Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24


 Reginald Terry
 3 years ago
 Views:
Transcription
1 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the numers re clled entries of the mtrix. Mtrices re usully denoted y uppercse letters:,, nd so on. Hence, 5 6 re mtrices. lerly, mtrices come in vrious shpe depending on the numer of rows nd columns. For exmple, the mtrix shown hs rows nd columns. In generl, mtrix with m rows nd n columns is referred to s n m n mtrix or s hving size m n. hus mtrices,, ove hve sizes,,, respectively. mtrix of size n is clled row mtrix, wheres one of size m is clled column mtrix. Ech entry of mtrix is identified y the row nd column in which it lies. he rows re numered from the top down, nd the columns re numered from left to right. hen the ( i, j)  entry of mtrix is the numer lying simultneously in row i nd column j. For exmple: he (, ) entry of is 5 6 Liner lger I 4
2 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition he (, ) entry of is specil nottion hs een devised for the entries of mtrix. If is n m n mtrix, nd if the ( i, j)  entry is denoted s ij s follows:, then is displyed m m n n mn his is usully denoted simplify s ij. n n n is clled squre mtrix. For squre mtrix, the entries : the min digonl of the mtrix.,, nn, re sid to lie on wo mtrices nd re clled equl ( written = ) if nd only if :. hey hve the sme size. orresponding entries re equl or cn e written s ij ij mens tht ij ij for ll i, j. Exmple Given, c d, 5 6, discuss the possiility tht,, : = is impossile, ecuse nd re of different sizes. Similrly, is impossile. is possile provided tht corresponding entries re equl: c d = mens,, c, d. Liner lger I 5
3 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix ddition If nd re mtrices of the sme size, their sum formed y dding corresponding entries. If ij nd ij is the mtrix, this tke the form: ij ij Note tht ddition is not defined for mtrices of different sizes. Exmple If nd 4, compute! 5 6 = Exmple Find,, c if c c dd the mtrices on the left side to otin: c c ecuse the corresponding entries must e equl, this gives three equtions: c,, c. Solving these yields,, c. he properties of Mtrix ddition If,, re ny mtrices of the sme size, then:. ( commuttive lw ). ( ) ( ) ( ssocitive lw ) he m n mtrix in which every entry is zero is clled the zero mtrix nd is denoted s, hence, Liner lger I 6
4 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition. X X he negtive of n m n mtrix ( written s  ) is defined to e m n mtrix otined y multiply ech entry of y. If ij, this ecomes ij, hence, 4. ( ) for ll mtrices ij where is the zero mtrix of the sme size s. closely relted notion is tht of sutrcting mtrices. If mtrices, their difference ( ), i.e. : is defined y: ij ij ij ij, re two m n Exmple 4,, ompute,, = 5 Exmple 5 Solve + X =, where X is mtrix. X must e mtrix. If X = x u y v, the eqution reds: Liner lger I 7
5 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition = + x u y v = x u y v he rule of mtrix equlity gives x, y, u, v. hus X =. We solve numericl eqution x y sutrcting the numer from oth sides to otin x. his lso works for mtrices. o solve + X =, simply sutrct the mtrix from oth sides to get: X =  Sclr Multipliction In Gussin Elimintion, multiplying row of mtrix y numer k mens multiplying every entry of tht row y k. More generlly, if is ny mtrix nd k is ny numer, the sclr multiple k is the mtrix otined from y multiplying ech entry of y. If ij,this is: k k ij We hve een using rel numers s sclrs, ut we could eqully well hve een using complex numers. Exmple 6 If,nd, compute 5, Liner lger I 8
6 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition 5 = ( ) = If is ny mtrix, note tht k is the sme size s for ll sclrs k. We lso hve: nd k ecuse the zero mtrix hs every entry zero. In other words, k if either k or. he properties of sclr multipliction Let, denote ritrry m n mtrices, where m, n re fixed, let k, l re denote ritrry rel numers. hen :. k ( ) k k. ( k l) k l. ( kl ) k( l) 4.. Exmple 7 Simplify ( ) ( ) [ ( 4) 4( )] where,, re ll mtrices of the sme size. ( ) ( ) [ ( 4) 4( )] rnspose Mny result out mtrix involve the rows of, nd the corresponding result for columns is derived in n nlogous wy, essentilly y replcing Liner lger I 9
