MATLAB Workshop 13  Linear Systems of Equations


 Berniece Robinson
 1 years ago
 Views:
Transcription
1 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 Fetures: colon (:) opertor  specil vector mtrix opertor Nottion Action Result :c cretes vector with elements prt [ c] :b:c cretes vector with elements b prt [ +b +2b... ( c)] A(m,:) selects m th row of A A(:,j) selects j th column of A A(m:n,:) selects m th through n th rows of A A(:,j:k) selects j th through k th columns of A A(m:n,j:k) selects m th through n th rows of A nd j th through k th columns of A mtrix functions number of rows nd columns workspce commnd Commnd vrnme = lod(filenme) [rows, cols] = size(x) Action retrieves the contents of file s ASCII text nd ssigns the contents to vrnme. filenme is string tht specifies the nme nd loction of the file. if the file is not in the current directory, the entire pth must be specified.
2 MATLAB: Workshop  Liner Systems of Equtions pge 2 Liner systems of equtions nd mtrix lgebr Mtrix lgebr provides convenient shorthnd nottion for liner systems of equtions. Consider the system of equtions = x + 2 x2 + x b 2x + 22 x2 + 2x b2 x + 2 x2 + x b = = where the ij represent the coefficients of the unknown vribles x i. This system of equtions cn be written in mtrix lgebr nottion s [ A ] N x N ( X ) N x = ( B) N x where [A] NxN is squre (sme number of columns s rows) mtrix of coefficients, (X) Nx is column vector of unknowns, nd (B) Nx is the forcing vector (lso column vector). Note tht the multipliction is defined within the rules of mtrix lgebr., i.e., n (NxN) mtrix times n (Nx) vector yields n (Nx) vector. In expnded nottion, the mtrix lgebr eqution cn be written s x x x 2 b = b b 2 You cn verify tht pplying the rules of mtrix multipliction to the bove mtrix lgebr eqution reproduces the originl set of equtions. Note tht the coefficient mtrix is obtined by tking the coefficients of ech eqution nd plcing them in the corresponding row. The following exmple illustrtes the use of mtrix lgebr for liner systems of equtions. The set of liner equtions 2 x + 47 y 2z = 25 5 x + 2z = 46 6 x 6y + 5z = 225 cn be written in mtrix lgebr form s x 25 2 y = 46 5 z 225 Note tht one of the coefficients in the second eqution is zero. This needs to be noted explicitly in the coefficient mtrix.
3 MATLAB: Workshop  Liner Systems of Equtions pge A Civil Engineering problem Liner systems of equtions rise frequently in every engineering discipline. For exmple, consider the following problem tht might rise in rod construction project. A civil engineer needs to purchse 85 tons of rod mteril consisting of 25 tons of grvel, 40 tons of pebbles, nd 20 tons of stones. She contcts three locl qurries nd obtins the following specifictions on their product mix. Qurry, wt% Product Mix 2 Grvel Pebbles Stones None of the qurries individully hs product vilble in the percentges tht re required for the project. So the next question she sks is whether she cn meet her requirements through buying some product from ech qurry. To do this, she sets up mteril blnces for the mounts of grvel, pebbles nd stones she requires. If she purchses Q tons from Qurry, Q 2 tons from Qurry 2, nd Q tons from Qurry, then mteril blnce for grvel, obtined by multiplying the frction of grvel from qurry by the totl tons to be purchsed from the qurry, becomes ( 0.40) Q + (0.20) Q2 + (0.00) Q = 25 where the totl mount of grvel purchsed from the three qurries must be equl to the mount needed for the project. Likewise, mteril blnces for pebbles nd stones yield the equtions ( 0.50) Q + (0.25) Q2 + (0.50) Q 0.0) Q + (0.55) Q + (0.50) Q = ( 2 = These equtions cn be put into mtrix lgebr form s Q 0.50 Q 0.50 Q 2 = MATLAB cn be used to solve this system of equtions for Q, Q 2, nd Q. Solution of liner equtions in MATLAB Solution of set of liner equtions in MATLAB is rther strightforwrd. In mtrix lgebr terminology the solution to the set of liner equtions [ A ]( x) = ( b) where A is the coefficient mtrix, x is the vector of unknowns, nd b is the right hnd side vector is given by (in MATLAB terminology)» x = A\b Tht is, the bckslsh opertor, \, is used to find the solution. () Solving system of liner equtions in MATLAB.
