INTRODUCTION TO LINEAR ALGEBRA
|
|
- Janel Hunter
- 6 years ago
- Views:
Transcription
1 ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & /
2 ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Mtri: A rectngulr rry of sclrs (numbers, vribles, or functions rel or comple). M m M m O n n M mn [ ] [ A] ij Rows Elements ; i,,...,m, j,,...,n Columns Totl no of rows Element i,j i th row j th column Size or dimension of mtri: m n Totl no of columns Prof. Dr. Bülent E. Pltin Spring Sections & /
3 ME Applied Mthemtics for Mechnicl Engineers Vector: A mtri with only one row (size n, row vector) or only one column (size m, column vector). Equlity of mtrices: Two mtrices nd [B] re sid to be equl to ech other if nd only if i) they hve the sme dimension m n, nd ii) their corresponding elements re equl; i.e., ij b ij for ll i,,, m nd j,,, n Addition/Subtrction of mtrices: Addition/subtrction is defined only for mtrices of the sme size nd result in nother mtri of the sme size. For two mtrices nd [B] of the sme size [C] ± [B] implies tht c ij ij ± b ij for ll i,,, m nd j,,, n Prof. Dr. Bülent E. Pltin Spring Sections & /
4 ME Applied Mthemtics for Mechnicl Engineers Emple: Given 6 nd [B] 6 6 [ A] [ B] [ A] - [ B] Multipliction/Division by sclr: Multipliction/division of mtri by sclr implies tht ll its elements re to be multiplied/divided by the sme sclr ; i.e., [ ij ] or / [ ij /] Prof. Dr. Bülent E. Pltin Spring Sections & /
5 ME Applied Mthemtics for Mechnicl Engineers Emple: Given Then nd 9 Some importnt properties of mtrices: Given mtrices, [B], nd [C] of the sme size nd set of sclr constnts,, nd, the following properties hold: [B] [B] commuttive ([B] [C]) ( [B]) [C]) ssocitive ( [B]) [B] distributive ( ) ( ) ( ) ( ) Prof. Dr. Bülent E. Pltin Spring Sections & /
6 ME Applied Mthemtics for Mechnicl Engineers Mtri multipliction: et be m n mtri nd [B] be p q mtri. The mtri product [B] is defined only if n p nd it gives mtri [C] of size m q nd shown s [C] [B] The elements of [C] re given s c ij i b j i b j.. in b nj Note tht the element c ij cn be interpreted s the dot product (inner product) of the i th row vector of nd the j th column vector of [B]. [B] n i b j Prof. Dr. Bülent E. Pltin Spring Sections & 6/
7 ME Applied Mthemtics for Mechnicl Engineers Emple: Given 6 nd [B] 6 Since number of columns of nd number of rows of [B] mtch, nd [B] mtrices re clled comptible s fr s [B] opertion is concerned. Hence, their product becomes 7 6 [ C] [ A][ B] Prof. Dr. Bülent E. Pltin Spring Sections & 7/
8 ME Applied Mthemtics for Mechnicl Engineers Note tht, in this emple, even though [B] multipliction is defined, [B] multipliction is not defined since the number of columns of [B] (which is ) nd number of rows of (which is ) do not mtch. Mtri multipliction is not commuttive. Even in cses where both [B] nd [B] multiplictions re defined, these multiplictions re usully not equl to ech other; i.e., [B] [B], in generl Note tht only if the mtri is of size m n nd [B] is n m, then both [C] [B] nd [D] [B] re defined where [C] nd [D] re squre mtrices of different sizes m nd n, respectively; therefore, not equl to ech other. If mtrices nd [B] re both squre nd of the sme size n, then both [C] [B] nd [D] [B] re defined where [C] nd [D] re squre mtrices of the sme size n; but not necessrily equl to ech other. Prof. Dr. Bülent E. Pltin Spring Sections & 8/
9 ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & 9/ Emple: Trnspose of mtri: The trnspose of n m n mtri is n n m mtri [B] whose rows re the columns nd columns re the rows of. It is denoted by T. Hence, [ ij ] T [ ji ] Given squre mtrices [B] nd 6 [B] 7 [B] Emple: Given squre mtrices [B] nd [B] [B]
10 ME Applied Mthemtics for Mechnicl Engineers Emple: T [b] [b] T [ 7 ] Note tht [b] T is convenient wy to describe row vector. The following re two importnt properties of trnspose opertion. ( [B]) T T [B] T ( [B]) T [B] T T Note the chnge of the order Prof. Dr. Bülent E. Pltin Spring Sections & /
11 ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & / Emples of sprse mtrices: Squre mtri: A mtri with equl number of rows nd columns. Principl (min) digonl Off-digonl elements [C] [B] Sprse mtri: A mtri with most of its elements zero [ ] nn n n n n A M O M M
12 ME Applied Mthemtics for Mechnicl Engineers Symmetric mtri: A squre mtri whose off digonl elements t symmetric loctions re equl; Emple: tht is ij ji for ll i, j (,,, n) Note tht the trnspose of symmetric mtri is equl to itself; i.e., T Sew-symmetric mtri: A squre mtri whose off digonl elements t symmetric loctions re Emple: equl in size but of opposite sign; tht is ij ji for ll i, j (,,, n) Note tht the elements of min digonl of [B] sew-symmetric re ll zero; i.e., ii for ll i,,, n Prof. Dr. Bülent E. Pltin Spring Sections & /
13 ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & / Note tht ny squre mtri,, cn be written s s s where is clled the symmetric prt of, nd is clled the sew-symmetric prt of. T s T s Emple: s 7 7 s
14 ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & / Tringulr mtri: A squre mtri whose off digonl elements bove or below the min digonl re ll zero. Emple: Upper tringulr mtri ower tringulr mtri Digonl mtri: A squre mtri whose off digonl elements re ll zero. Tht is ij for ll i, j,,, n, i j [B]
15 ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & / Identity (unity) mtri: A digonl mtri whose elements in the principl digonl re ll. Tht is ii for ll i,,, n Null mtri: A m n mtri whose elements re ll zero. Tht is, ij for ll i,,, m nd j,,, n [B] [C] [B]
16 ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & 6/ Note tht, contrry to the cse in multiplictions of sclrs, if the multipliction of two squre mtrices gives null mtri then this does not imply tht t lest one of the mtrices multiplied should be null mtri s well. Bnded mtri: A squre mtri; some of its digonls net to the min digonl re not zero, but the rest of the off-digonl elements re ll zero. [B] but [B] [B] &
17 ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & 7/ Emple: Consider the free (unforced) motion of the following model which my be representing the longitudinl vibrtions of long elstic body whose both ends re fied. Using the Newton s nd lw of motion for ech mss, the governing equtions of motion cn be written s: 6 m m m m m ) ( m ) ( ) ( m ) ( ) ( m ) ( ) ( m ) ( m 6 && && && && &&
18 ME Applied Mthemtics for Mechnicl Engineers Prof. Dr. Bülent E. Pltin Spring Sections & 8/ [ ] [ ] [ ] [ ] [] M K && Mss mtri digonl Position vector Stiffness mtri, symmetricl nd bnded Forcing vector null vector Governing eqution in mtri form: m m m m m [M] [] 6 [K] []
19 ME Applied Mthemtics for Mechnicl Engineers Determinnts, Minors, nd Cofctors Determinnt: It is sclr quntity nd defined only for squre rrys. Every squre rry A of size n hs unique determinnt vlue D nd it is shown s D M n M n O n n M nn A det Prof. Dr. Bülent E. Pltin Spring Sections & 9/