7 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition the word row y the word column throughout. he following definition is mde with such ppliction in mind. If is n m n mtrix, the trnspose of, written s mtrix whose rows re just the columns of in the sme order. In other words, the first row of of is the second column of, nd so on., is the is the first column of, the second row n m Exmple 8 Write down the trnspose of ech of the following mtrices:,,, D 4,,, D 4 If ij is mtrix, write ij. hen ij is the j th element of the i th row of nd so is the j th element of the i th column of. his mens ij ji so the definition of cn e stted s follows: If ij, then ji he Properties of trnsposition: heorem Let, denote mtrices of the sme size, nd let k denote sclr.. If is n m n mtrix, then ). ( is n n m mtrix.. ( k) k Liner lger I
8 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition 4. ( ) he mtrix D in Exmple 8 hs the property tht re importnt. mtrix is clled symmetric if D D.. Such mtrices symmetric mtrix is necessrily squre. he nme comes from the fct tht these mtrices exhiit symmetry out the min digonl. ht is, entries tht re directly cross the min digonl from ech other re equl. Exmple 9 If, re symmetrics n n, show tht is symmetric We hve ( ) nd, so y heorem, we otin Exercises.:. Find,, c, d if:. c d c d d d.. Let,, 4, D, 4 E. ompute the following ( if possile).. D E c. ( ) d. 4D e. D E. If X, Y, nd re mtrices of the sme size, solve the following equtions to otin X, Y in terms of, :. 5X X Y Y 4X. 5X Y 4Y Liner lger I
9 Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition 4. squre mtrix is clled skewsymmetric if. Show tht is skewsymmetric. Let e ny squre mtrix.. Find symmetric mtrix S nd skewsymmetric mtrix W such tht S W c. If W is skewsymmetric show tht the entries on the min digonl re zero 5. squre mtrix is clled digonl mtrix if ll entries off the min digonl re zero. If, re digonl mtrices show tht,, k re digonl mtrices. 6. Let e ny squre mtrix. If p nd q for some mtrix nd numers p, q, show tht either or p. q Liner lger I
Introduction To Matrices MCV 4UI Assignment #1
Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be
More informationChapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY
Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in
More informationMatrices. Elementary Matrix Theory. Definition of a Matrix. Matrix Elements:
Mtrices Elementry Mtrix Theory It is often desirble to use mtrix nottion to simplify complex mthemticl expressions. The simplifying mtrix nottion usully mkes the equtions much esier to hndle nd mnipulte.
More informationMATRIX DEFINITION A matrix is any doubly subscripted array of elements arranged in rows and columns.
4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DEFINITION mtrix is ny douly suscripted rry of elements rrnged in rows nd columns. m  Column Revised /0 n Row m,,,,,, n n mn ij nd Order
More informationECON 331 Lecture Notes: Ch 4 and Ch 5
Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose
More informationINTRODUCTION TO LINEAR ALGEBRA
ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR
More informationAlgebra Of Matrices & Determinants
lgebr Of Mtrices & Determinnts Importnt erms Definitions & Formule 0 Mtrix  bsic introduction: mtrix hving m rows nd n columns is clled mtrix of order m n (red s m b n mtrix) nd mtrix of order lso in
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors  Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
More informationLecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationThe Algebra (aljabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (ljbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN MARICES FOR ENGINEERING Dr Clum Mcdonld Contents Introduction Definitions Wht is mtri? Rows nd columns of mtri Order of mtri Element of mtri Equlity of mtrices Opertions
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationCHAPTER 2d. MATRICES
CHPTER d. MTRICES University of Bhrin Deprtment of Civil nd rch. Engineering CEG Numericl Methods in Civil Engineering Deprtment of Civil Engineering University of Bhrin Every squre mtrix hs ssocited
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd GussJordn elimintion to solve systems of liner
More informationMATRICES AND VECTORS SPACE
MATRICES AND VECTORS SPACE MATRICES AND MATRIX OPERATIONS SYSTEM OF LINEAR EQUATIONS DETERMINANTS VECTORS IN SPACE AND SPACE GENERAL VECTOR SPACES INNER PRODUCT SPACES EIGENVALUES, EIGENVECTORS LINEAR
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationGeometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.
