Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY


 Katrina Kathryn Harrington
 2 years ago
 Views:
Transcription
1 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 Solving Liner Eqution System Determinnt of 3 x 3 Mtrices 3. MTRIX NOTTION ND TERMINOLOGY Defnition. mtrix is rectngulr rry of numbers. The numbers in the rry re clled the entries in the mtrix. Exmples: Nottion Generlly written s: m n= ij = C = D = mtrix with 2 rows, nd 3 columns NOT mtrix mtrix with 3 rows, nd 3 columns 2 3 D 3 3
2 n rows nd n columns n n, is squre mtrix. SPECIL MTRICES The following re severl specil mtrices. Generl form Nme Properties Exmples 6 7 nd = n [ n] Row mtrix Only one row [ ] 2 [ ] 2 n = n Column mtrix Only one column 3 nd m n= Null mtrix ll entries re null , D I n d d dn 2 n = = 0 0 Digonl mtrix Identity mtrix squre mtrix in which ll of the entries off the min digonl re zero, i.e. = 0, i j ij squre mtrix, with ll digonl entries re, nd zero elsewhere , nd I I = 0, 0 0 =
3 n n n 22 2n = nn Upper tringulr mtrix squre mtrix in which ll the entries below the min digonl re zero , n n n n n n2 nn 2 22 = 2 n n = n n2... nn Lower tringulr mtrix Symmetric mtrix squre mtrix in which ll the entries bove the min digonl re zero squre mtrix in which ij = for ll i nd j. ji , OPERTIONS ON MTRICES Definition. Two mtrices re defined to be equl if they hve the sme size nd their corresponding entries re equl. In mtrix nottion, if only if ( ) ( B) ij ij = ij nd B bij =, or equivlently ij = bij for ll i nd j. = hve the sme size, then = B if nd Exmple: Consider the mtrices 3 3 = 2 x, B 3 4 = 2 7, nd C = Is = B? Why? 3. Is = C? Why? 2. Is B = C? Why? 3
4 Definition. If then: = ij nd B bij = re mtrices of the sme size, ( + B) = ( ) + ( B) = ij + bij, nd ( B) = ( ) ( B) = ij bij ij ij ij ij ij ij Exmple: Consider the mtrices 4 0 = 0 5, 9 7 B = 4 5 nd 3 9 C = 6 2 Then: ( 7) B= + = = , B=... Wht bout: + C? C B? Definition. If = is ny mtrix nd c is ny sclr, then the product c is the ij mtrix obtined by multiplying ech entry of by c. In mtrix nottion, if = ij then ( c) c( ) c ij ij ij = =. Exmple. For the mtrices We hve = 9 2 0,nd B = 4 5 2(3) 2( 8) 2(6) = = 2(9) 2(2) 2(0) , =
5 Wht bout: 5 = 3 = B = 2 B = 2+ 3B= + 2B = Definition. If = is n m x r mtrix nd B is n r x n mtrix, then the product ij B is the m x n mtrix whose entries re determined s follows. B = sum of multipliction of corresponding entries in row i of nd Entry of ( ) ij column j of B. The definition of mtrix multipliction requires tht the number of columns of the first fctor be the sme s the number of rows of the second fctor B in order to form the product B. If this condition is not stisfied, the produt is undefined. Exmple: For the mtrices = Let C = B. nd B b b b b b b = b3 b32 b33 b b b , 5
6 Then b b2 b b2 b22 b 23 C = B = c c2 c = 24 b3 b32 b33 c2 c22 c 23 b4 b42 b43 nd The entry of row 2 nd column 3 of C is: c23 = 2b3 + 22b b b43 Cn B be clculted? Exmple. Consider the mtrices = 3 7, nd B = Then: B = 2 8 4( 6) + ( 2)(2) + 5(3) 4(4) + ( 2)(8) + 5(9) 3 7 = 3( 6) (2) 7(3) 3(4) (8) 7(9) = = B = TRNSPOSE OF MTRIX T Definition. If is ny m x n mtrix, then the trnspose of, denoted by, is defined to be the n x m mtrix tht results from interchnging the rows nd the columns of. In mtrix nottion, ( T ) ( ) ij =. ji 6
7 Exmple. Consider the mtrices = 4 8 nd B = Then: T 0 4 = T B =... Exmine: Is T T =? Why? Is B = B? Why? TRCE OF SQURE MTRIX Definition. If is squre mtrix, then the trce of, denoted by tr( ), is defined to be the sum of the entries on the min digonl of. The trce of is undefined if is not squre mtrix. Exmple. The following re exmples of mtrices nd their trces = 4 0 5, then tr ( ) = = B = 5 83, then tr ( B ) = = 70 7
8 EXERCISE. Suppose tht, B, C, D, nd E re mtrices with the following sizes: 4 5 B4 5 C5 2 D4 2 E. 5 4 Determine which of the following mtrix expressions re defined. For those which re defined, give the size of the resulting mtrix.. B e. E(+B) b. C + D f. E(C) c. E + B g. E T d. B + B h. ( T +E)D 2. Solve the following mtrix eqution for, b, c, nd d. b b+ c 8 = 3d + c 2 4d Consider the mtrices =, B = 0 2, C = 3 5, D = 0, E = Compute the following (where possible).. D + E b. 2BC 8