4 MATLAB: Workshop  Liner Systems of Equtions pge 4 To solve system of liner equtions, one needs the coefficient mtrix, the right hnd side vector, nd the bckslsh opertor, \. To find the solution to the qurry problem, enter the following t the commnd prompt:» Coeff = [0.40, 0.20, 0.00; 0.50, 0.25, 0.50; 0.0, 0.55, 0.50];» mount = [25; 40; 20];» Q = Coeff\mount Q = The first line entered the coefficient mtrix, Coeff, by rows. The right hnd side vector, mount, ws entered s column vector. This is very importnt. Wht hppens if row vector is used insted? Try it! The solution vector, Q, ws obtined by using the bckslsh opertor. Note tht Q is column vector, s would be expected (why?). Creting n ugmented coefficient mtrix file While solution of liner systems of equtions is rther strightforwrd in MATLAB, doing so in the Commnd Window hs couple of drwbcks. The first drwbck is in entering the coefficient mtrix. This is rther esy for system of three equtions. However, systems of liner equtions frequently consist of hundreds of equtions. Entering the coefficient mtrix in the Commnd Window for such lrge system is not only tedious, but is frught with the potentil for error. Wht would hppen if one coefficient in the 0,000 required coefficients for system of 00 equtions is entered incorrectly? Would you even ctch such n error? Similr issues re encountered with entering the right hnd side vector in the Commnd Window. To void these problems, engineers (nd others) typiclly crete text files with the coefficients. Rther thn hve two text files for problem (one contining the coefficients nd the other contining the right hnd side vector), the two sets of numbers re generlly combined into single file clled the ugmented coefficient mtrix file. The ugmented coefficient mtrix file contins the coefficient mtrix in rows nd columns with the right hnd side vector ppended s the (n+) th column, where n is the number of equtions in the system. For exmple, the ugmented coefficient mtrix for the qurry problem would look like where the first three rows nd columns re the coefficient mtrix nd the fourth (lst) column is the right hnd side vector. A file contining these numbers hs ll of the informtion necessry to get find solution to the set of equtions. (2) Creting text file with the ugmented coefficient mtrix. There re severl methods to crete n ugmented coefficient mtrix file. You could use your fvorite text editor such s Notepd or Wordpd (these will utomticlly sve s n ASCII text file). If you use word processor, such s Word, you will need to specify tht the file is to be sved s ASCII text when you sve it. You could lso use spredsheet, such s Excel, to crete the file. Using spredsheet hs the dvntge tht ech coefficient cn be loded into seprte cell. You could then export the file s ASCII text.
5 MATLAB: Workshop  Liner Systems of Equtions pge 5 Pick method nd crete n ASCII file with the qurry problem coefficients. Sve it s qurry.dt in your ctive MATLAB directory. The.dt extension for file is frequently used to denote n ASCII file tht contins dt. Loding n ugmented file into the Workspce An ugmented coefficient mtrix is esily loded into the MATLAB Workspce by using vrition of the lod commnd described in Workshop 4. The prticulr form of the commnd to lod dt file is vrnme = lod(filenme); where vrnme is the nme ssigned to hold the ugmented coefficient mtrix vlues nd filenme is string tht identifies the nme nd loction of the file. () Loding text file contining n ugmented coefficient mtrix. Lod the ugmented coefficient mtrix vlues into the Workspce by entering» ugmt = lod('qurry.dt') ugmt = Why did the vlues disply? How would you suppress disply? Notes on filenme filenme cn be hrdwired s shown bove (not very useful). filenme cn be vrible nme tht contins string s shown here: fnme = 'qurry.dt'; ugmt = lod(fnme); or, better yet, fnme = input('plese enter ugmented mtrix file nme ==> ','s'); ugmt = lod(fnme); If the required file is locted in the current directory, only the file nme (with extension) is required. If the required file is not in the current directory, the entire pth must be specified, i.e., c:\temp\qurry.dt. Extrcting the coefficient nd right hnd side vector from the ugmented mtrix Once the ugmented coefficient mtrix is in the Workspce, we need to extrct the coefficient mtrix nd the right hnd side vector. Although we hve been working with system of three equtions nd three unknowns, in generl, system of liner equtions my be of ny size. Fortuntely, MATLAB provides the tools to hndle this. (4) Extrcting the coefficient mtrix nd right hnd side vector. Determine the size of the system of equtions by entering» [nrows, ncols] = size(ugmt) nrows =
