DOCSLIB.ORG

February '10 Tbl Math 28A — Lecture 13 — Outline Notes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
February '10 Tbl Math 28A — Lecture 13 — Outline Notes FEBRUARY '10 TBL MATH 28A | LECTURE 13 | OUTLINE NOTES We can now finish the job with unique factorization! Theorem 14. In any Euclidean domain R, every element can be factorized uniquely into irreducible elements. Proof. We know that every element can be factored into irreducible elements; the only point at issue is uniqueness. We will show by induction on N that all elements of size N have a unique factorization. Suppose that n 2 R can be factored into irreducibles in two different ways n = p1 : : : pk = q1 : : : ql we must show that k = l and, after reordering, and replacing elements with associates, p1 = q1, p2 = q2, etc. Since p1 is irreducible, it is prime, and since it divides q1 : : : ql, it must divide one of the qi. Re- ordering, we may suppose it divides q1. Then since q1 is prime, p1 and q1 must be assosciates. Since we are allowed to replace the qi with assosciates, we may suppose q1 = p1. Then dividing, p2 : : : pk = q2 : : : ql and both sides have smaller size than N. Thus we inductively know k = l and (after reordering and replacing with associates) p2 = q2; : : : ; pk = ql. Thus we are done. x4. Vector spaces| So far, we have been studying generalizations of rather familiar objects in math. The next kind of object we're going to study is meant to generalize a situation that's perhaps less familiar, and so it's very important that we look at an example first to get on the same page. The key example we will be looking at the following: } the collection of all polynomials p with real coefficients of degree ≤ 3 which satisfy p(1) = 0. Let us ask ourselves how such polynomials behave. We will make a series of observations. 1. We can certainly add such polynomials together, and the addition will respect the same associative and commutative properties that addition has in rings and fields. 2. It is therefore perhaps natural to ask ourselves which of the other properties of addition in rings will hold here. There were two: the existence of an additive identity, and the existence of additive inverses. There is certainly an additive identity: the zero polynomial. Also, there are inverses; we can take the negative such a polynomial and it is still a polynomial. Thus all four of the addition axioms for a ring are satisfied here too. 3. The situation for multiplication, however, is much less clear cut. We can't really multiply elements of the collection } together in general and expect to get another member of the collection. And though there are a few special cases where multiplication of elements does work, this turn out to be not a terribly fertile lead to pursue. So we say we can't usefully multiply an element of the collection } with each other. 4. Is there no role for multiplication here at all? No. There is some sense in which these elements can be multiplied|not with each other, but with real numbers. We can always multiply any element of the collection by any real number. 5. What properties does this multiplication have? Well, we do have that for any two real numbers a; b and any polynomial p from our collection, (ab) · p = a · (b · p). This is called compatibility of multiplication with real multiplication. It's also the case that 1 · p = p. 6. Are there any other nice properties that we could hope to think about? In rings, we had one more kind of property|the `distributive' one that includes addition and multiplication together. There are also distributive properties for the collection }. In fact there are two, one coming from addition in the reals and one coming from addition in the collection. So if a; b are reals and p; q are polynomials in our collection, (a + b) · p = a · p + bp;_ and a · (p + q) = a · p + a · q:.
Load more

Recommended publications
  • Chapter 1 the Field of Reals and Beyond
    Chapter 1 The Field of Reals and Beyond Our goal with this section is to develop (review) the basic structure that character- izes the set of real numbers. Much of the material in the ¿rst section is a review of properties that were studied in MAT108 however, there are a few slight differ- ences in the de¿nitions for some of the terms. Rather than prove that we can get from the presentation given by the author of our MAT127A textbook to the previous set of properties, with one exception, we will base our discussion and derivations on the new set. As a general rule the de¿nitions offered in this set of Compan- ion Notes will be stated in symbolic form this is done to reinforce the language of mathematics and to give the statements in a form that clari¿es how one might prove satisfaction or lack of satisfaction of the properties. YOUR GLOSSARIES ALWAYS SHOULD CONTAIN THE (IN SYMBOLIC FORM) DEFINITION AS GIVEN IN OUR NOTES because that is the form that will be required for suc- cessful completion of literacy quizzes and exams where such statements may be requested. 1.1 Fields Recall the following DEFINITIONS: The Cartesian product of two sets A and B, denoted by A B,is a b : a + A F b + B . 1 2 CHAPTER 1. THE FIELD OF REALS AND BEYOND A function h from A into B is a subset of A B such that (i) 1a [a + A " 2bb + B F a b + h] i.e., dom h A,and (ii) 1a1b1c [a b + h F a c + h " b c] i.e., h is single-valued.
