we have a 3 b 2 a 2 a 3 b 1 b 2 a 1 a 2 b b 1 b 2 a a 1 b b 1 a b c 1 c d 2 d 1 d =. (8) The matrix of the linear system is called the Sylvester matrix of f and g. Only if its determinant is zero can the linear system have a nontrivial solution. Observe that the determinant leads to a polynomial in the coefficients of the polynomials. This polynomial is called the resultant. With the computer algebra system Maple, we can eliminate x from two polynomials f and g, via the following commands: [> sm := LinearAlgebra[SylvesterMatrix](f,g,x); [> rs := LinearAlgebra[Determinant](sm); If we are given a polynomial system, we can eliminate one variable by viewing the equations as polynomials in that one variable while hiding the other variables as part of the symbolic coefficients. This is the basic principle of elimination methods. Geometrically, the process of elimination corresponds to projecting the solution set onto a space with fewer variables. While dealing with fewer variables sounds attractive, it is not too hard to find examples where the elimination introduces additional singular solutions. While the setup in this lecture is symbolic, for problems with approximate coefficients, the determinant computation is replaced by determining the numerical rank via a singular value decomposition or via a QR decomposition. Jan Verschelde UIC, Dept of Math, Stat & CS Lecture 2, page 3

4 4 Cascading Resultants to Compute Discriminants Resultants can be used to define discriminants. For a polynomial with indeterminate coefficients we may ask for a condition on the coefficients for which there are multiple roots, i.e.: where both the polynomial and its derivative vanish. The discriminant of f can thus be seen as the resultant of f and its derivative. The notion of a discriminant is not limited to polynomials in one variable. We can say that the system x f(x, y) = 2 + y 2 1 = (x c) 2 + y 2 r 2 (9) = has exactly two solutions [ x = r2 + c r4 + 2c, y = ± 2 r 2 + 2r 2 c 4 + 2c 2 1 2c 2c ] (1) except for those c and r satisfying D(c, r) = 256(c r 1) 2 (c r + 1) 2 (c + r 1) 2 (c + r + 1) 2 c 4 =. (11) The polynomial D(c, r) is the discriminant for the system. The discriminant variety is the solution set of the discriminant. For the example, it is shown below: By symmetry, we can classify the regular solutions into four different categories. Given a system f with parameters, a straightforward method to compute the discriminant takes 2 steps: 1. Let E be (f, det(j f )), where J f is the Jacobian matrix of f. 2. Eliminate from E all the indeterminates, using resultants. What remains after elimination is an expression in the parameters: the discriminant. For the example, we consider: [ ] 2x 2y J f = 2(x c) 2y det(j f ) = 4xy 4(x c)y x 2 + y 2 1 = E(x, y, c, r) = (x c) 2 + y 2 r 2 = 4cy = (12) With Sage, we proceed as follows (note that sage: is the prompt in a terminal session): Jan Verschelde UIC, Dept of Math, Stat & CS Lecture 2, page 4

5 sage: x,y,c,r = var( x,y,c,r ) sage: f1 = x^2 + y^2-1 sage: f2 = (x-c)^2 + y^2 - r^2 sage: J = matrix([[diff(f1,x),diff(f1,y)],[diff(f2,x),diff(f2,y)]]) sage: dj = det(j) sage: print f1, f2, dj 2 2 y + x y + (x - c) - r 4 x y - 4 (x - c) y To use the resultants of Singular, we declare a ring of rational numbers and convert the three polynomials: f1, f2, and dj to this ring. sage: R.<x,y,c,r> = QQ[] sage: E = [R(f1),R(f2),R(dJ)] sage: print E [x^2 + y^2-1, x^2 + y^2-2*x*c + c^2 - r^2, 4*y*c] Cascading resultants, we first eliminate x and then eliminate y. To eliminate x we combine the equations pairwise: sage: e1 = singular.resultant(e[],e[1],x) sage: e12 = singular.resultant(e[1],e[2],x) sage: print e1, e12 4*y^2*c^2+c^4-2*c^2*r^2+r^4-2*c^2-2*r^2+1 16*y^2*c^2 After elimination of y we obtain the discriminant: sage: discriminant = singular.resultant(e1,e12,y) sage: factor(r(discriminant)) (256) * (c - r - 1)^2 * (c - r + 1)^2 * (c + r - 1)^2 * (c + r + 1)^2 * c^4 Notice the ring conversion before the factorization. 5 The Main Theorem of Elimination Theory Resultants provide a constructive proof for the following theorem. Theorem 5.1 Let V P n be the solution set of a polynomial system. Then the projection of V onto P n 1 is again the solution set of a system of polynomial equations. The use of projective coordinates is needed. Consider for instance x 1 x 2 1 =. The projection of this hyperbola onto the coordinate axes leads to a line minus a point. A line minus a point is not the solution set of a polynomial. This observation points at another drawback of resultants (in addition to the potential introduction of singular solutions): the solutions of the resultant may include solutions at infinity which are spurious to the original problem. In [2, Theorem 14.1] we find the main theorem of elimination theory formulated in more general terms, but also with an outline of a constructive proof using resultants. Jan Verschelde UIC, Dept of Math, Stat & CS Lecture 2, page 5

