# Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the physicists. In mny engineering opertions, we need to invoke the concept of mtrix, nd sometimes, it is more of routine work wherein the physicl quntities re expressed by liner opertors on vector spce. Very often, there is quite close correspondence between experimentl vlues with those clculted by the use of this forml or rther bstrct method, which nturlly gives lot of credibility to mtrix formlism. Hence, cler concept of mtrix lgebr should be given in the first chpter of this book. In quntum mechnics nd in mny other modern brnches of physics, we hve to operte through mtrix, nd the mtrix representtion is more often necessry, i.e., how Heisenberg Mtrix formlism cme into existence. 1.1 MATRIX ALGEBRA The study of mtrices is mostly motivted by the necessity to solve system of liner simultneous equtions of the form: 11 x x n x n = b 1 21 x x n x n = b 2 n1 x 1 + n2 x nn x n = b n (1.1) where, x 1, x 2,.... x n re the unknowns. The eqution (1.1) cn be esily expressed in the mtrix form s: Ax = b (1.2) where, A is squre mtrix of dimension (n n). Here, x nd b re vectors of dimension (n 1). All these quntities re given by the following reltions: é n éx1 éb1 A= n, x = x 2, b = b ën1 n2... nn ëx n ëb n From the bove mtrices, we notice tht mtrix is simply n rry of elements, clled mtrix elements in the usul prlnce in the field of physics nd engineering. The bove

2 2 Mthemticl Physics for Engineers mtrix A cn lso be denoted s [A]. There re different elements in different positions of the rows nd columns, e.g., n element locted t the i th row nd j th column of the mtrix A is simply denoted by ij. If ij = 0 for ll i nd j, then A is clled null mtrix. The mtrix is lso denoted by [ ij ]. The question of multipliction of two mtrices A nd x is implicit in the bove equtions. The dot product of the i th row of A with the vector x is equl to b i giving rise to i th eqution in (1.1), s lso shown little lter in this section Row nd Column Vectors A mtrix of dimension (1 n) is clled row vector, wheres mtrix of dimension (m 1) is clled column vector. Let us tke some exmples s: 1. c = [1 1 3] is (1 3) row vector, nd é d = is (4 1) column vector. 4 ë MATRIX OPERATIONS After properly defining the bsics of mtrix lgebr, it is useful to describe importnt opertions for mtrices s below: Addition nd Subtrction Let us consider two mtrices A nd B with the dimensions of both (m n). Then, the sum C = A + B is defined s: c ij = ij + b ij (1.3) It mens tht the (ij) th component of C cn be obtined by dding the (ij) th component of A to the (ij) th component of B. é 3 2 é1 2 + ë 2 4 ë4 0 = é 4 0 (1.4) ë2 4 Similrly, the subtrction of the mtrices cn be defined nd explined. The following lws re lso vlid for ddition of the mtrices of the sme order in two different cses s: () For Commuttive Cse: A + B = B + A (b) For Associtive Cse: (A + B) + C = A + (B + C) Multipliction by Sclr The multipliction of mtrix A by sclr c cn be defined s: ca = [c ij ] (1.5) As n exmple, let us write the following mtrix s: é ë = 3 é ë 5 7 (1.6)

3 1.2.3 Mtrix Multipliction Mtrix Algebr 3 Let us tke (m n) mtrix A nd nother mtrix B of dimension (n p). The product of these two mtrices results in mtrix C of dimension (m p), s shown below: A B = C (1.7) (m n)(n p) (m p) The (ij) th component of C is obtined by tking the dot product s: c ij = (i th row of A). ( j th column of B) (1.8) As n exmple, let us show tht: é1 4 é ë = é (1.9) ë 0 3 ë 10 7 It hs to be noted tht AB ¹ BA. Actully, BA my not even be defined, since the number of columns of B my not equl the number of rows of A. The commuttive lw of multipliction is not vlid (in generl) for mtrix product, i.e., AB ¹ BA. But, the ssocitive lw of multipliction is vlid for the mtrix product, i.e., A(BC) = (AB)C Differentition nd Integrtion It is known tht the components of mtrix do not hve to be sclrs. They my lso be functions. As n exmple, let us tke: A= é 2 x + y x 2xy (1.10) ë4 + x y In such sitution, the mtrix cn be differentited nd integrted. The derivtive or the integrl of given mtrix is simply the derivtive or the integrl of ech component of the mtrix. Therefore, we cn write these mtrices s: d édij () x A( x) = (1.11) dx ë dx ò Adxdy = é ëò ij dxdy (1.12) If A be the mtrix of constnts hving dimension (n n) nd x = [x 1, x 2,.... x n ] be column vector of n vribles, then we cn cite n interesting cse by using eqution (1.12), whereby the derivtive of Ax with respect to the vrible x p is given s: d dx p (A x) = p (1.13) Here, p is the p th column of the mtrix A, which cn be esily verified by writing down the whole eqution Ax in the mtrix form. 1.3 PROPERTIES For n rbitrry mtrix A, it is quite importnt to know some bsics of their properties like trnsposition, complex conjugtion, etc. Now, some of these importnt properties of the rbitrry mtrices will be discussed here.

