Matrices. Elementary Matrix Theory. Definition of a Matrix. Matrix Elements:
|
|
- Sarah Bradley
- 5 years ago
- Views:
Transcription
1 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. Let consider the following set of n simultneous lgebric equtions: x x 2 2 x n n y x x x 2n n y 2 x x x y n n2 2 nn n n We my simplify to X defined to be: Y ;, X, nd Y re defined s mtrices. These three mtrices re n 2 n n nn X x x 2 x n Y y y 2 y n (D-) where the brcketed rrys re simplified of coefficients nd vribles. Definition of Mtrix mtrix is collection of elements rrnged in rectngulr or squre rry. Severl wys of representing mtrix re s follows: ,2 (D-2) Mtrix Elements:
2 is defined s the element in the i th row nd the j th column of the mtrix (Row st nd column lst). Order of Mtrix: refers to the totl number of rows nd columns of the mtrix. In generl, mtrix with n rows nd m columns is termed "n x m" or "n by m". Squre Mtrix: one tht hs one column nd more thn one row: mx mtrix m >. Row Mtrix: is one tht hs one row nd more thn one column: xn mtrix n >. Digonl Mtrix: is squre mtrix with 0 for ll i j (D-3) Unit Mtrix (Identity Mtrix): is digonl mtrix with ll the elements on the min digonl i j equl to. unity mtrix is often designted by either I or U. Unity exmple is I (D-4) Null Mtrix: is one whose elements re equl to zero for exmple: O (D-5) Symmetric Mtrix: is squre mtrix tht stisfies the condition ; 4 7. For exmple: ji
3 Determinnt of Mtrix: with ech squre mtrix determinnt hving the sme elements nd order. The determinnt of squre mtrix is designted by: Det D Determinnt of the mtrix c b d is define s c b d d bc Consider the mtrix: Then: ( ) 0 0 (0 2) 0(0 2) ( )(0 3) Singulr Mtrix: is sid to be singulr if the vlue of its determinnt is zero. Where mtrix is singulr, it usully mens tht not ll the rows or not ll the columns of the mtrix re independent of ech other. Let consider the following set of equtions: 2x 3x x x x x x 2x 2x The third eqution is equl to the sum of the first two equtions. Therefore these three equtions re not completely independent. In mtrix form, these equtions my be represented by X = 0
4 2 3 Where 2 2 & X x x x 2 3 Determinnt of : Therefore the mtrix is singulr. Trnspose of Mtrix: The trnspose of mtrix is defined s the mtrix tht is obtined by interchnging the corresponding rows nd columns in. Let be n nxm mtrix which is represented by. For exmple: nm, Then the trnspose of, denoted by ' is given by ' = trnspose of. mn, Skew-Symmetric Mtrix: is squre mtrix tht equls its negtive trnspose, tht is: Some Opertions of Mtrix Trnspose: k k ; where k is sclr B B B B djoint of Mtrix: Let be squre mtrix of order n. The djoint mtrix of denoted by dj( ) is defined s dj( ) cofctor of Det, ; where the nn cofctor of the determinnt of is the determinnt obtined by
5 omitting the i th row nd the j th column of nd then multiplying by i j. Exmple D-: Determine the djoint mtrix of: c b d The determinnt of is c b d d b dj c Exmple D-2: dj Summtion or Subtrction of Mtrix: Given two m x n mtrices nd B b their sum is nd their difference is B b B b Sclr Multipliction: The sclr product of number k nd mtrix is denoted by k. k k 2 2 k k k k
6 Multipliction of Mtrices: Let be n m x n mtrix nd let B be n n x k mtrix. To find the element in the i th row nd j th column of the product mtrix B, multiply ech element in the i th row of by the corresponding element in the j th column of B, nd then dd these products. The product mtrix B is n m x k mtrix. Mtrix m x n Inner must be equl Mtrix B n x k Outer: Order of B is m x k B = b e f b e f e bg _. b e f _ f bh. b e f. ce dg _ b e f. _ cf dh b e f e bg f bh. ce dg cf dh C B b m, n n, k c mk, c k b in nj for i,2,, k nd j,2,, k n
7 Exmple D-3: Given the mtrices 2 3 b B b b b b b B b b b Exmple D-4: B () ( 3)3 (0) ( 3) ( ) ( 3)4 (2) ( 3)( ) 7() 2(3) 7(0) 2() 7( ) 2(4) 7(2) 2( ) Inverse of mtrix: is dj Properties of the Inverse Mtrix: Exmple D-5: I B B ;
8 Properties of Mtrix ddition nd Sclr Multipliction B B Commuttive Property of ddition ( B C) ( B) C ssocitive Property of ddition ( kl) k( l) ssocitive Property of Sclr Multipliction k( B) k kb Distributive Property ( k l) k l Distributive Property 0 0 dditive Identity Property ( ) ( ) 0 dditive Inverse Property Multipliction ( BC) ( B) C ssocitive Property of Multipliction ( B C) B C Distributive Property ( B C) B C Distributive Property
Chapter 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 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 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 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 informationMatrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24
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
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 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 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 informationIntroduction 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 informationThe Algebra (al-jabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) 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 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 Guss-Jordn elimintion to solve systems of liner
More informationChapter 2. Determinants
Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if d-bc0. The expression d-bc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is
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 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 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 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 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 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 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 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 informationEngineering Analysis ENG 3420 Fall Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00
Engineering Anlysis ENG 3420 Fll 2009 Dn C. Mrinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00 Lecture 13 Lst time: Problem solving in preprtion for the quiz Liner Algebr Concepts Vector Spces,