6 6 Exercises 1. For the system in (1) determine the exceptional values for the parameters c and r for which there are infinitely many solutions. Justify your answer. 2. Derive the formulas for the transformation which uses half angles, used to transform (5) into (6). 3. Consider f(x) = ax 2 + bx + c. Use resultant methods available in Maple or Sage to compute the discriminant of this polynomial. 4. Use Maple or Sage to compute resultants to solve the system x1 x 2 1 = x x =. 5. Consider the system x 2 + y 2 2 = x 2 + y2 5 1 =. Show geometrically by making a plot of the two curves using Maple or Sage that elimination will lead to polynomials with double roots. Compute resultants (as univariate polynomials in x or y) and show algebraically that they have double roots. 6. Consider as choice in system (6) for the β ij coefficients the matrix (second instance in [3]): Try to solve the system using these coefficients. How many solutions do you find? 7. Extend the discriminant computation for the ellipse ( x a )2 + ( y b )2 = 1 and the circle (x c) 2 + y 2 = r 2. Use Sage or Maple. Interpret the result by appropriate projections of the four dimensional parameter space onto a plane. 8. Examine the intersection of three spheres. What are the components of its discriminant variety? References [1] D. Cox, J. Little, and D. O Shea. Ideals, Varieties and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics. Springer Verlag, second edition, [2] D. Eisenbud. Commutative Algebra with a View Toward Algebraic Geometry, volume 15 of Graduate Texts in Mathematics. Springer-Verlag, [3] I.Z. Emiris and B. Mourrain. Computer algebra methods for studying and computing molecular conformations. Algorithmica, 25(2-3):372 42, Special issue on algorithmic research in Computational Biology, edited by D. Gusfield and M.-Y. Kao. Jan Verschelde UIC, Dept of Math, Stat & CS Lecture 2, page 6

### MCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 9 April. Hilbert Polynomials

Hilbert Polynomials For a monomial ideal, we derive the dimension counting the monomials in the complement, arriving at the notion of the Hilbert polynomial. The first half of the note is derived from

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

### 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

### MATH 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

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### CONICS ON THE PROJECTIVE PLANE

CONICS ON THE PROJECTIVE PLANE CHRIS CHAN Abstract. In this paper, we discuss a special property of conics on the projective plane and answer questions in enumerative algebraic geometry such as How many

### Multiplicity. Chapter 6

Chapter 6 Multiplicity The fundamental theorem of algebra says that any polynomial of degree n 0 has exactly n roots in the complex numbers if we count with multiplicity. The zeros of a polynomial are

### MATHEMATICS (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)

### Basics of Polynomial Theory

3 Basics of Polynomial Theory 3.1 Polynomial Equations In geodesy and geoinformatics, most observations are related to unknowns parameters through equations of algebraic (polynomial) type. In cases where

### Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

### Math Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices.

Math Placement Test Study Guide General Characteristics of the Test 1. All items are to be completed by all students. The items are roughly ordered from elementary to advanced. The expectation is that

### Appendix A. Appendix. A.1 Algebra. Fields and Rings

Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### Roots and Coefficients of a Quadratic Equation Summary

Roots and Coefficients of a Quadratic Equation Summary For a quadratic equation with roots α and β: Sum of roots = α + β = and Product of roots = αβ = Symmetrical functions of α and β include: x = and

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### 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

### Algebra 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 real-life problem. algebraic expression - An expression with variables.

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### GRE 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

### Class XI Chapter 5 Complex Numbers and Quadratic Equations Maths. Exercise 5.1. Page 1 of 34

Question 1: Exercise 5.1 Express the given complex number in the form a + ib: Question 2: Express the given complex number in the form a + ib: i 9 + i 19 Question 3: Express the given complex number in

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### 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

### Computer Algebra for Computer Engineers

p.1/14 Computer Algebra for Computer Engineers Preliminaries Priyank Kalla Department of Electrical and Computer Engineering University of Utah, Salt Lake City p.2/14 Notation R: Real Numbers Q: Fractions

### x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### Quotient 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.

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### RESULTANT AND DISCRIMINANT OF POLYNOMIALS

RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks

Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Algebra 2! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

### 28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z

28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition

### Figure 1.1 Vector A and Vector F

CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

### Lecture 2: Homogeneous Coordinates, Lines and Conics

Lecture 2: Homogeneous Coordinates, Lines and Conics 1 Homogeneous Coordinates In Lecture 1 we derived the camera equations λx = P X, (1) where x = (x 1, x 2, 1), X = (X 1, X 2, X 3, 1) and P is a 3 4

### MATH 65 NOTEBOOK CERTIFICATIONS

MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Introduction to Algebraic Geometry. Bézout s Theorem and Inflection Points

Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### 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

### Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

### 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

### Solutions 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

### 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

### MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

### 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

### Alex, I will take congruent numbers for one million dollars please

Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohio-state.edu One of the most alluring aspectives of number theory

### Prompt Students are studying multiplying binomials (factoring and roots) ax + b and cx + d. A student asks What if we divide instead of multiply?

