CHAPTER 4. Vector Spaces

Size: px
Start display at page:

Download "CHAPTER 4. Vector Spaces"

Transcription

1 man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear mathematcs. In Chapter 2 we studed sstems of lnear equatons, and the theor underlng the soluton of a sstem of lnear equatons can be consdered as a specal case of a general mathematcal framework for lnear problems. To llustrate ths framework, we dscuss an eample. Consder the homogeneous lnear sstem A = 0, where A = It s straghtforward to show that ths sstem has soluton set S = {(r 2s, r, s) : r, s R}. Geometrcall we can nterpret each soluton as definng the coordnates of a pont n space or, equvalentl, as the geometrc vector wth components v = (r 2s, r, s). Usng the standard operatons of vector addton and multplcaton of a vector b a real number, t follows that v can be wrtten n the form v = r(1, 1, 0) + s( 2, 0, 1). We see that ever soluton to the gven lnear problem can be epressed as a lnear combnaton of the two basc solutons (see Fgure 4.0.1): v1 = (1, 1, 0) and v2 = ( 2, 0, 1). 234

2 4.1 Vectors n R n v 2 ( 2, 0, 1) v rv 1 + sv v 1 (1, 1, 0) Fgure 4.0.1: Two basc solutons to A = 0 and an eample of an arbtrar soluton to the sstem. We wll observe a smlar phenomenon n Chapter 6, when we establsh that ever soluton to the homogeneous second-order lnear dfferental equaton can be wrtten n the form + a 1 + a 2 = 0 () = c 1 1 () + c 2 2 (), where 1 () and 2 () are two nonproportonal solutons to the dfferental equaton on the nterval of nterest. In each of these problems, we have a set of vectors V (n the frst problem the vectors are ordered trples of numbers, whereas n the second, the are functons that are at least twce dfferentable on an nterval I) and a lnear vector equaton. Further, n both cases, all solutons to the gven equaton can be epressed as a lnear combnaton of two partcular solutons. In the net two chapters we develop ths wa of formulatng lnear problems n terms of an abstract set of vectors, V, and a lnear vector equaton wth solutons n V. We wll fnd that man problems ft nto ths framework and that the solutons to these problems can be epressed as lnear combnatons of a certan number (not necessarl two) of basc solutons. The mportance of ths result cannot be overemphaszed. It reduces the search for all solutons to a gven problem to that of fndng a fnte number of solutons. As specfc applcatons, we wll derve the theor underlng lnear dfferental equatons and lnear sstems of dfferental equatons as specal cases of the general framework. Before proceedng further, we gve a word of encouragement to the more applcatonorented reader. It wll probabl seem at tmes that the deas we are ntroducng are rather esoterc and that the formalsm s pure mathematcal abstracton. However, n addton to ts nherent mathematcal beaut, the formalsm ncorporates deas that pervade man areas of appled mathematcs, partcularl engneerng mathematcs and mathematcal phscs, where the problems under nvestgaton are ver often lnear n nature. Indeed, the lnear algebra ntroduced n the net two chapters should be consdered an etremel mportant addton to one s mathematcal repertore, certanl on a par wth the deas of elementar calculus. 4.1 Vectors n R n In ths secton, we use some famlar deas about geometrc vectors to motvate the more general and abstract dea of a vector space, whch wll be ntroduced n the net secton. We begn b recallng that a geometrc vector can be consdered mathematcall as a drected lne segment (or arrow) that has both a magntude (length) and a drecton attached to t. In calculus courses, we defne vector addton accordng to the parallelogram law (see Fgure 4.1.1); namel, the sum of the vectors and s the dagonal of