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 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 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 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 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 informationDETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ
DETERMINANTS Chpter 4 All Mthemticl truths re reltive nd conditionl. C.P. STEINMETZ 4. Introduction In the previous chpter, we hve studied bout mtrices nd lgebr of mtrices. We hve lso lernt tht sstem of
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 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 two-member set G = {0, is
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 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 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 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 non-empty 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 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 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 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 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 informationMatrices. Introduction
Mtrices Introduction Mtrices - Introduction Mtrix lgebr hs t lest two dvntges: Reduces complicted systems of equtions to simple expressions Adptble to systemtic method of mthemticl tretment nd well suited
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 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 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 informationYear 11 Matrices. A row of seats goes across an auditorium So Rows are horizontal. The columns of the Parthenon stand upright and Columns are vertical
Yer 11 Mtrices Terminology: A single MATRIX (singulr) or Mny MATRICES (plurl) Chpter 3A Intro to Mtrices A mtrix is escribe s n orgnise rry of t. We escribe the ORDER of Mtrix (it's size) by noting how
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 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 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 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 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 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 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 informationDonnishJournals
DoishJournls 2041-1189 Doish Journl of Eductionl Reserch nd Reviews Vol 2(1) pp 001-007 Jnury, 2015 http://wwwdoishjournlsorg/djerr Copyright 2015 Doish Journls Originl Reserch Article Algebr of Mtrices
More informationMatrices 13: determinant properties and rules continued
Mtrices : determinnt properties nd rules continued nthony Rossiter http://controleduction.group.shef.c.uk/indexwebbook.html http://www.shef.c.uk/cse Deprtment of utomtic Control nd Systems Engineering
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 informationVectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3-dimensional vectors:
Vectors 1-23-2018 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 2-dimensionl vectors: (2, 3), ( )
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 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 informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
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 n-dimensionl
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 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
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 informationHAND BOOK OF MATHEMATICS (Definitions and Formulae) CLASS 12 SUBJECT: MATHEMATICS
HAND BOOK OF MATHEMATICS (Definitions nd Formule) CLASS 12 SUBJECT: MATHEMATICS D.SREENIVASULU PGT(Mthemtics) KENDRIYA VIDYALAYA D.SREENIVASULU, M.Sc.,M.Phil.,B.Ed. PGT(MATHEMATICS), KENDRIYA VIDYALAYA.
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 informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationResults on Planar Near Rings
Interntionl Mthemticl Forum, Vol. 9, 2014, no. 23, 1139-1147 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/imf.2014.4593 Results on Plnr Ner Rings Edurd Domi Deprtment of Mthemtics, University
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 informationOptimization Lecture 1 Review of Differential Calculus for Functions of Single Variable.
Optimiztion Lecture 1 Review of Differentil Clculus for Functions of Single Vrible http://users.encs.concordi.c/~luisrod, Jnury 14 Outline Optimiztion Problems Rel Numbers nd Rel Vectors Open, Closed nd
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 informationLecture 0. MATH REVIEW for ECE : LINEAR CIRCUIT ANALYSIS II
Lecture 0 MATH REVIEW for ECE 000 : LINEAR CIRCUIT ANALYSIS II Aung Kyi Sn Grdute Lecturer School of Electricl nd Computer Engineering Purdue University Summer 014 Lecture 0 : Mth Review Lecture 0 is intended
More informationSemigroup of generalized inverses of matrices
Semigroup of generlized inverses of mtrices Hnif Zekroui nd Sid Guedjib Abstrct. The pper is divided into two principl prts. In the first one, we give the set of generlized inverses of mtrix A structure
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 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 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 informationMATHEMATICS AND STATISTICS 1.2
MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr
More informationScientific notation is a way of expressing really big numbers or really small numbers.
Scientific Nottion (Stndrd form) Scientific nottion is wy of expressing relly big numbers or relly smll numbers. It is most often used in scientific clcultions where the nlysis must be very precise. Scientific
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 informationFACULTY OF ENGINEERING TECHNOLOGY GROUP T LEUVEN CAMPUS INTRODUCTORY COURSE MATHEMATICS
FACULTY OF ENGINEERING TECHNOLOGY GROUP T LEUVEN CAMPUS INTRODUCTORY COURSE MATHEMATICS Algebr Content. Rel numbers. The power of rel number with n integer eponent. The n th root of rel number 4. The power
More informationLINEAR ALGEBRA AND MATRICES. n ij. is called the main diagonal or principal diagonal of A. A column vector is a matrix that has only one column.
