has the desired form. On the other hand, its product with z is 1. So the inverse x


 Beatrice Walton
 2 years ago
 Views:
Transcription
1 First homework ssignment p. 5 Exercise. Verify tht the set of complex numers of the form x + y 2, where x nd y re rtionl, is sufield of the field of complex numers. Solution: Evidently, this set contins 0 nd. It is lso esy to check tht it is closed under the ddition nd multipliction. Then most of the xioms of the field follows directly from the corresponding properties of complex numers. The nontrivil prt of this exercise is to prove tht for ny element z = x + y 2, where x nd y re rtionl, its inverse lso hs this form. Consider n element z = x y 2. Then the product zz = x 2 2y 2 z is rtionl. Hence, the numer x 2 2y 2 hs the desired form. On the other hnd, its product with z is. So the inverse x is given y the formul y x 2 2y 2 x 2 2y 2. 2 Exercise 3. Are the following two systems of liner equtions equivlent? If so, express ech eqution in ech system s liner comintion of the equtions in the other system. x + x 2 + 4x 3 = 0 x + 3x 2 + 8x 3 = x 2 + 5x 2 3 = 0 x x 3 = 0 x 2 +3x 3 = 0 Solution: For instnce, let us express the first eqution of the first system s the liner comintion of the equtions of the second system. Use the method of undetermined coefficients: x + x 2 + 4x 3 = (x x 3 ) + (x 2 + x 3 ), where, re the numers tht we wnt to find. Compring coefficients with x in oth sides, one gets =, nd compring coefficients with x 2, one gets =. Then it is esy to check tht the coefficients with x 3 in oth sides coincide. Hence, the first eqution of the first system is the liner comintion of the equtions of the second system with coefficients nd. To complete the solution one just need to pply the sme method nd express the other equtions of the first system vi the equtions of the second one, nd vice vers. Exercise 5 Let F e set which contins exctly two elements, 0 nd. Define n ddition nd multipliction y the tles:
2 Verify tht the set F, together with these two opertions, is field. Solution The solution ws explined in the first recittion. Exercise Prove tht ech sufield of the complex numers contins every rtionl numer. Solution: Let F e sufield of C. Since F is field it must contin distinguished element 0 which plys the role of the dditive identity. But this element is unique (regrded s complex numer) therefore 0 = 0, where 0 is the usul 0 of the complex numers. Therefore 0 F. Anlogously we my conclude tht F. But if F then we must hve tht + = 2 F (notice tht + = 2 since the ddition inside of F is the sme ddition s the usul ddition of complex numers). Using induction we conclude tht N F (if n F then since ddition is closed n + F ). But F must contin ll the dditive inverses of ll its elements, in prticulr, since N F we must hve Z F. Also, since F is field, it must contin ll multiplictive inverses of its elements, therefore Z := { n Z} F. Now let with, Z, 0 e ny rtionl numer, we n know Z F nd F therefore = F. Tht is, F contins every Z rtionl numer. pp. 0 Exercise 2. If 3 2 A = find ll solutions of AX = 0 y rowreducing A. Solution: Denote with S(A, B) the opertion of swpping rows A nd B. And denote with c A + B the opertion of multiplying row A times c nd dding it to row B S(I,III) 2 2 I+II 0 3 I+III II II+III III 6 0 0