Geometric Sequences Geometric Sequence sequence whose consecutive terms hve common rtio. Geometric Sequence A sequence is geometric if the rtios of consecutive terms re the sme. 2 3 4... 2 3 The number
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationChapter 2. Determinants
Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if dbc0. The expression dbc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is
More informationA Matrix Algebra Primer
A Mtrix Algebr Primer Mtrices, Vectors nd Sclr Multipliction he mtrix, D, represents dt orgnized into rows nd columns where the rows represent one vrible, e.g. time, nd the columns represent second vrible,
More informationIntroduction to Group Theory
Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements
More informationCHAPTER 1 PROGRAM OF MATRICES
CHPTER PROGRM OF MTRICES  INTRODUCTION definition of engineering is the science y which the properties of mtter nd sources of energy in nture re mde useful to mn. Thus n engineer will hve to study the
More informationAnalytically, vectors will be represented by lowercase boldface Latin letters, e.g. a, r, q.
1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples
More informationBases for Vector Spaces
Bses for Vector Spces 22625 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
More informationElements of Matrix Algebra
Elements of Mtrix Algebr Klus Neusser Kurt Schmidheiny September 30, 2015 Contents 1 Definitions 2 2 Mtrix opertions 3 3 Rnk of Mtrix 5 4 Specil Functions of Qudrtic Mtrices 6 4.1 Trce of Mtrix.........................
More informationHere we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.
Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous
More information308K. 1 Section 3.2. Zelaya Eufemia. 1. Example 1: Multiplication of Matrices: X Y Z R S R S X Y Z. By associativity we have to choices:
8K Zely Eufemi Section 2 Exmple : Multipliction of Mtrices: X Y Z T c e d f 2 R S X Y Z 2 c e d f 2 R S 2 By ssocitivity we hve to choices: OR: X Y Z R S cr ds er fs X cy ez X dy fz 2 R S 2 Suggestion
More informationVectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3dimensional vectors:
Vectors 1232018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2dimensionl vectors: (2, 3), ( )
More informationNumerical Linear Algebra Assignment 008
Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More informationMatrix & Vector Basic Linear Algebra & Calculus
Mtrix & Vector Bsic Liner lgebr & lculus Wht is mtrix? rectngulr rry of numbers (we will concentrte on rel numbers). nxm mtrix hs n rows n m columns M x4 M M M M M M M M M M M M 4 4 4 First row Secon row
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationOn the diagram below the displacement is represented by the directed line segment OA.
Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples
More informationRudimentary Matrix Algebra
Rudimentry Mtrix Alger Mrk Sullivn Decemer 4, 217 i Contents 1 Preliminries 1 1.1 Why does this document exist?.................... 1 1.2 Why does nyone cre out mtrices?................ 1 1.3 Wht is mtrix?...........................
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More informationInfinite Geometric Series
Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to
More information8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.
8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More informationIntroduction to Determinants. Remarks. Remarks. The determinant applies in the case of square matrices
Introduction to Determinnts Remrks The determinnt pplies in the cse of squre mtrices squre mtrix is nonsingulr if nd only if its determinnt not zero, hence the term determinnt Nonsingulr mtrices re sometimes
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationOperations with Matrices
Section. Equlit of Mtrices Opertions with Mtrices There re three ws to represent mtri.. A mtri cn be denoted b n uppercse letter, such s A, B, or C.. A mtri cn be denoted b representtive element enclosed
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 200910 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationM A T H F A L L CORRECTION. Algebra I 2 1 / 0 9 / U N I V E R S I T Y O F T O R O N T O
M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #1 CORRECTION Alger I 2 1 / 0 9 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O 1. Suppose nd re nonzero elements of field F. Using only the field xioms,
More informationLecture 2e Orthogonal Complement (pages )
Lecture 2e Orthogonl Complement (pges ) We hve now seen tht n orthonorml sis is nice wy to descrie suspce, ut knowing tht we wnt n orthonorml sis doesn t mke one fll into our lp. In theory, the process
More informationLesson Notes: Week 40Vectors
Lesson Notes: Week 40Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples
More informationDeterminants Chapter 3
Determinnts hpter Specil se : x Mtrix Definition : the determinnt is sclr quntity defined for ny squre n x n mtrix nd denoted y or det(). x se ecll : this expression ppers in the formul for x mtrix inverse!