    [Show full text]
  • Real Numbers and Their Properties
    Real Numbers and their Properties Types of Numbers + • Z Natural numbers - counting numbers - 1, 2, 3,... The textbook uses the notation N. • Z Integers - 0, ±1, ±2, ±3,... The textbook uses the notation J. • Q Rationals - quotients (ratios) of integers. • R Reals - may be visualized as correspond- ing to all points on a number line. The reals which are not rational are called ir- rational. + Z ⊂ Z ⊂ Q ⊂ R. R ⊂ C, the field of complex numbers, but in this course we will only consider real numbers. Properties of Real Numbers There are four binary operations which take a pair of real numbers and result in another real number: Addition (+), Subtraction (−), Multiplication (× or ·), Division (÷ or /). These operations satisfy a number of rules. In the following, we assume a, b, c ∈ R. (In other words, a, b and c are all real numbers.) • Closure: a + b ∈ R, a · b ∈ R. This means we can add and multiply real num- bers. We can also subtract real numbers and we can divide as long as the denominator is not 0. • Commutative Law: a + b = b + a, a · b = b · a. This means when we add or multiply real num- bers, the order doesn’t matter. • Associative Law: (a + b) + c = a + (b + c), (a · b) · c = a · (b · c). We can thus write a + b + c or a · b · c without having to worry that different people will get different results. • Distributive Law: a · (b + c) = a · b + a · c, (a + b) · c = a · c + b · c. The distributive law is the one law which in- volves both addition and multiplication.
    [Show full text]
  • Ring (Mathematics) 1 Ring (Mathematics)
    Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right.
    [Show full text]
  • Math 11200/20 Lectures Outline I Will Update This
    Math 11200/20 lectures outline I will update this document after every lecture to keep track of what we covered. \Textbook" refers to the book by Diane Herrmann and Paul Sally. \Intro to Cryptography" refers to: • Introduction to Cryptography with Coding Theory, Second Edition, by Wade Trappe and Lawrence Washington. • http://calclab.math.tamu.edu/~rahe/2014a_673_700720/textbooks.html. week 1 9/26/16. Reference: Textbook, Chapter 0 (1) introductions (2) syllabus (boring stuff) (3) Triangle game (a) some ideas: parity, symmetry, balancing large/small, largest possible sum (b) There are 6 \symmetries of the triangle." They form a \group." (just for fun) 9/28/16. Reference: Textbook, Chapter 1, pages 11{19 (1) Russell's paradox (just for fun) (2) Number systems: N; Z; Q; R. (a) π 2 R but π 62 Q. (this is actually hard to prove) (3) Set theory (a) unions, intersections. (remember: \A or B" means \at least one of A, B is true") (b) drawing venn diagrams is very useful! (c) empty set. \vacuously true" (d) commutative property, associative, identity (the empty set is the identity for union), inverse, (distributive?) (i) how to show that there is no such thing as \distributive property of addi- tion over multiplication" (a + (b · c) = (a · b) + (a · c)) (4) Functions 9/30/16. Reference: Textbook, Chapter 1, pages 20{32 (1) functions (recap). f : A ! B. A is domain, B is range. (2) binary operations are functions of the form f : S × S ! S. (a) \+ is a function R × R ! R." Same thing as \+ is a binary operation on R." Same thing as \R is closed under +." (closed because you cannot escape!) a c ad+bc (b) N; Z are also closed under addition.