3 Therefore the only solution of AX = 0 is X = 0. Exercise 3. If A = find ll solutions of AX = 2X nd ll solutions of AX = 3X. Solution Notice tht the eqution AX = 2X is equivlent to the eqution AX 2X = 0 which is equivlent to AX 2IX = 0 where I is the 3 3 identity mtrix. This lst eqution cn e fctored s (A 2I)X = 0. Writing explicitly A 2I we get A 2I = which is row equivlent to thus the solutions re the elements in the set{(x, x 2, x 3 ) x = x 2 = x 3 } Anlogously for the system AX = 3X we get tht the solutions re the elements in the set {(x, x 2, x 3 ) x = x 2 = 0}. Exercise 8. Consider the system of equtions AX = 0 where ( ) A = c d is 2 2 mtrix over the field F. Prove the following. () If every entry of A is 0, then every pir (x, x 2 ) is solution of AX = 0. Solution: 0 x + 0 x 2 = 0. () If d c 0, then the system AX = 0 hs only the trivil solution x = x 2 = 0. Solution I: Define A in the following wy: A = d c ( d ) c Notice tht A A = I therefore, if AX = 0 then AA X = 0, i.e. IX = X = 0 Solution II: Since t this point we re not supposed to know how to multiply mtrices let us work horrily complicted solution. goes y cses: The proof 3
4 . = 0 in this cse, since d c 0 we know tht c 0 therefore 0 nd c 0. Therefore we cn perform the following reduction: 0 S(I,II) c d ( ) c I d c II d d c c II+I 0 c d Thus AX = 0 nd IX = 0 hve the sme solutions, ut the ltter system only hs the trivil solution In this cse we cn perform the following reduction: ( ) I ci+ii d c II c d c d 0 d c II+I Thus AX = 0 nd IX = 0 hve the sme solutions, ut the ltter system only hs the trivil solution. (c) If d c = 0 nd some entry of A is different from 0, then there is solution (x 0, x 0 2) such tht (x, x 2 ) is solution if nd only if there is sclr y such tht x = yx 0, x 2 = yx 0 2. Solution: Suppose tht is the nonzero entry. Since d c = 0 we know tht d = c. Thus we cn perform the following reduction of A:: ( ) I ci+ii c d c d 0 0 Thus AX = 0 if nd only if x = x 2, letting x 0 = nd x0 2 = we otin wht we wnt. An nlogous rgument cn e given if we ssume ny other of the entries to e different from 0. pp. 56 Exercise 3. Descrie explicitly ll 2 2 rowreduced echelon mtrices. Solution: The first element in the first row of rowreduced echelon mtrix is either or 0. In either cse the first element in the second row is 0. The second entry in the second row is gin either or 0. However, the former is possile only if the first entry in the first row is not 0. So there re 3 possiilities to fill out ll entries except the upper right: ( ) x x 0 x,, In the first cse, x is necessrily 0, in the third cse x is either or 0, while in the second cse x cn e ritrry. Thus n infinite fmily of 2 2 rowreduced echelon mtrices prmeterized y n element c F ( ) c 0 0 4
5 nd three other 2 2 rowreduced echelon mtrices ( ) ,, exhust ll possiilities. Exercise. Find ll solutions of 2x 3x 2 x 3 + 5x 4 + 2x 5 = 2 x 2x 2 4x 3 + 3x 4 + x 5 = 2 2x 4x 3 + 2x 4 + x 5 = 3 x 5x 2 x 3 + 6x 4 + 2x 5 =. Solution: Consider the ugmented mtrix of this system It is esy to check tht the sum of the first nd the second rows is equl to the sum of the third nd the fourth rows. Hence, the fourth row is the liner comintion of the other rows nd cn e eliminted. Now sutrct the second row times 2 from the first nd the second rows. We get Interchnge the first nd the second rows , nd sutrct the second row times 4 from the third one Consider the system corresponding to the ltter mtrix x 2x 2 4x 3 +3x 4 +x 5 = 2 x 2 +x 3 x 4 = x 5 = 5
6 We get tht x 5 =, x 2 = x 3 +x 4 +2, x = 2x 2 +4x 3 3x 4 3. Hence, for ny choice of (x 3, x 4 ) there is unique pir (x, x 2 ) such tht (x, x 2, x 3, x 4, ) stisfy the system. Then the solutions re prmeterized y two elements, F : x = + 2 x 2 = 2 + x 3 = x 4 = x 5 =. Exercise 0. Suppose R nd R re 2 3 rowreduced echelon mtrices nd tht the systems RX = 0 nd R X = 0 hve exctly the sme solutions. Prove tht R = R. Solution: The min ide is to prove tht ech row of ech mtrix is liner comintion of the rows of the other mtrix. First, suppose tht one of the mtrices R, R hs one zero row. Then the solutions