4 4 Mthemticl Physics for Engineers Trnsposition If the mtrix A = [ ij ], then the trnspose of n rbitrry mtrix A, usully denoted by A T, is defined by A T = [ ji ]. Therefore, the rows of A becomes the columns of A T. Let us tke n exmple s: é A = then, A T é = 0 5 ë ë 3 2 Generlly speking, if A is of dimension (m n), then A T hs to be of dimension (n m), i.e., the trnsposition is chieved by interchnging corresponding rows nd columns of A. The trnspose of product is defined s the product of the trnsposes in reverse order s: (ABC) T = C T B T A T (1.14) Complex Conjugte Mtrix The complex conjugte of n rbitrry mtrix A is formulted by tking the complex conjugte of ech element. Let us tke n exmple s: A* = ij * for ll i nd j. é3 + 2i 4 6i * é3 2i 4 + 6i A=,A = ë 4 3i ë 4 3i If A* = A, then A is rel mtrix, which is importnt in physicl situtions. In this ctegory, there is nother importnt mtrix clled Hermitin conjugte, denoted by Ay, which is obtined by tking the complex conjugte of the mtrix, nd then the trnspose of this complex conjugte mtrix, which is useful in certin pplictions. 1.4 SQUARE MATRICES The squre mtrix is so simple tht it is not discussed here with ny detils, since we mostly come cross such mtrices in the engineering problems. Any mtrix whose number of columns equls the number of rows is clled squre mtrix, e.g., simple (2 2) mtrix or (4 4) mtrix commonly occurring routinely in our problems. Certin importnt squre mtrices re described below Digonl Mtrix After understnding simple squre mtrix, it is esy to grsp digonl mtrix, which is ctully squre mtrix with nonzero mtrix elements only long the principl digonl. This type of mtrix is commonly encountered in mny problems of solid stte physics. The typicl exmple of digonl mtrix is shown s: é3 0 0 A= (1.15) ë0 0 6 Here, it is clerly seen tht the nondigonl mtrix elements re ll zero, wheres the digonl elements re ll finite numbers. It hs to be noted tht if the vlue of the determinnt of the mtrix A, i.e., det A = 0, then this mtrix is sid to be singulr mtrix.

5 1.4.2 Identity Mtrix Mtrix Algebr 5 This is lso clled unit mtrix. This identity or unit mtrix is digonl mtrix with 1 s long the principl digonl, s shown below: é I= (1.16) ë If I is of dimension (n n) nd x is vector of dimension (n 1), then we cn write: Ix = x (1.17) This unit mtrix cn be generlly described by Kronecker delt (d ij ) Symmetric Mtrix A symmetric mtrix is gin squre mtrix, whose mtrix elements stisfy the following: ij = ji (1.18) or, on n equivlent bsis, for A to be symmetric mtrix, we cn write it s: A= A T (1.19) It mens tht ll the mtrix elements, which re locted symmetriclly with respect to both sides of the principl digonl, re equl, s evident below: é A= (1.20) ë If A T = A, then A is clled n ntisymmetric or skew mtrix. These mtrices hve importnt pplictions s Puli mtrices to describe the spin properties of n electron. Typiclly, this cn be shown s: é0 i T é 0 i x 2 =, x2 = ëi 0 i 0 = x 2 ë Upper Tringulr Mtrix This is simple mtrix, but it is quite uncommon in engineering problems. In this cse, it is mtrix whose mtrix elements below the principl digonl re ll zero, s show below: UT = é ë (1.21) Determinnt of Mtrix Here, gin we del with squre mtrix. The determinnt of squre mtrix A is sclr quntity, which is denoted by det A. There is method of cofctors, which re used here to show the determinnts of (2 2) nd (3 3) mtrix s:

6 6 Mthemticl Physics for Engineers é det ë é det ë = (1.22) = ( ) ( ) + ( )(1.23) Mtrix Inversion This is perhps one of the most importnt of ll mtrices discussed so fr due to its sheer importnce in the pplictions of mtrix lgebr in the field of engineering nd mthemticl physics. Here, gin we del with squre mtrix A. Now, if det A ¹ 0 (to void infinity ), then the mtrix A hs n inverse, which is simply denoted by A 1. The inverse mtrix stisfies the following reltion s: A 1 A = A A 1 = I (1.24) If the det A ¹ 0, then the question of nonsingulrity comes, nd we cn sy tht the mtrix A is nonsingulr. But, if det A = 0, then we get singulr mtrix A, nd in this cse the inverse cnnot be defined for obvious resons. By eliminting the ith row nd the jth column of squre mtrix A, we get the minor M ij, which is the determinnt of (n 1 n 1) mtrix. Here, the cofctors C ij of mtrix A cn be written s: C ij = ( 1) i+j M ij (1.25) Adjoint of Mtrix The mtrix elements C ij mkes the mtrix C, which is clled cofctor mtrix, which hs reltion with the djoint of mtrix. The djoint of this mtrix A cn be defined s: Adj A = C T (1.26) But, this djoint mtrix A hs lso reltion with the inverse of squre mtrix A, which cn be described s: A 1 = Adj A / det A (1.27) As n exmple, we cn write the inverse of (2 2) mtrix A s: é ë = 1 é det A ë (1.28) This mtrix long with selfdjoint mtrix is very importnt in quntum mechnics nd in other fields of mthemticl physics. If Adj A = A, then A is sid to be selfdjoint. Here, it is importnt to mention bout nother importnt mtrix in physics, i.e., Hermitin mtrix, i.e., Ay = A, then A is sid to be Hermitin mtrix, which is lwys rel, which is necessry nd lso useful in quntum mechnics to find the vlue of different mesurble vlues of the observbles. The other importnt squre mtrices like Orthogonl mtrix re not discussed here, since the entire gmut of squre mtrices described bove lredy give enough insight into different types of useful mtrices in mthemticl physics.

7 Mtrix Algebr EIGENVALUES AND EIGENVECTORS This is the most importnt topic in the mtrix lgebr. It is very useful in quntum mechnics nd host of other subjects in physics nd engineering. First of ll, we hve to pose problem s n eigenvlue problem. Let us consider the eigenvlue problem s: Ay = ly (1.29) where, A is the usul squre mtrix (n n) signifying liner opertor, s described bove, y is n eigenvector or eigenfunction nd l is the chrcteristic or eigenvlue. Here, if we desire nontrivil solution, i.e. we wnt nonzero eigenvector y nd the consequent eigenvlue l, which must stisfy the bove eqution (1.29). We cn lso write the mtrix form of n eigenvlue problem by using eqution (1.29) s: (A li)y = 0 (1.30) It is very esily noted tht nonzero solution for y will be obtined when (A li) is singulr mtrix, or it cn be rticulted s: det (A li) = 0 (1.31) This eqution (1.31) is normlly clled the seculr or chrcteristic eqution of A. This eqution cn be solved for the n roots or rther different eigenvlues l 1, l 2,....l n. For ech of these eigenvlues (l i ) obtined by expnding the determinnt in eqution (1.31), the corresponding eigenvectors (y i ) cn then be obtined s: (A l i I)y i = 0 (1.32) It hs to be noted tht the bove eigenvectors (y i ) cn be determined only to within multiplictive constnt, since (A l i I) is singulr mtrix. It is better to tke n exmple to explin the eigenvlue problem s: Exmple 1 A= é 3 4 ë4 3 In order to mke n eigenvlue problem, the bove mtrix cn be written in the usul determinnt form s: 3 l l = 0 or, 9 + l 2 16 = 0 l = ± 5, l 1 = 5, l 2 = + 5 So, we get the eigenvlues (l 1 nd l 2 ). Now, we hve to find the corresponding eigenvectors s: é1 x 1 = ë 2 nd, x 2 = é 2 ë1 Let us tke é1 2 P= ë 2 1 Then, we cn write the inverse of this mtrix s:

8 8 Mthemticl Physics for Engineers P 1 1 é 1 2 = 5 ë 2 1 Therefore, finlly, we cn write it s: P 1 1 é 1 2é3 4é AP = ë 2 1ë4 3ë 2 1 = 1 é 1 2 é ë 2 1 ë é25 0 é5 0 = = 5 ë 0 25 ë 0 5 = é l1 0 ë 0 l2 The bove sums up the eigenvlue problem, which hs myrids of useful exmples (s described briefly in the Premble), in different brnches of physics nd engineering.

### Lecture 2: Matrix Algebra. General

Lecture 2: Mtrix Algebr Generl Definitions Algebric Opertions Vector Spces, Liner Independence nd Rnk of Mtrix Inverse Mtrix Liner Eqution Systems, the Inverse Mtrix nd Crmer s Rule Chrcteristic Roots

### DETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.

Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of

### LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

### INVERSE OF A MATRIX AND ITS APPLICATION O Q

Inverse f A tri And Its Appliction DUE - III IVERSE F A ATRIX AD ITS AICATI ET US CSIDER A EXAE: Abhinv spends Rs. in buying pens nd note books wheres Shntnu spends Rs. in buying 4 pens nd note books.we

### and thus, they are similar. If k = 3 then the Jordan form of both matrices is

Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

### MODULE 3. 0, y = 0 for all y

Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)

### In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.

Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix

### Generalized Inverses: How to Invert a Non-Invertible Matrix

Generlized Inverses: How to Invert Non-Invertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax

### 4.11 Inner Product Spaces

314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

### Novel Methods of Generating Self-Invertible Matrix for Hill Cipher Algorithm

Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy Novel Methods of Generting Self-Invertible Mtrix for Hill Cipher lgorithm Bibhudendr chry Deprtment of Electronics & Communiction Engineering

### 9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

### MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions Section 3 Integrl Equtions Integrl Opertors nd Liner Integrl Equtions As we sw in Section on opertor nottion, we work with functions

### Algebra Review. How well do you remember your algebra?

Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

### Uniform convergence and its consequences

Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,

### has the desired form. On the other hand, its product with z is 1. So the inverse x

First homework ssignment p. 5 Exercise. Verify tht the set of complex numers of the form x + y 2, where x nd y re rtionl, is sufield of the field of complex numers. Solution: Evidently, this set contins

### 4.0 5-Minute Review: Rational Functions

mth 130 dy 4: working with limits 1 40 5-Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two

### Vectors 2. 1. Recap of vectors

Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

### Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

### Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

### Addition and subtraction of rational expressions

Lecture 5. Addition nd subtrction of rtionl expressions Two rtionl expressions in generl hve different denomintors, therefore if you wnt to dd or subtrct them you need to equte the denomintors first. The

### UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES

UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES Solution to exm in: FYS30, Quntum mechnics Dy of exm: Nov. 30. 05 Permitted mteril: Approved clcultor, D.J. Griffiths: Introduction to Quntum

### Solutions to Section 1

Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this

### 1. Inverse of a tridiagonal matrix

Pré-Publicções do Deprtmento de Mtemátic Universidde de Coimbr Preprint Number 05 16 ON THE EIGENVALUES OF SOME TRIDIAGONAL MATRICES CM DA FONSECA Abstrct: A solution is given for problem on eigenvlues

### 5.1 Second-Order linear PDE

5.1 Second-Order liner PDE Consider second-order liner PDE L[u] = u xx + 2bu xy + cu yy + du x + eu y + fu = g, (x,y) U (5.1) for n unknown function u of two vribles x nd y. The functions,b nd c re ssumed

### Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.

Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:

### 1B METHODS LECTURE NOTES. PART I: Fourier series, Self adjoint ODEs

1B Methods 1. 1B METHODS ECTURE NOTES Richrd Jozs, DAMTP Cmbridge rj31@cm.c.uk October 213 PART I: Fourier series, Self djoint ODEs 1B Methods 2 PREFACE These notes (in four prts cover the essentil content

### 11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.

. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for

### 2m + V ( ˆX) (1) 2. Consider a particle in one dimensions whose Hamiltonian is given by

Teoretisk Fysik KTH Advnced QM SI2380), Exercise 8 12 1. 3 Consider prticle in one dimensions whose Hmiltonin is given by Ĥ = ˆP 2 2m + V ˆX) 1) with [ ˆP, ˆX] = i. By clculting [ ˆX, [ ˆX, Ĥ]] prove tht

### NUMBER SYSTEMS CHAPTER 1. (A) Main Concepts and Results

CHAPTER NUMBER SYSTEMS Min Concepts nd Results Rtionl numbers Irrtionl numbers Locting irrtionl numbers on the number line Rel numbers nd their deciml expnsions Representing rel numbers on the number line

### Chapter 6 Solving equations

Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign

### Number Systems & Working With Numbers

Presenting the Mths Lectures! Your best bet for Qunt... MATHS LECTURE # 0 Number Systems & Working With Numbers System of numbers.3 0.6 π With the help of tree digrm, numbers cn be clssified s follows

### Physics 6010, Fall 2010 Symmetries and Conservation Laws: Energy, Momentum and Angular Momentum Relevant Sections in Text: 2.6, 2.

Physics 6010, Fll 2010 Symmetries nd Conservtion Lws: Energy, Momentum nd Angulr Momentum Relevnt Sections in Text: 2.6, 2.7 Symmetries nd Conservtion Lws By conservtion lw we men quntity constructed from

### Lecture 3 Basic Probability and Statistics

Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The

### EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

### Factoring Polynomials

Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

### 1 Numerical Solution to Quadratic Equations

cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll

### Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

### PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

### 4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS

4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem

Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.

### Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra

Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to

### SPECIAL PRODUCTS AND FACTORIZATION

MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

### 6.2 Volumes of Revolution: The Disk Method

mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

### Operations with Polynomials

38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

### Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables

The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long

### A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

### Introduction to Linear Algebra using MATLAB Tutorial on Material Covered in ENG EK 127 Relevant to Linear Algebra.

Introduction to Liner Algebr using MATLAB Tutoril on Mteril Covered in ENG EK 7 Relevnt to Liner Algebr By Stormy Attwy Reference: Stormy Attwy, MATLAB: A Prcticl Introduction to Progrmming nd Problem

### g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany

Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required

### Binary Representation of Numbers Autar Kaw

Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

### Education Spending (in billions of dollars) Use the distributive property.

0 CHAPTER Review of the Rel Number System 96. An pproximtion of federl spending on eduction in billions of dollrs from 200 through 2005 cn be obtined using the e xpression y = 9.0499x - 8,07.87, where

### VECTOR-VALUED FUNCTIONS

VECTOR-VALUED FUNCTIONS MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the book: Sections 13.1. 13.2. Wht students should definitely get: Definition of vector-vlued function, reltion with prmetric

### Curve Sketching. 96 Chapter 5 Curve Sketching

96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of

### Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

### Math Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.

Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while

### Section A-4 Rational Expressions: Basic Operations

A- Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr open-topped bo is to be constructed out of 9- by 6-inch sheets of thin crdbord by cutting -inch squres out of ech corner nd bending the

### Exponents base exponent power exponentiation

Exonents We hve seen counting s reeted successors ddition s reeted counting multiliction s reeted ddition so it is nturl to sk wht we would get by reeting multiliction. For exmle, suose we reetedly multily

### Review guide for the final exam in Math 233

Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

### Vector differentiation. Chapters 6, 7

Chpter 2 Vectors Courtesy NASA/JPL-Cltech Summry (see exmples in Hw 1, 2, 3) Circ 1900 A.D., J. Willird Gis invented useful comintion of mgnitude nd direction clled vectors nd their higher-dimensionl counterprts

### Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity

Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University

### Rational Numbers - Grade 10 [CAPS]

OpenStx-CNX module: m848 Rtionl Numers - Grde 0 [CAPS] Free High School Science Texts Project Bsed on Rtionl Numers y Rory Adms Free High School Science Texts Project Mrk Horner Hether Willims This work

### Math 135 Circles and Completing the Square Examples

Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

### Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

### Finite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh

Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is

### Section 4.3. By the Mean Value Theorem, for every i = 1, 2, 3,..., n, there exists a point c i in the interval [x i 1, x i ] such that

Difference Equtions to Differentil Equtions Section 4.3 The Fundmentl Theorem of Clculus We re now redy to mke the long-promised connection between differentition nd integrtion, between res nd tngent lines.