3 236 CHAPTER 4 Vector Spaces the parallelogram formed b and. We denote the sum b +. It can then be shown geometrcall that for all vectors,, z, Fgure 4.1.1: Parallelogram law of vector addton. and + = + (4.1.1) + ( + z) = ( + ) + z. (4.1.2) These are the statements that the vector addton operaton s commutatve and assocatve. The zero vector, denoted 0, s defned as the vector satsfng + 0 =, (4.1.3) for all vectors. We consder the zero vector as havng zero magntude and arbtrar drecton. Geometrcall, we pcture the zero vector as correspondng to a pont n space. Let denote the vector that has the same magntude as, but the opposte drecton. Then accordng to the parallelogram law of addton, + ( ) = 0. (4.1.4) k, k 0 k, k 0 Fgure 4.1.2: Scalar multplcaton of b k. The vector s called the addtve nverse of. Propertes (4.1.1) (4.1.4) are the fundamental propertes of vector addton. The basc algebra of vectors s completed when we also defne the operaton of multplcaton of a vector b a real number. Geometrcall, f s a vector and k s a real number, then k s defned to be the vector whose magntude s k tmes the magntude of and whose drecton s the same as f k > 0, and opposte to f k<0. (See Fgure ) If k = 0, then k = 0. Ths scalar multplcaton operaton has several mportant propertes that we now lst. Once more, each of these can be establshed geometrcall usng onl the foregong defntons of vector addton and scalar multplcaton. For all vectors and, and all real numbers r, s and t, 1 =, (4.1.5) (st) = s(t), (4.1.6) r( + ) = r + r, (4.1.7) (s + t) = s + t. (4.1.8) It s mportant to realze that, n the foregong development, we have not defned a multplcaton of vectors. In Chapter 3 we dscussed the dea of a dot product and cross product of two vectors n space (see Equatons (3.1.4) and (3.1.5)), but for the purposes of dscussng abstract vector spaces we wll essentall gnore the dot product and cross product. We wll revst the dot product n Secton 4.11, when we develop nner product spaces. We wll see n the net secton how the concept of a vector space arses as a drect generalzaton of the deas assocated wth geometrc vectors. Before performng ths abstracton, we want to recall some further features of geometrc vectors and gve one specfc and mportant etenson. We begn b consderng vectors n the plane. Recall that R 2 denotes the set of all ordered pars of real numbers; thus, R 2 ={(, ) : R, R}. The elements of ths set are called vectors n R 2, and we use the usual vector notaton to denote these elements. Geometrcall we dentf the vector v = (, ) n R 2 wth

4 (0, ) v (, ) (, 0) Fgure 4.1.3: Identfng vectors n R 2 wth geometrc vectors n the plane. 4.1 Vectors n R n 237 the geometrc vector v drected from the orgn of a Cartesan coordnate sstem to the pont wth coordnates (, ). Ths dentfcaton s llustrated n Fgure The numbers and are called the components of the geometrc vector v. The geometrc vector addton and scalar multplcaton operatons are consstent wth the addton and scalar multplcaton operatons defned n Chapter 2 va the correspondence wth row (or column) vectors for R 2 : If v = ( 1, 1 ) and w = ( 2, 2 ), and k s an arbtrar real number, then v + w = ( 1, 1 ) + ( 2, 2 ) = ( 1 + 2, ), (4.1.9) kv = k( 1, 1 ) = (k 1,k 1 ). (4.1.10) These are the algebrac statements of the parallelogram law of vector addton and the scalar multplcaton law, respectvel. (See Fgure ) Usng the parallelogram law of vector addton and Equatons (4.1.9) and (4.1.10), t follows that an vector v = (, ) can be wrtten as v = + j = (1, 0) + (0, 1), where = (1, 0) and j = (0, 1) are the unt vectors pontng along the postve - and -coordnate aes, respectvel. ( 1 2, 1 2 ) ( 2, 2 ) w v w v ( 1, 1 ) kv (k 1, k 1 ) Fgure 4.1.4: Vector addton and scalar multplcaton n R 2. The propertes (4.1.1) (4.1.8) are now easl verfed for vectors n R 2. In partcular, the zero vector n R 2 s the vector 0 = (0, 0). Furthermore, Equaton (4.1.9) mples that (, ) + (, ) = (0, 0) = 0, so that the addtve nverse of the general vector v = (, ) s v = (, ). It s straghtforward to etend these deas to vectors n 3-space. We recall that R 3 ={(,,z): R, R,z R}. As llustrated n Fgure 4.1.5, each vector v = (,,z)n R 3 can be dentfed wth the geometrc vector v that jons the orgn of a Cartesan coordnate sstem to the pont wth coordnates (,,z). We call,, and z the components of v.