    [Show full text]
  • Vector Spaces
    Chapter 1 Vector Spaces Linear algebra is the study of linear maps on finite-dimensional vec- tor spaces. Eventually we will learn what all these terms mean. In this chapter we will define vector spaces and discuss their elementary prop- erties. In some areas of mathematics, including linear algebra, better the- orems and more insight emerge if complex numbers are investigated along with real numbers. Thus we begin by introducing the complex numbers and their basic✽ properties. 1 2 Chapter 1. Vector Spaces Complex Numbers You should already be familiar with the basic properties of the set R of real numbers. Complex numbers were invented so that we can take square roots of negative numbers. The key idea is to assume we have i − i The symbol was√ first a square root of 1, denoted , and manipulate it using the usual rules used to denote −1 by of arithmetic. Formally, a complex number is an ordered pair (a, b), the Swiss where a, b ∈ R, but we will write this as a + bi. The set of all complex mathematician numbers is denoted by C: Leonhard Euler in 1777. C ={a + bi : a, b ∈ R}. If a ∈ R, we identify a + 0i with the real number a. Thus we can think of R as a subset of C. Addition and multiplication on C are defined by (a + bi) + (c + di) = (a + c) + (b + d)i, (a + bi)(c + di) = (ac − bd) + (ad + bc)i; here a, b, c, d ∈ R. Using multiplication as defined above, you should verify that i2 =−1.
    [Show full text]
  • Properties of the Real Numbers Follow from the Ones Given in Table 1.2
    Properties of addition and multiplication of real numbers, and some consequences Here is a list of properties of addition and multiplication in R. They are given in Table 1.2 on page 18 of your book. It is good practice to \not accept everything that you are told" and to see if you can see why other \obvious sounding" properties of the real numbers follow from the ones given in Table 1.2. This type of critical thinking lead to the development of axiomatic systems; the earliest (and most widely known) being Euclid's axioms for geometry. The set of real numbers R is closed under addition (denoted by +) and multiplication (denoted by × or simply by juxtaposition) and satisfies: 1. Addition is commutative. x + y = y + x for all x; y 2 R. 2. Addition is associative. (x + y) + z = x + (y + z) for all x; y; z 2 R. 3. There is an additive identity element. There is a real number 0 with the property that x + 0 = x for all x 2 R. 4. Every real number has an additive inverse. Given any real number x, there is a real number (−x) so that x + (−x) = 0 5. Multiplication is commutative. xy = yx for all x; y 2 R. 6. Multiplication is associative. (xy)z = x(yz) for all x; y; z 2 R. 7. There is a multiplicative identity element. There is a real number 1 with the property that x:1 = x for all x 2 R. Furthermore, 1 6= 0. 8. Every non-zero real number has a multiplicative inverse.
    [Show full text]
  • 10.1 ­ Groups ­ Filled In.Notebook December 01, 2015
    10.1 ­ Groups ­ filled in.notebook December 01, 2015 10.1 ­ Groups 1 10.1 ­ Groups ­ filled in.notebook December 01, 2015 A mathematical system consists of a set of elements and at least one binary operation. 2 10.1 ­ Groups ­ filled in.notebook December 01, 2015 A mathematical system consists of a set of elements and at least one binary operation. A binary operation is an operation, or rule, that can be performed on two and only two elements of a set with the result being a single element. 3 10.1 ­ Groups ­ filled in.notebook December 01, 2015 A mathematical system consists of a set of elements and at least one binary operation. A binary operation is an operation, or rule, that can be performed on two and only two elements of a set with the result being a single element. Examples of binary operations: Addition, Subtraction, Multiplication, Division Examples of non­binary operations: Finding the reciprocal, inverse, identity 4 10.1 ­ Groups ­ filled in.notebook December 01, 2015 Commutative Property Addition Multiplication Associative Property Addition Multiplication 5 10.1 ­ Groups ­ filled in.notebook December 01, 2015 If a binary operation is performed on any two elements of a set and the result is an element of the set, then that set is closed, or we say it has closure, under the given binary operation. 6 10.1 ­ Groups ­ filled in.notebook December 01, 2015 If a binary operation is performed on any two elements of a set and the result is an element of the set, then that set is closed, or we say it has closure, under the given binary operation.