6 MATLAB: Workshop  Liner Systems of Equtions pge 6 ncols = 4 The number of rows, nrows, tells the number of equtions in the ugmented coefficient mtrix. ncols should be one greter thn nrows. Complete the following MATLAB commnds to extrct the coefficient mtrix nd right hnd side vector from the ugmented mtrix» Coeff = ugmt(????) Coeff = » rhs = ugmt(????) rhs = Refer to Workshop 2 if necessry. Crete script for solving systems of equtions Systems of liner equtions occur so frequently in engineering tht hving user friendly script for their solution would be useful. Remember the generl prdigm for script development: () get informtion (2) do computtions () report results. Becuse MATLAB cn solve system of liner equtions with single ssignment sttement, designing function to get the necessry informtion (i.e., problem nme, coefficient mtrix nd right hnd side vector) is the most complex prt of designing the script. The output function design is lso rther strightforwrd. (5) Design n informtion input function for liner eqution solver script. Following the design methodology of the previous workshops, design function tht will ) sk the user for the nme of the problem to be solved 2) sk the user for the nme/loction of the ugmented coefficient mtrix file ) lod the ugmented coefficient mtrix file 4) extrct the coefficient mtrix 5) extrct the rhs vector The function should return the problem nme, coefficient mtrix, nd rhs vector (6) Solve liner system of equtions. Becuse MATLAB solves system of liner equtions with single ssignment sttement, there is no need to design seprte function for this tsk. (7) Disply the results.
7 MATLAB: Workshop  Liner Systems of Equtions pge 7 Design function tht will disply the results. To be user friendly, the results from solving the system of liner equtions should hve lbel prior to disply. From where might n pproprite lbel come? How might it be pssed to this function for use? (8) Crete the script lineqsolver. You now hve the requisite functions to crete script tht puts it ll together. Crete script, lineqsolver.m, tht does the following displys script heder informtion (cll to lineqsolver_heder) gets the necessry input informtion (cll to lineqsolver_info) solves the system of equtions (simple ssignment sttement) displys the results (cll to lineqsolver_results) Be sure your script hs n pproprite heder section nd vrible dictionry. Bonus: Add while loop to your script so tht the script sks the user whether to repet for nother system of liner equtions nd, if the nswer is yes, repets the input, clcultion, disply portion of the script. Exercises: Use your script, lineqsolver, to solve the following problems.. Six people, Alice, Brb, Crol, Den, Eric, nd Fred, ech hve bg of money. Find the mount in ech bg if: The totl of ll six bgs is $.56; Crol hs twice s much money s Alice; Brb nd Crol together hve s much s Den; Alice nd Brb together hve s much s Eric; Eric s mount subtrcted from Fred s mount is twice Alice s mount; nd $.5 is left fter subtrcting Brb s mount nd Eric s mount from Fred s mount. 2. Find the fivedigit number tht stisfies the following properties: The sum of ll the digits is 8; The third digit is the sum of the first nd second digits; Subtrcting the third digit from the fourth digit yields 4; The first nd third digits dded together give the fifth digit; nd The first digit is twice the second digit.. Find the sixdigit number tht stisfies the following properties: The first nd third digit sum to 0; The sum of the second, third, nd fifth digits equls the fourth digit; The third nd lst digits re the sme; The sum of the second nd third digit equls the fifth digit; The fourth digit is the sme s the lst digit minus the first digit; nd The sum of ll the digits is Five people, Alice, Ben, Cindy, Den, nd Evn, decide to invest in the stock mrket. Wht is ech person s profit fter one yer if:
8 MATLAB: Workshop  Liner Systems of Equtions pge 8 The sum of ll their profits is $299.25; Evn s profit is three times s lrge s Den s profit; Ben s nd Cindy s profits together totl to Den s profit; Alice nd Cindy together mde $89.20 profit; nd Alice s profit ws $0.55 more thn Den s profit. Recp: You should hve lerned Wht n ugmented coefficient mtrix is nd its reltion to system of liner equtions. How to use the lod commnd to retrieve n ugmented coefficient file. How to use the size commnd to determine the number of equtions. How to extrct the coefficient mtrix nd rhs vector from the ugmented coefficient mtrix. How to use the bckslsh opertor, \, to solve system of equtions.