More informationA matrix is a set of numbers or symbols arranged in a square or rectangular array of m rows and n columns as
RMI University ENDIX MRIX GEBR INRDUCIN Mtrix lgebr is powerful mthemticl tool, which is extremely useful in modern computtionl techniques pplicble to sptil informtion science. It is neither new nor difficult,
More informationSTRAND J: TRANSFORMATIONS, VECTORS and MATRICES
Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors
More informationLecture Solution of a System of Linear Equation
ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville  D. Keffer, 5/9/98 (updted /) Lecture 8  Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner
More informationModule 6: LINEAR TRANSFORMATIONS
Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for
More informationMATHEMATICS FOR MANAGEMENT BBMP1103
T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP Ojectives: TOPIC : MTRIX. Define mtri. ssess the clssifictions of mtrices s well s know how to perform its opertions. Clculte the determinnt for squre mtri
More informationLinear Algebra Introduction
Introdution Wht is Liner Alger out? Liner Alger is rnh of mthemtis whih emerged yers k nd ws one of the pioneer rnhes of mthemtis Though, initilly it strted with solving of the simple liner eqution x +
More informationMath 154B Elementary Algebra2 nd Half Spring 2015
Mth 154B Elementry Alger nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities RentHep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationMatrix Solution to Linear Equations and Markov Chains
Trding Systems nd Methods, Fifth Edition By Perry J. Kufmn Copyright 2005, 2013 by Perry J. Kufmn APPENDIX 2 Mtrix Solution to Liner Equtions nd Mrkov Chins DIRECT SOLUTION AND CONVERGENCE METHOD Before
More informationMTH 5102 Linear Algebra Practice Exam 1  Solutions Feb. 9, 2016
Nme (Lst nme, First nme): MTH 502 Liner Algebr Prctice Exm  Solutions Feb 9, 206 Exm Instructions: You hve hour & 0 minutes to complete the exm There re totl of 6 problems You must show your work Prtil
More informationChapter 9 Definite Integrals
Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished
More information4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX. be a real symmetric matrix. ; (where we choose θ π for.
4.5 JACOBI ITERATION FOR FINDING EIGENVALUES OF A REAL SYMMETRIC MATRIX Some reliminries: Let A be rel symmetric mtrix. Let Cos θ ; (where we choose θ π for Cos θ 4 purposes of convergence of the scheme)
More informationChapter 5 Determinants
hpter 5 Determinnts 5. Introduction Every squre mtri hs ssocited with it sclr clled its determinnt. Given mtri, we use det() or to designte its determinnt. We cn lso designte the determinnt of mtri by
More informationMathCity.org Merging man and maths
MthCity.org Merging mn nd mths Exercise.8 (s) Pge 46 Textbook of Algebr nd Trigonometry for Clss XI Avilble online @ http://, Version: 3.0 Question # Opertion performed on the twomember set G = {0, is
More informationMatrix Eigenvalues and Eigenvectors September 13, 2017
Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues
More informationChapter 2. Vectors. 2.1 Vectors Scalars and Vectors
Chpter 2 Vectors 2.1 Vectors 2.1.1 Sclrs nd Vectors A vector is quntity hving both mgnitude nd direction. Emples of vector quntities re velocity, force nd position. One cn represent vector in ndimensionl
More informationa a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.
Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting
More informationDEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1.
398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 11.1 ORTHOGONAL FUNCTIONS REVIEW MATERIAL The notions of generlized vectors nd vector spces cn e found in ny liner lger text. INTRODUCTION The concepts
More informationParse trees, ambiguity, and Chomsky normal form
Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for contextfree grmmrs
More informationN 0 completions on partial matrices
N 0 completions on prtil mtrices C. Jordán C. Mendes Arújo Jun R. Torregros Instituto de Mtemátic Multidisciplinr / Centro de Mtemátic Universidd Politécnic de Vlenci / Universidde do Minho Cmino de Ver
More informationTHE DISCRIMINANT & ITS APPLICATIONS
THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationBoolean algebra.