### 1.2 The Integers and Rational Numbers

.2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl

### All pay auctions with certain and uncertain prizes a comment

CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin

### Mathematics Higher Level

Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

### Lecture 3 Gaussian Probability Distribution

Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

### CHAPTER 5 Spline Approximation of Functions and Data

CHAPTER 5 Spline Approximtion of Functions nd Dt This chpter introduces number of methods for obtining spline pproximtions to given functions, or more precisely, to dt obtined by smpling function. In Section

### CHAPTER 5a. SIMULTANEOUS LINEAR EQUATIONS

CHAPTER 5. SIMULTANEOUS LINEAR EQUATIONS A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering by Dr. Ibrhim A. Asskkf Spring 00 ENCE 0 - Computtion Methods in Civil Engineering

### Control Systems

6.5 Control Systems Lst Time: trix Opertions Fundmentl to Liner Algebr Determinnt trix ultipliction Eigenvlue nk th. Descriptions of Systems ~ eview LTI Systems: Stte Vrible Description Lineriztion Tody:

### Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: x n+ x n n + + C, dx = ln x + C, if n if n = In prticulr, this mens tht dx = ln x + C x nd x 0 dx = dx = dx = x + C Integrl of Constnt:

### Numerical Solutions of Linear Systems of Equations

EE 6 Clss Notes Numericl Solutions of Liner Systems of Equtions Liner Dependence nd Independence An eqution in set of equtions is linerly independent if it cnnot e generted y ny liner comintion of the

### Ae2 Mathematics : Fourier Series

Ae Mthemtics : Fourier Series J. D. Gibbon (Professor J. D Gibbon, Dept of Mthemtics j.d.gibbon@ic.c.uk http://www.imperil.c.uk/ jdg These notes re not identicl word-for-word with my lectures which will

### Algorithms Chapter 4 Recurrences

Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering

### Quadratic Equations. Math 99 N1 Chapter 8

Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree

### Basic Analysis of Autarky and Free Trade Models

Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

### Reasoning to Solve Equations and Inequalities

Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

### Sequences and Series

Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic

### Graphs on Logarithmic and Semilogarithmic Paper

0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

### Tests for One Poisson Mean

Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution

### Numerical integration

Chpter 4 Numericl integrtion Contents 4.1 Definite integrls.............................. 4. Closed Newton-Cotes formule..................... 4 4. Open Newton-Cotes formule...................... 8 4.4

### Lectures 8 and 9 1 Rectangular waveguides

1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

### Net Change and Displacement

mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the

### 9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

### CHAPTER 11 Numerical Differentiation and Integration

CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods

### Scalar Line Integrals

Mth 3B Discussion Session Week 5 Notes April 6 nd 8, 06 This week we re going to define new type of integrl. For the first time, we ll be integrting long something other thn Eucliden spce R n, nd we ll

### Sect 8.3 Triangles and Hexagons

13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed two-dimensionl geometric figure consisting of t lest three line segments for its

### The Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx

The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the single-vrible chin rule extends to n inner

### Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

### r 2 F ds W = r 1 qe ds = q

Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study

### MATLAB Workshop 13 - Linear Systems of Equations

MATLAB: Workshop - Liner Systems of Equtions pge MATLAB Workshop - Liner Systems of Equtions Objectives: Crete script to solve commonly occurring problem in engineering: liner systems of equtions. MATLAB

Section.3 - Properties of rphs of Qudrtic Functions Specific Curriculum Outcomes covered C3 Anlyze nd describe the chrcteristics of qudrtic functions C3 Solve problems involving qudrtic equtions F Anlyze

Alger Module A60 Qudrtic Equtions - 1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions - 1 Sttement of Prerequisite

### Fourier Analysis. Chapter 3

Chpter 3 Fourier Anlysis Fourier nlysis is concerned with the decomposition of periodic functions into sine nd cosine components. A series of sine nd cosine functions often form prt of generl solution

### Integration. 148 Chapter 7 Integration

48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

### Matrix Inverse and Condition

Mtrix Inverse nd Condition Berlin Chen Deprtment of Computer Science & Informtion Engineering Ntionl Tiwn Norml University Reference: 1. Applied Numericl Methods with MATLAB for Engineers, Chpter 11 &