20 ME Applied Mthemtics for Mechnicl Engineers Minor: It is determinnt of n rry, which is one lower size thn the rry of n originl determinnt. A minor is lwys ssocited with only one of the elements of n originl rry. Its rry is obtined by deleting the row nd column of the originl rry contining tht prticulr element; tht is the minor M ij of the element ij of n rry of size n is the determinnt of the rry of size n, which is formed by deleting the i th row nd j th column of the originl rry. Emple: A M M Prof. Dr. Bülent E. Pltin Spring Sections & /
21 ME Applied Mthemtics for Mechnicl Engineers Cofctor: It is lso ssocited with only one of the elements of n originl rry nd defined s C ij ( ) ij M ij Emple: A C ( ) M C ( ) M Evlution of Determinnts: The vlue of the determinnt of size one is sme s the sclr involved; tht is The vlue of the determinnt of size two is obtined s Prof. Dr. Bülent E. Pltin Spring Sections & /
22 ME Applied Mthemtics for Mechnicl Engineers Determinnts of size three nd higher cn be evluted by epnding it with respect to one of its rows or columns s D M n M n O n n M nn n j ny i ij C ij n i ny j ij C ij Emple: 6 6 D 6 ( ) ( ) ep nsion w. r. t. st row Prof. Dr. Bülent E. Pltin Spring Sections & /
23 ME Applied Mthemtics for Mechnicl Engineers END OF WEEK Prof. Dr. Bülent E. Pltin Spring Sections & /
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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
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 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 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 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 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 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 & 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
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 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 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 Guss-Siedel Associte Prof Mzen Abultyef Civil Engineering Deprtment, The Islmic
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 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 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 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 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: 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 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 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 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 informationLesson 1: Quadratic Equations
Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring
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 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 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 informationPre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs
Pre-Session 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 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 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 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 informationBoolean Algebra. Boolean Algebras
Boolen Algebr Boolen Algebrs A Boolen lgebr is set B of vlues together with: - two binry opertions, commonly denoted by + nd, - unry opertion, usully denoted by or ~ or, - two elements usully clled zero
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 information1. Extend QR downwards to meet the x-axis 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 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 informationDEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS
3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationMATH , Calculus 2, Fall 2018
MATH 36-2, 36-3 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
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 informationA - INTRODUCTION AND OVERVIEW
MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS A - INTRODUCTION AND OVERVIEW INTRODUCTION AND OVERVIEW M.N. Tmin, CSMLb, UTM MMJ5 COMPUTATIONAL METHOD IN SOLID MECHANICS Course Content: A INTRODUCTION AND
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of 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 out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
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 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 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
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 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 informationDo the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent?
1 Problem 1 Do the one-dimensionl kinetic energy nd momentum opertors commute? If not, wht opertor does their commuttor represent? KE ˆ h m d ˆP i h d 1.1 Solution This question requires clculting the
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
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 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 informationEquations and Inequalities
Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in
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 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 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 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 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 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, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
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 informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
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 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 informationUnit 1 Exponentials and Logarithms
HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)
More informationIn Section 5.3 we considered initial value problems for the linear second order equation. y.a/ C ˇy 0.a/ D k 1 (13.1.4)
678 Chpter 13 Boundry Vlue Problems for Second Order Ordinry Differentil Equtions 13.1 TWO-POINT BOUNDARY VALUE PROBLEMS In Section 5.3 we considered initil vlue problems for the liner second order eqution
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 informationIn Mathematics for Construction, we learnt that
III DOUBLE INTEGATION THE ANTIDEIVATIVE OF FUNCTIONS OF VAIABLES In Mthemtics or Construction, we lernt tht the indeinite integrl is the ntiderivtive o ( d ( Double Integrtion Pge Hence d d ( d ( The ntiderivtive
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 2009-10 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 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 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 information