9 c. 3(D+2E) g. (3E)D d. 4tr(7B) h. B e. 2 T +C i. B f. 2E T 3D T j. (2D T E) 4. Find the 3 x 3 mtrix = ij whose entries stisfy the stted condition., if i j >. ij = i+ j b. ij =, if i j 5. Given tht 3 + b b is symmetric mtrix, find nd b
10 DETERMINNT OF 2 X 2 MTRIX Definition. Let b = c d, then the determinnt of is d bc, tht is det() = d bc In mtrix nottion, the determinnt of cn be written s, b det() = = = d bc c d Exmple: Consider the following mtrices: = 4 b. B = 4 3 Then: c. 3 6 C = 2. det() = 4 3() 9(4) = = = b. det(b) =... c. det(c) =... Exmple: Find x so tht the determinnt of x 2 4 = x is 8. Solution: x 2 4 Det() = = ( x 2)( 3) 4( x+ 3) = 3x+ 6 4x 2= 7x 6 x Since the determinnt is 8, then 7x 6= 8 7x = 4 x = 2 0
11 . Find x so tht the determinnt of x 2 4 = x is Find x so tht the determinnt of x + 2 B = 3 x 5 is EXERCISE. Evlute the determinnt d b e c f Find x so tht the determinnt of 3 x 4 is 8. x 2
12 3. Find ll possible vlues for so tht the determinnt of is Consider the mtrices 2 4 =, B =. Evlute: 4 3. det() d. det() + det(b) b. det(b) e. det(b) c. det( + B) f. det()det(b) INVERSE OF 2 x 2 MTRIX Definition. If is squre mtrix, nd if mtrix B of the sme size cn be found such tht B = B = I, then is sid to be invertible nd B is clled n inverse of. In mtrix nottion, the inverse of is, so if B = B = I, then = B Hence, = = I 2
13 Exmple: The mtrix = 2 is n inverse of B = 3, since nd B = I 2 = = = B= = = = I Theorem. The mtrix b = c d is invertible if d bc 0, in which cse the inverse is given by the formul d b = d bc c noninvertible mtrix, or mtrix with the determinnt = 0 is clled singulr mtrix. Wht is nonsingulr mtrix? Exmple. Consider the following mtrices: 3 5 = 2 B 6 4 = C = 6 9 Then the inverses re: = =. = B =... C =... 3
14 Properties of n inverse If nd B re nonsingulr squre mtrices of the sme size, then ( B) = B Exmple: 2 Let = 2 3 nd Then, B 2 0 = 3 On the other hnd, B = 2 3 = nd 3 2 ( B) = 2 ( 0) = = = = 3 ( 4) B 2 nd B = = = = 3 2 = Both give the sme result, which verify tht ( B) = B EXERCISE. Determine whether is the inverse of B, or vise vers =, B 4 3 = b. =, B = 3 5 4
15 2. Evlute whether the following mtrices re singulr or nonsingulr. If it is nonsingulr mtrix, find the inverse d b. 5 4 e Let 3 4 =, B =. Find: 4 0. nd B c. (B) b. B d. B 5
16 Let =. Find: 2 4. T c. ( T ) b. d. ( ) T 5. Given the following informtion, find. 9. = 4 b. 6 7 (3 ) = 7 8 6
Matrices. 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 25pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
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 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 informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
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 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 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
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 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 informationPreSession Review. Part 1: Basic Algebra; Linear Functions and Graphs
PreSession Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
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 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 informationComputing The Determinants By Reducing The Orders By Four
Applied Mthemtics ENotes, 10(2010), 151158 c ISSN 16072510 Avilble free t mirror sites of http://wwwmthnthuedutw/ men/ Computing The Determinnts By Reducing The Orders By Four Qefsere Gjonblj, Armend
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 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 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 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 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 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 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 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 information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