    [Show full text]
  • Section 10.1 Groups Guided Notes
    Section 10.1 Groups Guided Notes What You Will Learn Mathematical Systems Commutative Property Associative Property Closure Identity Element Inverses Groups Let’s begin with, a Mathematical System: A mathematical system consists of a set of elements and at least one binary operation. Binary Operation A binary operation is an operation, or rule, that can be performed on two and only two elements of a set. The result is a single element. Addition, multiplication, subtraction and division are all binary operations. The set of integers and the binary operation of addition make a mathematical system. Commutative Property For any elements a, b, and c Addition a + b = b + a Multiplication a • b = b • a Associative Property For any elements a, b, and c Addition (a + b) + c = a + (b + c) Multiplication (a • b) • c = a • (b • c) Closure If a binary operation is performed on any two elements of a set and the result is an element of the set, then that set is closed (or has closure) under the given binary operation. Identity Element An identity element is an element in a set such that when a binary operation is performed on it and any given element in the set, the result is the given element. Additive identity element is 0. Multiplicative identity element is 1. Inverses When a binary operation is performed on two elements in a set and the result is the identity element for the binary operation, each element is said to be the inverse of the other. The additive inverse of a nonzero integer, a, is –a.
    [Show full text]
  • MAT 240 - Algebra I Fields Definition
    MAT 240 - Algebra I Fields Definition. A field is a set F , containing at least two elements, on which two operations + and · (called addition and multiplication, respectively) are defined so that for each pair of elements x, y in F there are unique elements x + y and x · y (often written xy) in F for which the following conditions hold for all elements x, y, z in F : (i) x + y = y + x (commutativity of addition) (ii) (x + y) + z = x + (y + z) (associativity of addition) (iii) There is an element 0 ∈ F , called zero, such that x+0 = x. (existence of an additive identity) (iv) For each x, there is an element −x ∈ F such that x+(−x) = 0. (existence of additive inverses) (v) xy = yx (commutativity of multiplication) (vi) (x · y) · z = x · (y · z) (associativity of multiplication) (vii) (x + y) · z = x · z + y · z and x · (y + z) = x · y + x · z (distributivity) (viii) There is an element 1 ∈ F , such that 1 6= 0 and x·1 = x. (existence of a multiplicative identity) (ix) If x 6= 0, then there is an element x−1 ∈ F such that x · x−1 = 1. (existence of multiplicative inverses) Remark: The axioms (F1)–(F-5) listed in the appendix to Friedberg, Insel and Spence are the same as those above, but are listed in a different way. Axiom (F1) is (i) and (v), (F2) is (ii) and (vi), (F3) is (iii) and (vii), (F4) is (iv) and (ix), and (F5) is (vii). Proposition. Let F be a field.