Prompt Students are studying multiplying binomials (factoring and roots) ax + b and cx + d. A student asks What if we divide instead of multiply? Commentary In our foci, we are assuming that we have a

### 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 +

### 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)

### Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

### The Math Circle, Spring 2004

The Math Circle, Spring 2004 (Talks by Gordon Ritter) What is Non-Euclidean Geometry? Most geometries on the plane R 2 are non-euclidean. Let s denote arc length. Then Euclidean geometry arises from the

### ME 115(b): Solution to Homework #1

ME 115(b): Solution to Homework #1 Solution to Problem #1: To construct the hybrid Jacobian for a manipulator, you could either construct the body Jacobian, JST b, and then use the body-to-hybrid velocity

### The Cycloheptane Molecule

The Cycloheptane Molecule A Challenge to Computer Algebra (Invited lecture, ISSAC 97, Maui, Hawaii, USA, (version Jan. 98)) A.H.M. Levelt University of Nijmegen, The Netherlands ahml@sci.kun.nl 1 Thanks

### MyMathLab ecourse for Developmental Mathematics

MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and

### 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

### Computer Aided Geometric Design. Axial moving planes and singularities of rational space curves

Computer Aided Geometric Design 26 (2009) 300 316 Contents lists available at ScienceDirect Computer Aided Geometric Design www.elsevier.com/locate/cagd Axial moving planes and singularities of rational

### Mathematical 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,

### MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

### Computational algebraic geometry

Computational algebraic geometry Learning coefficients via symbolic and numerical methods Anton Leykin Georgia Tech AIM, Palo Alto, December 2011 Let k be a field (R or C). Ideals varieties Ideal in R

### 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

### Overview of Math Standards

Algebra 2 Welcome to math curriculum design maps for Manhattan- Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse

### Unique Factorization

Unique Factorization Waffle Mathcamp 2010 Throughout these notes, all rings will be assumed to be commutative. 1 Factorization in domains: definitions and examples In this class, we will study the phenomenon

### 1 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

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Situation: Dividing Linear Expressions

Situation: Dividing Linear Expressions Date last revised: June 4, 203 Michael Ferra, Nicolina Scarpelli, Mary Ellen Graves, and Sydney Roberts Prompt: An Algebra II class has been examining the product

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### Trigonometric Functions and Equations

Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### Double integrals. Notice: this material must not be used as a substitute for attending the lectures

ouble integrals Notice: this material must not be used as a substitute for attending the lectures . What is a double integral? Recall that a single integral is something of the form b a f(x) A double integral

### Math 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

### MATRIX 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

### Algebra II. Weeks 1-3 TEKS

Algebra II Pacing Guide Weeks 1-3: Equations and Inequalities: Solve Linear Equations, Solve Linear Inequalities, Solve Absolute Value Equations and Inequalities. Weeks 4-6: Linear Equations and Functions:

### TOPIC 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

### 1.3 Algebraic Expressions

1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

### LINEAR EQUATIONS IN TWO VARIABLES

66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that

### α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

### DESARGUES THEOREM DONALD ROBERTSON

DESARGUES THEOREM DONALD ROBERTSON Two triangles ABC and A B C are said to be in perspective axially when no two vertices are equal and when the three intersection points AC A C, AB A B and BC B C are

### Prentice Hall Algebra 2 2011 Correlated to: Colorado P-12 Academic Standards for High School Mathematics, Adopted 12/2009

Content Area: Mathematics Grade Level Expectations: High School Standard: Number Sense, Properties, and Operations Understand the structure and properties of our number system. At their most basic level

### Vector Algebra II: Scalar and Vector Products

Chapter 2 Vector Algebra II: Scalar and Vector Products We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define

### Factoring Polynomials

Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### 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

### Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper

Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### Mathematics Georgia Performance Standards

Mathematics Georgia Performance Standards K-12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by

### Introduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test

Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are

### Torgerson s Classical MDS derivation: 1: Determining Coordinates from Euclidean Distances

Torgerson s Classical MDS derivation: 1: Determining Coordinates from Euclidean Distances It is possible to construct a matrix X of Cartesian coordinates of points in Euclidean space when we know the Euclidean

### Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula

Lecture 5 : Solving Equations, Completing the Square, Quadratic Formula An equation is a mathematical statement that two mathematical expressions are equal For example the statement 1 + 2 = 3 is read as

### Algebra II: Strand 7. Conic Sections; Topic 1. Intersection of a Plane and a Cone; Task 7.1.2

1 TASK 7.1.2: THE CONE AND THE INTERSECTING PLANE Solutions 1. What is the equation of a cone in the 3-dimensional coordinate system? x 2 + y 2 = z 2 2. Describe the different ways that a plane could intersect

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional 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 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Math 215 HW #6 Solutions

Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

### 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