of the corresponding system depend on 2 prmeters. Hence, the other mtrix lso hs zero row, nd the second rows of R, R coincide (they re oth zero). Consider 2 3mtrix A whose first row is the first row of R nd whose second row is the first row of R. The corresponding system gin hs the sme 2prmeter fmily of solutions s the systems RX = 0, R X = 0. Thus rowreduced echelon mtrix equivlent to A should hve one zero row. This is possile only if the rows of the mtrix A coincide (since oth rows of A hve leding coefficients ). Now suppose tht ll rows of mtrices R, R re nonzero. Then the solutions of the corresponding system depend on prmeter. Form 3 4mtrix B whose first two rows re the rows of R nd whose lst two rows re the rows of R. Then B should e row equivlent to mtrix with two zero rows (otherwise the system BX = 0 would not hve prmeter fmily of solutions). Hence, the rows of of R cn e represented s liner comintions of the rows of R nd vice vers. This is possile only if the leding coefficients of their first rows occur t the sme plce. Indeed, if for instnce, the first row R of R strts with more zeros thn the first row of R, then so do the second row R 2 of R nd ny liner comintion of R, R 2. Using similr rguments one cn prove tht the leding coefficients of the second rows of R, R occur t the sme plce. Then it is esy to see tht the only wy for row of R to e liner comintion of the rows of R is to coincide with the respective row of R. 6
and thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationAssuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;
B26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndomnumer genertor supplied s stndrd with ll computer systems Stn KellyBootle,
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationMath 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationFormal Languages and Automata Exam
Forml Lnguges nd Automt Exm Fculty of Computers & Informtion Deprtment: Computer Science Grde: Third Course code: CSC 34 Totl Mrk: 8 Dte: 23//2 Time: 3 hours Answer the following questions: ) Consider
More information1.2 The Integers and Rational Numbers
.2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationA new algorithm for generating Pythagorean triples
A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf
More informationMultiplication and Division  Left to Right. Addition and Subtraction  Left to Right.
Order of Opertions r of Opertions Alger P lese Prenthesis  Do ll grouped opertions first. E cuse Eponents  Second M D er Multipliction nd Division  Left to Right. A unt S hniqu Addition nd Sutrction
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions Section 3 Integrl Equtions Integrl Opertors nd Liner Integrl Equtions As we sw in Section on opertor nottion, we work with functions
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationLecture 2: Matrix Algebra. General
Lecture 2: Mtrix Algebr Generl Definitions Algebric Opertions Vector Spces, Liner Independence nd Rnk of Mtrix Inverse Mtrix Liner Eqution Systems, the Inverse Mtrix nd Crmer s Rule Chrcteristic Roots
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting
More informationPentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simplelooking set of objects through which some powerful
Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationFUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation
FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationCalculus of variations with fractional derivatives and fractional integrals
Anis do CNMAC v.2 ISSN 1984820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810193 Aveiro, Portugl
More informationMA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!
MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more
More informationRational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and
Rtionl Functions Rtionl unctions re the rtio o two polynomil unctions. They cn be written in expnded orm s ( ( P x x + x + + x+ Qx bx b x bx b n n 1 n n 1 1 0 m m 1 m + m 1 + + m + 0 Exmples o rtionl unctions
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics. W02D3_0 Group Problem: Pulleys and Ropes Constraint Conditions
MSSCHUSES INSIUE OF ECHNOLOGY Deprtment of hysics 8.0 W02D3_0 Group roblem: ulleys nd Ropes Constrint Conditions Consider the rrngement of pulleys nd blocks shown in the figure. he pulleys re ssumed mssless
More informationVersion 001 CIRCUITS holland (1290) 1
Version CRCUTS hollnd (9) This printout should hve questions Multiplechoice questions my continue on the next column or pge find ll choices efore nswering AP M 99 MC points The power dissipted in wire
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More information1. Description of Linear Prediction
Liner Prediction nd LevinsonDurbin lgorithm Cedric Collomb http://ccollomb.free.fr/ Copyright 9. ll ights eserved. Creted: Februry 3, 9 Lst Modified: ovember, 9 Contents. Description of Liner Prediction....
More informationAnswer, Key Homework 10 David McIntyre 1
Answer, Key Homework 10 Dvid McIntyre 1 This printout should hve 22 questions, check tht it is complete. Multiplechoice questions my continue on the next column or pge: find ll choices efore mking your
More informationPHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
More information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More informationLec 2: Gates and Logic
Lec 2: Gtes nd Logic Kvit Bl CS 34, Fll 28 Computer Science Cornell University Announcements Clss newsgroup creted Posted on wepge Use it for prtner finding First ssignment is to find prtners Due this
More informationTHE RATIONAL NUMBERS CHAPTER
CHAPTER THE RATIONAL NUMBERS When divided by b is not n integer, the quotient is frction.the Bbylonins, who used number system bsed on 60, epressed the quotients: 0 8 s 0 60 insted of 8 s 7 60,600 0 insted
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationMATLAB Workshop 13  Linear Systems of Equations
MATLAB: Workshop  Liner Systems of Equtions pge MATLAB Workshop  Liner Systems of Equtions Objectives: Crete script to solve commonly occurring problem in engineering: liner systems of equtions. MATLAB
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More informationArithmetic Sequences
Arithmetic equeces A simple wy to geerte sequece is to strt with umber, d dd to it fixed costt d, over d over gi. This type of sequece is clled rithmetic sequece. Defiitio: A rithmetic sequece is sequece
More informationMatrix Inverse and Condition
Mtrix Inverse nd Condition Berlin Chen Deprtment of Computer Science & Informtion Engineering Ntionl Tiwn Norml University Reference: 1. Applied Numericl Methods with MATLAB for Engineers, Chpter 11 &
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationVariable Dry Run (for Python)
Vrile Dr Run (for Pthon) Age group: Ailities ssumed: Time: Size of group: Focus Vriles Assignment Sequencing Progrmming 7 dult Ver simple progrmming, sic understnding of ssignment nd vriles 2050 minutes
More information19. The FermatEuler Prime Number Theorem
19. The FermtEuler Prime Number Theorem Every prime number of the form 4n 1 cn be written s sum of two squres in only one wy (side from the order of the summnds). This fmous theorem ws discovered bout
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More information200506 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 256 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationNovel Methods of Generating SelfInvertible Matrix for Hill Cipher Algorithm
Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy Novel Methods of Generting SelfInvertible Mtrix for Hill Cipher lgorithm Bibhudendr chry Deprtment of Electronics & Communiction Engineering
More informationA Note on Complement of Trapezoidal Fuzzy Numbers Using the αcut Method
Interntionl Journl of Applictions of Fuzzy Sets nd Artificil Intelligence ISSN  Vol.  A Note on Complement of Trpezoidl Fuzzy Numers Using the αcut Method D. Stephen Dingr K. Jivgn PG nd Reserch Deprtment
More informationDouble Integrals over General Regions
Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing
More informationOne Minute To Learn Programming: Finite Automata
Gret Theoreticl Ides In Computer Science Steven Rudich CS 15251 Spring 2005 Lecture 9 Fe 8 2005 Crnegie Mellon University One Minute To Lern Progrmming: Finite Automt Let me tech you progrmming lnguge
More informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationCS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationFAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University
SYSTEM FAULT AND Hrry G. Kwtny Deprtment of Mechnicl Engineering & Mechnics Drexel University OUTLINE SYSTEM RBD Definition RBDs nd Fult Trees System Structure Structure Functions Pths nd Cutsets Reliility
More informationLaplace s Equation on a Disc
LECTURE 15 Lplce s Eqution on Disc Lst time we solved the Diichlet poblem fo Lplce s eqution on ectngul egion. Tody we ll look t the coesponding Diichlet poblem fo disc. Thus, we conside disc of dius 1
More informationFORMAL LANGUAGES, AUTOMATA AND THEORY OF COMPUTATION EXERCISES ON REGULAR LANGUAGES
FORMAL LANGUAGES, AUTOMATA AND THEORY OF COMPUTATION EXERCISES ON REGULAR LANGUAGES Introduction This compendium contins exercises out regulr lnguges for the course Forml Lnguges, Automt nd Theory of Computtion
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls : The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N):  counting numers. {,,,,, } Whole Numers (W):  counting numers with 0. {0,,,,,, } Integers (I): 
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More information1. In the Bohr model, compare the magnitudes of the electron s kinetic and potential energies in orbit. What does this imply?
