INTRODUCTION TO LINEAR ALGEBRA

Size: px
Start display at page:

Download "INTRODUCTION TO LINEAR ALGEBRA"

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:

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 information

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

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 information

Chapter 5 Determinants

Chapter 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 information

The Algebra (al-jabr) of Matrices

The 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 information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

SCHOOL 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 information

MATRICES AND VECTORS SPACE

MATRICES 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 information

Introduction To Matrices MCV 4UI Assignment #1

Introduction To Matrices MCV 4UI Assignment #1 Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be

More information

Matrices and Determinants

Matrices 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 information

Geometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.

Geometric 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 information

CHAPTER 2d. MATRICES

CHAPTER 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 information

Algebra Of Matrices & Determinants

Algebra 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 information

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

Matrix 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 information

ECON 331 Lecture Notes: Ch 4 and Ch 5

ECON 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 information

September 13 Homework Solutions

September 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 information

THE DISCRIMINANT & ITS APPLICATIONS

THE 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 information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT

SCHOOL 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 information

Chapter 2. Determinants

Chapter 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 information

a 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.

a 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 information

DETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ

DETERMINANTS. 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 information

Elements of Matrix Algebra

Elements 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 information

Engineering Analysis ENG 3420 Fall Dan C. Marinescu Office: HEC 439 B Office hours: Tu-Th 11:00-12:00

Engineering 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 information

MATRIX DEFINITION A matrix is any doubly subscripted array of elements arranged in rows and columns.

MATRIX 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 information

Matrix Eigenvalues and Eigenvectors September 13, 2017

Matrix 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 information

HW3, Math 307. CSUF. Spring 2007.

HW3, 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 information

Matrix & Vector Basic Linear Algebra & Calculus

Matrix & 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 information

Operations with Matrices

Operations 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 information

Multivariate problems and matrix algebra

Multivariate 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 information

Determinants Chapter 3

Determinants 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 information

A Matrix Algebra Primer

A 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 information

Lecture Solution of a System of Linear Equation

Lecture 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 information

Operations with Polynomials

Operations 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 information

308K. 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:

308K. 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 information

Introduction to Determinants. Remarks. Remarks. The determinant applies in the case of square matrices

Introduction 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 information

MATHEMATICS FOR MANAGEMENT BBMP1103

MATHEMATICS 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 information

Matrices 13: determinant properties and rules continued

Matrices 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 information

Things to Memorize: A Partial List. January 27, 2017

Things 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 information

CSCI 5525 Machine Learning

CSCI 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 information

Module 6: LINEAR TRANSFORMATIONS

Module 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 information

Chapter 2. Vectors. 2.1 Vectors Scalars and Vectors

Chapter 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 information

A matrix is a set of numbers or symbols arranged in a square or rectangular array of m rows and n columns as

A 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 information

Before 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!!!

Before 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 information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC 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 information

How 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?

How 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 information

The Islamic University of Gaza Faculty of Engineering Civil Engineering Department. Numerical Analysis ECIV Chapter 11

The 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 information

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 2

CS434a/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 information

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

Analytically, 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 information

Quadratic Forms. Quadratic Forms

Quadratic 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 information

LINEAR 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.

LINEAR 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 information

Introduction to Group Theory

Introduction 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

Chapter 1: Fundamentals

Chapter 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 information

DonnishJournals

DonnishJournals 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 information

DETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ

DETERMINANTS. 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 information

fractions Let s Learn to

fractions 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 information

Lesson 1: Quadratic Equations

Lesson 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 information

Theoretical foundations of Gaussian quadrature

Theoretical 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 information

N 0 completions on partial matrices

N 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 information

Numerical Linear Algebra Assignment 008

Numerical 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 information

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-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 information

AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system

AQA 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 information

On the diagram below the displacement is represented by the directed line segment OA.

On 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 information

Here we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.

Here 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 information

Boolean Algebra. Boolean Algebras

Boolean 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 information

Computing The Determinants By Reducing The Orders By Four

Computing 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 information

1. Extend QR downwards to meet the x-axis at U(6, 0). y

1. 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 information

Matrices. Introduction

Matrices. 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 information

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS

DEFINITION 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 information

Calculus 2: Integration. Differentiation. Integration

Calculus 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 information

MATH , Calculus 2, Fall 2018

MATH , 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 information

Rudimentary Matrix Algebra

Rudimentary 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 information

A - INTRODUCTION AND OVERVIEW

A - 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 information

Definite integral. Mathematics FRDIS MENDELU

Definite 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 information

How 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?

How 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 information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher 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 information

4.5 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. 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 information

Chapter 1: Logarithmic functions and indices

Chapter 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 information

Elementary Linear Algebra

Elementary 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 information

Math 4310 Solutions to homework 1 Due 9/1/16

Math 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 information

Do the one-dimensional kinetic energy and momentum operators commute? If not, what operator does their commutator represent?

Do 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 information

Definite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30

Definite 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 information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND 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 information

FACULTY OF ENGINEERING TECHNOLOGY GROUP T LEUVEN CAMPUS INTRODUCTORY COURSE MATHEMATICS

FACULTY 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 information

Equations and Inequalities

Equations 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 information

CHAPTER 1 PROGRAM OF MATRICES

CHAPTER 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 information

Math 75 Linear Algebra Class Notes

Math 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 information

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.

Lecture 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 information

Boolean Algebra. Boolean Algebra

Boolean 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 information

Mathematics. Area under Curve.

Mathematics. 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 information

Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Math 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 information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 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 information

Summary: Method of Separation of Variables

Summary: 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 information

Review of Gaussian Quadrature method

Review 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 information

MathCity.org Merging man and maths

MathCity.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 information

Linear Algebra Introduction

Linear 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 information

Unit 1 Exponentials and Logarithms

Unit 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 information

In 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)

In 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 information

Matrix Solution to Linear Equations and Markov Chains

Matrix 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 information

In Mathematics for Construction, we learnt that

In 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 information

Jim Lambers MAT 169 Fall Semester Lecture 4 Notes

Jim 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 information

Optimization Lecture 1 Review of Differential Calculus for Functions of Single Variable.

Optimization 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 information

MATHEMATICS AND STATISTICS 1.2

MATHEMATICS 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