    [Show full text]
  • Algebra Notes Sept
    Algebra Notes Sept. 14: Definition and Examples of Rings Geoffrey Scott Rings: Motivation and Definition Throughout your education, you've encountered many different mathematical objects that can be \added" and \multiplied": Integers, rationals, real numbers: You learned what it meant to \add" and \multiply" numbers in primary school. Polynomials: You learned what it meant to \add" and \multiply" polynomials with real coefficients in high school Square matricies: You learned to \add" square matrices by adding the corresponding terms of the matrices, and to \multiply" square matrices by the matrix multiplication algorithm. Functions: You \add" functions by constructing a new function whose output values are the sum (as numbers) of the output values of the original functions, and you \multiply" func- tions by constructing a new function whose output values are the product (as numbers) of the output values of the original functions. If you were an apt student, you might have complained to your teachers that the terms \add" and \multiply" are really overloaded! Why use the same two words to describe such different operations on such different kinds of mathematical objects? One reason is that there are properties of these \adding" and \multiplying" operations that are true in all of these examples { properties such as a+b = b+a and a(bc) = (ab)c and a(b+c) = ab + ac and (b + c)a = ba + ca. In this way, the process of \adding" and \multiplying" square matrices (or polynomials, or functions) behaves like the process of \adding" and \multiplying" numbers. As you learn more mathematics, you will encounter even more mathematical objects that have \addition" and \multiplication" operations with these properties, so mathematicians have decided to give a name to these objects: a ring.
    [Show full text]
  • Definition and Examples of Rings 50
    LECTURE 14 Definition and Examples of Rings Definition 14.1. A ring is a nonempty set R equipped with two operations ⊕ and ⊗ (more typically denoted as addition and multiplication) that satisfy the following conditions. For all a; b; c 2 R: (1) If a 2 R and b 2 R, then a ⊕ b 2 R. (2) a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c (3) a ⊕ b = b ⊕ a (4) There is an element 0R in R such that a ⊕ 0R = a ; 8 a 2 R: (5) For each a 2 R, the equation a ⊕ x = 0R has a solution in R. (6) If a 2 R, and b 2 R, then ab 2 R. (7) a ⊗ (b ⊗ c) = (a ⊗ b) ⊗ c. (8) a ⊗ (b ⊕ c) = (a ⊗ b) ⊕ (b ⊗ c) Definition 14.2. A commutative ring is a ring R such that (14.1) a ⊗ b = b ⊗ a ; 8 a; b 2 R: Definition 14.3. A ring with identity is a ring R that contains an element 1R such that (14.2) a ⊗ 1R = 1R ⊗ a = a ; 8 a 2 R: Let us continue with our discussion of examples of rings. Example 1. Z, Q, R, and C are all commutative rings with identity. Example 2. Let I denote an interval on the real line and let R denote the set of continuous functions f : I ! R. R can be given the structure of a commutative ring with identity by setting [f ⊕ g](x) = f(x) + g(x) [f ⊗ g](x) = f(x)g(x) 0R ≡ function with constant value 0 1R ≡ function with constant value 1 and then verifying that properties (1)-(10) hold.
    [Show full text]
  • Worksheet 1/22. Math 110, Spring 2014. Solutions
    Worksheet 1/22. Math 110, Spring 2014. Solutions These problems are intended as supplementary material to the homework exercises and will hopefully give you some more practice with actual examples. In particular, they may be easier/harder than homework. Send me an email if you have any questions! Vector spaces; subspaces 1. Which of the following sets V are vector spaces over R (Note: this is changed from the worksheet handed out in discussion.), with the given vector addition and scalar multiplica- tion? You will need to check all of the axioms! (Sorry...) 3 3 i) V = f(x, y, z) 2 R j x + y + z = 0g ⊂ R , with the `usual' addition of vectors and 3 scalar multiplication inherited from R . 3 3 ii) V = f(x, y, z) 2 R j x + y + z = πg ⊂ R , with the `usual' addition of vectors and 3 scalar multiplication inherited from R . 0 1 iii) Let B = and 0 0 V = f2 × 2 matrices A with real entries j AB = 0 (the zero 2 × 2 matrix)g ⊂ Mat2(R), with the `usual' addition of vectors and scalar multiplication inherited from Mat2(R), the set of all 2 × 2 matrices with real entries. 3 3 iv) (Tricky!) V = f(1, y, z) 2 R g ⊂ R , where we define vector addition as (1, y, z) + (1, u, v) = (1, y + u, z + v), and scalar multiplication as c · (1, y, z) = (1, cy, cz), c 2 R. Note: For this example you will need to say what an appropriate zero vector in V should be.
    [Show full text]