PART 1 LINEAR ALGEBRA AND MATRICES Generl Nottions Mtri (denoted by cpitl boldfce letter) A is n m n mtri. 11 1... 1 n 1... n A ij...... m1 m... mn ij denotes the component t row i nd column j of A. If
More informationEcuaciones Algebraicas lineales
Ecuciones Algebrics lineles An eqution of the form x+by+c=0 or equivlently x+by=c is clled liner eqution in x nd y vribles. x+by+cz=d is liner eqution in three vribles, x, y, nd z. Thus, liner eqution
More informationSave from: 1 st. class Mathematics الرياضيات استاذ الماده: م.م. سرى علي مجيد علي
Sve from: www.uotechnology.edu.iq st clss Mthemtics الرياضيات استاذ الماده: م.م. سرى علي مجيد علي Chpter One Consider n rbitrry system of eqution in unknown s: A B.( n mn m m m n n n n B r bm n n m m m
More informationDownloaded from
POLYNOMIALS UNIT- It is not once nor twice but times without number tht the sme ides mke their ppernce in the world.. Find the vlue for K for which x 4 + 0x 3 + 5x + 5x + K exctly divisible by x + 7. Ans:
More informationfractions Let s Learn to
5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin
More informationElementary Linear Algebra
Elementry Liner Algebr Anton & Rorres, 1 th Edition Lecture Set 5 Chpter 4: Prt II Generl Vector Spces 163 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 163 คณตศาสตรวศวกรรม 3 สาขาวชาวศวกรรมคอมพวเตอร
More informationREVIEW Chapter 1 The Real Number System
Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole
More informationCS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 2
CS434/54: Pttern Recognition Prof. Olg Veksler Lecture Outline Review of Liner Algebr vectors nd mtrices products nd norms vector spces nd liner trnsformtions eigenvlues nd eigenvectors Introduction to
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 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 informationELE B7 Power System Engineering. Unbalanced Fault Analysis
Power System Engineering Unblnced Fult Anlysis Anlysis of Unblnced Systems Except for the blnced three-phse fult, fults result in n unblnced system. The most common types of fults re single lineground
More informationINJNTU.COM LECTURE NOTES
LECTURE NOTES ON LINEAR ALGEBRA AND ORDINARY DIFFERENTIAL EQUATIONS I B. Tech I semester UNIT-I THEORY OF MATRICES Solution for liner systems Mtri : A system of mn numbers rel (or) comple rrnged in the
More informationComputing The Determinants By Reducing The Orders By Four
Applied Mthemtics E-Notes, 10(2010), 151-158 c ISSN 1607-2510 Avilble free t mirror sites of http://wwwmthnthuedutw/ men/ Computing The Determinnts By Reducing The Orders By Four Qefsere Gjonblj, Armend
More information11-755/ Machine Learning for Signal Processing. Algebra. Class August Instructor: Bhiksha Raj
-755/8-797 Mchine Lerning for Signl Processing Fundmentls of Liner Alger Clss 6 August 9 Instructor: Bhiksh Rj 6 Aug -755/8-797 Administrivi Registrtion: Anyone on witlist still? Our TA is here Sourish
More informationAnalytically, vectors will be represented by lowercase bold-face 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 informationThe Dirac distribution
A DIRAC DISTRIBUTION A The Dirc distribution A Definition of the Dirc distribution The Dirc distribution δx cn be introduced by three equivlent wys Dirc [] defined it by reltions δx dx, δx if x The distribution
More informationMath 75 Linear Algebra Class Notes
Mth 75 Liner Algebr Clss Notes Prof. Erich Holtmnn For use with Elementry Liner Algebr, 7 th ed., Lrson Revised -Nov-5 p. i Contents Chpter : Systems of Liner Equtions. Introduction to Systems of Equtions..
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 informationad = cb (1) cf = ed (2) adf = cbf (3) cf b = edb (4)
10 Most proofs re left s reding exercises. Definition 10.1. Z = Z {0}. Definition 10.2. Let be the binry reltion defined on Z Z by, b c, d iff d = cb. Theorem 10.3. is n equivlence reltion on Z Z. Proof.
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 doc-69 Activity -B- More integer multipliction doc-698 Activity -B- Advnced integer multipliction doc-699 Multiplying integers FLUENCY
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 informationThe usual algebraic operations +,, (or ), on real numbers can then be extended to operations on complex numbers in a natural way: ( 2) i = 1
Mth50 Introduction to Differentil Equtions Brief Review of Complex Numbers Complex Numbers No rel number stisfies the eqution x =, since the squre of ny rel number hs to be non-negtive. By introducing
More informationThe use of a so called graphing calculator or programmable calculator is not permitted. Simple scientific calculators are allowed.
ERASMUS UNIVERSITY ROTTERDAM Informtion concerning the Entrnce exmintion Mthemtics level 1 for Interntionl Bchelor in Communiction nd Medi Generl informtion Avilble time: 2 hours 30 minutes. The exmintion
More information