Assignment 3: Bohr s model nd lser fundmentls 1. In the Bohr model, compre the mgnitudes of the electron s kinetic nd potentil energies in orit. Wht does this imply? When n electron moves in n orit, the
More informationPure C4. Revision Notes
Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd
More informationVector differentiation. Chapters 6, 7
Chpter 2 Vectors Courtesy NASA/JPLCltech Summry (see exmples in Hw 1, 2, 3) Circ 1900 A.D., J. Willird Gis invented useful comintion of mgnitude nd direction clled vectors nd their higherdimensionl counterprts
More informationCOMPLEX FRACTIONS. section. Simplifying Complex Fractions
58 (66) Chpter 6 Rtionl Epressions undles tht they cn ttch while working together for 0 hours. 00 600 6 FIGURE FOR EXERCISE 9 95. Selling. George sells one gzine suscription every 0 inutes, wheres Theres
More informationTwo hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00
COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationDATABASDESIGN FÖR INGENJÖRER  1056F
DATABASDESIGN FÖR INGENJÖRER  06F Sommr 00 En introuktionskurs i tssystem http://user.it.uu.se/~ul/tsommr0/ lt. http://www.it.uu.se/eu/course/homepge/esign/st0/ Kjell Orsorn (Rusln Fomkin) Uppsl Dtse
More informationBabylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity
Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationVolumes of solids of revolution
Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the xxis. There is strightforwrd technique which enbles this to be done, using
More informationFinite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh
Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is
More informationUnambiguous Recognizable Twodimensional Languages
Unmbiguous Recognizble Twodimensionl Lnguges Mrcell Anselmo, Dor Gimmrresi, Mri Mdoni, Antonio Restivo (Univ. of Slerno, Univ. Rom Tor Vergt, Univ. of Ctni, Univ. of Plermo) W2DL, My 26 REC fmily I REC
More informationRepeated multiplication is represented using exponential notation, for example:
Appedix A: The Lws of Expoets Expoets re shorthd ottio used to represet my fctors multiplied together All of the rules for mipultig expoets my be deduced from the lws of multiplictio d divisio tht you
More information0.1 Basic Set Theory and Interval Notation
0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is welldefined
More information2.016 Hydrodynamics Prof. A.H. Techet
.01 Hydrodynics Reding #.01 Hydrodynics Prof. A.H. Techet Added Mss For the cse of unstedy otion of bodies underwter or unstedy flow round objects, we ust consider the dditionl effect (force) resulting
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationLinear Equations in Two Variables
Liner Equtions in Two Vribles In this chpter, we ll use the geometry of lines to help us solve equtions. Liner equtions in two vribles If, b, ndr re rel numbers (nd if nd b re not both equl to 0) then
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More information