In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.
Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix
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 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 informationand 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 informationMATLAB: Mfiles; Numerical Integration Last revised : March, 2003
MATLAB: Mfiles; Numericl Integrtion Lst revised : Mrch, 00 Introduction to Mfiles In this tutoril we lern the bsics of working with Mfiles in MATLAB, so clled becuse they must use.m for their filenme
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 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 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 informationChapter 6 Solving equations
Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign
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 informationQuadratic Functions. Analyze and describe the characteristics of quadratic functions
Section.3  Properties of rphs of Qudrtic Functions Specific Curriculum Outcomes covered C3 Anlyze nd describe the chrcteristics of qudrtic functions C3 Solve problems involving qudrtic equtions F Anlyze
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 informationAddition and subtraction of rational expressions
Lecture 5. Addition nd subtrction of rtionl expressions Two rtionl expressions in generl hve different denomintors, therefore if you wnt to dd or subtrct them you need to equte the denomintors first. The
More informationTests for One Poisson Mean
Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution
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 informationhas the desired form. On the other hand, its product with z is 1. So the inverse x
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
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 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 informationNet Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
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 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 informationExponents base exponent power exponentiation
Exonents We hve seen counting s reeted successors ddition s reeted counting multiliction s reeted ddition so it is nturl to sk wht we would get by reeting multiliction. For exmle, suose we reetedly multily
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 informationQuadratic Equations. Math 99 N1 Chapter 8
Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More information4.0 5Minute Review: Rational Functions
mth 130 dy 4: working with limits 1 40 5Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two
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 information2 If a branch is prime, no other factors
Chpter 2 Multiples, nd primes 59 Find the prime of 50 by drwing fctor tree. b Write 50 s product of its prime. 1 Find fctor pir of the given 50 number nd begin the fctor tree (50 = 5 10). 5 10 2 If brnch
More informationMatrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA
CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the
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 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 informationEducation Spending (in billions of dollars) Use the distributive property.
0 CHAPTER Review of the Rel Number System 96. An pproximtion of federl spending on eduction in billions of dollrs from 200 through 2005 cn be obtined using the e xpression y = 9.0499x  8,07.87, where
More informationSequences and Series
Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationGeneralized Inverses: How to Invert a NonInvertible Matrix
Generlized Inverses: How to Invert NonInvertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
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 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 informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology
More information1.00/1.001 Introduction to Computers and Engineering Problem Solving Fall 2011  Final Exam
1./1.1 Introduction to Computers nd Engineering Problem Solving Fll 211  Finl Exm Nme: MIT Emil: TA: Section: You hve 3 hours to complete this exm. In ll questions, you should ssume tht ll necessry pckges
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 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 informationDETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.
Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of
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 informationQuadratic Equations  1
Alger Module A60 Qudrtic Equtions  1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions  1 Sttement of Prerequisite
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
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 informationCHAPTER 5a. SIMULTANEOUS LINEAR EQUATIONS
CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering by Dr. Ibrhim A. Asskkf Spring 00 ENCE 0  Computtion Methods in Civil Engineering
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
More informationChapter 9: Quadratic Equations
Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.
More informationRational Expressions
C H A P T E R Rtionl Epressions nformtion is everywhere in the newsppers nd mgzines we red, the televisions we wtch, nd the computers we use. And I now people re tlking bout the Informtion Superhighwy,
More informationUniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More informationNotes for Thurs 8 Sept Calculus II Fall 2005 New York University Instructor: Tyler Neylon Scribe: Kelsey Williams
Notes for Thurs 8 Sept Clculus II Fll 00 New York University Instructor: Tyler Neylon Scribe: Kelsey Willims 8. Integrtion by Prts This section is primrily bout the formul u dv = uv v ( ) which is essentilly
More informationExponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.
Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:
More informationName: Lab Partner: Section:
Chpter 4 Newton s 2 nd Lw Nme: Lb Prtner: Section: 4.1 Purpose In this experiment, Newton s 2 nd lw will be investigted. 4.2 Introduction How does n object chnge its motion when force is pplied? A force
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 information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationSmall Business Networking
Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology
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 informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationLecture 3 Basic Probability and Statistics
Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The
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 informationSection 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables
The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long
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 informationMath Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
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 informationIntroduction to Linear Algebra using MATLAB Tutorial on Material Covered in ENG EK 127 Relevant to Linear Algebra.