http://en.wikipedi.org/wiki/elementry_boolen_lger Boolen lger www.tudorgir.com Computer science is not out computers, it is out computtion nd informtion. computtion informtion computer informtion Turing
More informationMultiplying integers EXERCISE 2B INDIVIDUAL PATHWAYS. 6 ì 4 = 6 ì 0 = 4 ì 0 = 6 ì 3 = 5 ì 3 = 4 ì 3 = 4 ì 2 = 4 ì 1 = 5 ì 2 = 6 ì 2 = 6 ì 1 =
EXERCISE B INDIVIDUAL PATHWAYS Activity B Integer multipliction doc69 Activity B More integer multipliction doc698 Activity B Advnced integer multipliction doc699 Multiplying integers FLUENCY
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 3: Creating Quadratic Equations in Two or More Variables Instruction
Lesson 3: Creting Qudrtic Equtions in Two or More Vriles Prerequisite Skills This lesson requires the use of the following skill: solving equtions with degree of Introduction 1 The formul for finding the
More informationChapters Five Notes SN AA U1C5
Chpters Five Notes SN AA U1C5 Nme Period Section 5: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
More informationMATH1131 Mathematics 1A Algebra
MATH1131 Mthemtics 1A Alger UNSW Sydney Semester 1, 017 Mike Mssierer Mike is pronounced like Mich mike@unsweduu Plese emil me if you hve ny questions or comments Office hours TBA (week ), or emil me to
More informationNatural examples of rings are the ring of integers, a ring of polynomials in one variable, the ring
More generlly, we define ring to be nonempty set R hving two binry opertions (we ll think of these s ddition nd multipliction) which is n Abelin group under + (we ll denote the dditive identity by 0),
More informationMultivariate problems and matrix algebra
University of Ferrr Stefno Bonnini Multivrite problems nd mtrix lgebr Multivrite problems Multivrite sttisticl nlysis dels with dt contining observtions on two or more chrcteristics (vribles) ech mesured
More informationLinear Algebra 1A  solutions of ex.4
Liner Algebr A  solutions of ex.4 For ech of the following, nd the inverse mtrix (mtritz hofkhit if it exists  ( 6 6 A, B (, C 3, D, 4 4 ( E i, F (inverse over C for F. i Also, pick n invertible mtrix
More informationBoolean Algebra. Boolean Algebra
Boolen Alger Boolen Alger A Boolen lger is set B of vlues together with:  two inry opertions, commonly denoted y + nd,  unry opertion, usully denoted y ˉ or ~ or,  two elements usully clled zero nd
More informationCOSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III)  Gauss Quadrature and Adaptive Quadrature
COSC 336 Numericl Anlysis I Numericl Integrtion nd Dierentition III  Guss Qudrture nd Adptive Qudrture Edgr Griel Fll 5 COSC 336 Numericl Anlysis I Edgr Griel Summry o the lst lecture I For pproximting
More informationEngineering Analysis ENG 3420 Fall Dan C. Marinescu Office: HEC 439 B Office hours: TuTh 11:0012:00
Engineering Anlysis ENG 3420 Fll 2009 Dn C. Mrinescu Office: HEC 439 B Office hours: TuTh 11:0012:00 Lecture 13 Lst time: Problem solving in preprtion for the quiz Liner Algebr Concepts Vector Spces,
More information1 Linear Least Squares
Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving
More informationCSCI 5525 Machine Learning
CSCI 555 Mchine Lerning Some Deini*ons Qudrtic Form : nn squre mtri R n n : n vector R n the qudrtic orm: It is sclr vlue. We oten implicitly ssume tht is symmetric since / / I we write it s the elements
More informationInner Product Space. u u, v u, v u, v.
Inner Product Spce Definition Assume tht V is ector spce oer field of sclrs F in our usge this will e. Then we define inry opertor.. :V V F [once gin in our usge this will e ] so tht the following properties
More informationDETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ
All Mthemticl truths re reltive nd conditionl. C.P. STEINMETZ 4. Introduction DETERMINANTS In the previous chpter, we hve studied bout mtrices nd lgebr of mtrices. We hve lso lernt tht system of lgebric
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More informationDonnishJournals
DoishJournls 20411189 Doish Journl of Eductionl Reserch nd Reviews Vol 2(1) pp 001007 Jnury, 2015 http://wwwdoishjournlsorg/djerr Copyright 2015 Doish Journls Originl Reserch Article Algebr of Mtrices
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationNetwork Analysis and Synthesis. Chapter 5 Two port networks
Network Anlsis nd Snthesis hpter 5 Two port networks . ntroduction A one port network is completel specified when the voltge current reltionship t the terminls of the port is given. A generl two port on
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More information