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 informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rellife exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
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 informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a nonconstant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More information1. Extend QR downwards to meet the xaxis at U(6, 0). y
In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions
More informationThe Islamic University of Gaza Faculty of Engineering Civil Engineering Department. Numerical Analysis ECIV Chapter 11
The Islmic University of Gz Fculty of Engineering Civil Engineering Deprtment Numericl Anlysis ECIV 6 Chpter Specil Mtrices nd GussSiedel Associte Prof Mzen Abultyef Civil Engineering Deprtment, The Islmic
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 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 informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
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 informationChapter 1 Cumulative Review
1 Chpter 1 Cumultive Review (Chpter 1) 1. Simplify 7 1 1. Evlute (0.7). 1. (Prerequisite Skill) (Prerequisite Skill). For Questions nd 4, find the vlue of ech expression.. 4 6 1 4. 19 [(6 4) 7 ] (Lesson
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
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 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 informationIs there an easy way to find examples of such triples? Why yes! Just look at an ordinary multiplication table to find them!
PUSHING PYTHAGORAS 009 Jmes Tnton A triple of integers ( bc,, ) is clled Pythgoren triple if exmple, some clssic triples re ( 3,4,5 ), ( 5,1,13 ), ( ) fond of ( 0,1,9 ) nd ( 119,10,169 ). + b = c. For
More informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
More informationTABLE OF CONTENTS 3 CHAPTER 1
TABLE OF CONTENTS 3 CHAPTER 1 Set Lnguge & Nottion 3 CHAPTER 2 Functions 3 CHAPTER 3 Qudrtic Functions 4 CHAPTER 4 Indices & Surds 4 CHAPTER 5 Fctors of Polynomils 4 CHAPTER 6 Simultneous Equtions 4 CHAPTER
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationGeneralized Fano and nonfano networks
Generlized Fno nd nonfno networks Nildri Ds nd Brijesh Kumr Ri Deprtment of Electronics nd Electricl Engineering Indin Institute of Technology Guwhti, Guwhti, Assm, Indi Emil: {d.nildri, bkri}@iitg.ernet.in
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 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 informationOXFORD H i g h e r E d u c a t i o n Oxford University Press, All rights reserved.
Renshw: Mths for Econoics nswers to dditionl exercises Exercise.. Given: nd B 5 Find: () + B + B 7 8 (b) (c) (d) (e) B B B + B T B (where 8 B 6 B 6 8 B + B T denotes the trnspose of ) T 8 B 5 (f) (g) B
More informationNOTE ON TRACES OF MATRIX PRODUCTS INVOLVING INVERSES OF POSITIVE DEFINITE ONES
Journl of pplied themtics nd Computtionl echnics 208, 7(), 2936.mcm.pcz.pl pissn 22999965 DOI: 0.752/jmcm.208..03 eissn 23530588 NOE ON RCES OF RIX PRODUCS INVOLVING INVERSES OF POSIIVE DEFINIE ONES
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 informationChapter 3 Polynomials
Dr M DRAIEF As described in the introduction of Chpter 1, pplictions of solving liner equtions rise in number of different settings In prticulr, we will in this chpter focus on the problem of modelling
More informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
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 informationDiscrete Leastsquares Approximations
Discrete Lestsqures Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve
More information(4.1) D r v(t) ω(t, v(t))
1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution
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 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 informationLinearity, linear operators, and self adjoint eigenvalue problems
Linerity, liner opertors, nd self djoint eigenvlue problems 1 Elements of liner lgebr The study of liner prtil differentil equtions utilizes, unsurprisingly, mny concepts from liner lgebr nd liner ordinry
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 information