Introduction to Liner Algebr using MATLAB Tutoril on Mteril Covered in ENG EK 7 Relevnt to Liner Algebr By Stormy Attwy Reference: Stormy Attwy, MATLAB: A Prcticl Introduction to Progrmming nd Problem
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 informationOn the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding
Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl
More informationContent Objectives: After completing the activity, students will gain experience of informally proving Pythagoras Theorem
Pythgors Theorem S Topic 1 Level: Key Stge 3 Dimension: Mesures, Shpe nd Spce Module: Lerning Geometry through Deductive Approch Unit: Pythgors Theorem Student ility: Averge Content Ojectives: After completing
More informationFor the Final Exam, you will need to be able to:
Mth B Elementry Algebr Spring 0 Finl Em Study Guide The em is on Wednesdy, My 0 th from 7:00pm 9:0pm. You re lloed scientific clcultor nd " by 6" inde crd for notes. On your inde crd be sure to rite ny
More information10.3 Systems of Linear Equations: Determinants
758 CHAPTER 10 Systems of Equtions nd Inequlities 10.3 Systems of Liner Equtions: Determinnts OBJECTIVES 1 Evlute 2 y 2 Determinnts 2 Use Crmer s Rule to Solve System of Two Equtions Contining Two Vriles
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 informationNumber Systems & Working With Numbers
Presenting the Mths Lectures! Your best bet for Qunt... MATHS LECTURE # 0 Number Systems & Working With Numbers System of numbers.3 0.6 π With the help of tree digrm, numbers cn be clssified s follows
More information11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
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 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 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 informationRational Numbers  Grade 10 [CAPS]
OpenStxCNX module: m848 Rtionl Numers  Grde 0 [CAPS] Free High School Science Texts Project Bsed on Rtionl Numers y Rory Adms Free High School Science Texts Project Mrk Horner Hether Willims This work
More informationAntiSpyware Enterprise Module 8.5
AntiSpywre Enterprise Module 8.5 Product Guide Aout the AntiSpywre Enterprise Module The McAfee AntiSpywre Enterprise Module 8.5 is n ddon to the VirusScn Enterprise 8.5i product tht extends its ility
More informationArc Length. P i 1 P i (1) L = lim. i=1
Arc Length Suppose tht curve C is defined by the eqution y = f(x), where f is continuous nd x b. We obtin polygonl pproximtion to C by dividing the intervl [, b] into n subintervls with endpoints x, x,...,x
More informationSmall Business Networking
Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer
More informationg(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany
Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required
More informationSirindhorn International Institute of Technology Thammasat University at Rangsit
Sirindhorn Interntionl Institute of Technology Thmmst University t Rngsit School of Informtion, Computer nd Communiction Technology COURSE : ECS 204 Bsic Electricl Engineering L INSTRUCTOR : Asst. Prof.
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
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 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 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 informationnot to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions
POLYNOMIALS (A) Min Concepts nd Results Geometricl mening of zeroes of polynomil: The zeroes of polynomil p(x) re precisely the xcoordintes of the points where the grph of y = p(x) intersects the xxis.
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationAlgorithms Chapter 4 Recurrences
Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More informationAnswer, Key Homework 8 David McIntyre 1
Answer, Key Homework 8 Dvid McIntyre 1 This printout should hve 17 questions, check tht it is complete. Multiplechoice questions my continue on the net column or pge: find ll choices before mking your
More informationQuick Reference Guide: Onetime Account Update
Quick Reference Guide: Onetime Account Updte How to complete The Quick Reference Guide shows wht existing SingPss users need to do when logging in to the enhnced SingPss service for the first time. 1)
More informationSimplifying Algebraic Expressions
Lesson Objectives Simplify lgebric epressions Vocbulry term (p. 42) coefficient (p. 42) Additionl Emples Emple 1 Identify like terms in the list. 3t 5w 2 7t 9v 4w 2 8v Look for like with like. 3t 5w 2
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 informationPrealgebra 7* In your group consider the following problems:
Prelger * Group Activit # Group Memers: In our group consider the following prolems: 1) If ever person in the room, including the techer, were to shke hnds with ever other person ectl one time, how mn
More informationMathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
More information