5 238 CHAPTER 4 Vector Spaces z (0, 0, z) (,, z) v (0,, 0) (, 0, 0) (,, 0) Fgure 4.1.5: Identfng vectors n R 3 wth geometrc vectors n space. Recall that f v = ( 1, 1,z 1 ), w = ( 2, 2,z 2 ), and k s an arbtrar real number, then addton and scalar multplcaton were gven n Chapter 2 b v + w = ( 1, 1,z 1 ) + ( 2, 2,z 2 ) = ( 1 + 2, 1 + 2,z 1 + z 2 ), (4.1.11) kv = k( 1, 1,z 1 ) = (k 1,k 1,kz 1 ). (4.1.12) Once more, these are, respectvel, the component forms of the laws of vector addton and scalar multplcaton for geometrc vectors. It follows that an arbtrar vector v = (,,z)can be wrtten as v = + j + zk = (1, 0, 0) + (0, 1, 0) + z(0, 0, 1), where = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1) denote the unt vectors whch pont along the postve -, -, and z-coordnate aes, respectvel. We leave t as an eercse to check that the propertes (4.1.1) (4.1.8) are satsfed b vectors n R 3, where 0 = (0, 0, 0), and the addtve nverse of v = (,,z)s v = (,, z). We now come to our frst major abstracton. Whereas the sets R 2 and R 3 and ther assocated algebrac operatons arse naturall from our eperence wth Cartesan geometr, the motvaton behnd the algebrac operatons n R n for larger values of n does not come from geometr. Rather, we can vew the addton and scalar multplcaton operatons n R n for n>3as the natural etenson of the component forms of addton and scalar multplcaton n R 2 and R 3 n (4.1.9) (4.1.12). Therefore, n R n we have that f v = ( 1, 2,..., n ), w = ( 1, 2,..., n ), and k s an arbtrar real number, then v + w = ( 1 + 1, 2 + 2,..., n + n ), (4.1.13) kv = (k 1,k 2,...,k n ). (4.1.14) Agan, these defntons are drect generalzatons of the algebrac operatons defned n R 2 and R 3, but there s no geometrc analog when n>3. It s easl establshed that these operatons satsf propertes (4.1.1) (4.1.8), where the zero vector n R n s 0 = (0, 0,...,0), and the addtve nverse of the vector v = ( 1, 2,..., n ) s v = ( 1, 2,..., n ). The verfcaton of ths s left as an eercse.

6 4.1 Vectors n R n 239 Eample If v = (1.2, 3.5, 2, 0) and w = (12.23, 19.65, 23.22, 9.76), then v + w = (1.2, 3.5, 2, 0) + (12.23, 19.65, 23.22, 9.76) = (13.43, 23.15, 25.22, 9.76) and 2.35v = (2.82, 8.225, 4.7, 0). Eercses for 4.1 Ke Terms Vectors n R n, Vector addton, Scalar multplcaton, Zero vector, Addtve nverse, Components of a vector. Sklls Be able to perform vector addton and scalar multplcaton for vectors n R n gven n component form. Understand the geometrc perspectve on vector addton and scalar multplcaton n the cases of R 2 and R 3. Be able to formall verf the aoms (4.1.1) (4.1.8) for vectors n R n. True-False Revew For Questons 1 12, decde f the gven statement s true or false, and gve a bref justfcaton for our answer. If true, ou can quote a relevant defnton or theorem from the tet. If false, provde an eample, llustraton, or bref eplanaton of wh the statement s false. 1. The vector (, ) n R 2 s the same as the vector (,, 0) n R Each vector (,,z) n R 3 has eactl one addtve nverse. 3. The soluton set to a lnear sstem of 4 equatons and 6 unknowns conssts of a collecton of vectors n R For ever vector ( 1, 2,..., n ) n R n, the vector ( 1) ( 1, 2,..., n ) s an addtve nverse. 5. A vector whose components are all postve s called a postve vector. 6. If s and t are scalars and and are vectors n R n, then (s + t)( + ) = s + t. 7. For ever vector n R n, the vector 0 s the zero vector of R n. 8. The parallelogram whose sdes are determned b vectors and n R 2 have dagonals determned b the vectors + and. 9. If s a vector n the frst quadrant of R 2, then an scalar multple k of s stll a vector n the frst quadrant of R The vector 5 6j + 2k n R 3 s the same as (5, 6, 2). 11. Three vectors,, and z n R 3 alwas determne a 3-dmensonal sold regon n R If and are vectors n R 2 whose components are even ntegers and k s a scalar, then + and k are also vectors n R 2 whose components are even ntegers. Problems 1. If = (3, 1), = ( 1, 2), determne the vectors v 1 = 2, v 2 = 3, v 3 = Sketch the correspondng ponts n the -plane and the equvalent geometrc vectors. 2. If = ( 1, 4) and = ( 5, 1), determne the vectors v 1 = 3, v 2 = 4, v 3 = 3+( 4). Sketch the correspondng ponts n the -plane and the equvalent geometrc vectors. 3. If = (3, 1, 2, 5), = ( 1, 2, 9, 2), determne v = 5 + ( 7) and ts addtve nverse. 4. If = (1, 2, 3, 4, 5) and z = ( 1, 0, 4, 1, 2), fnd n R 5 such that 2 + ( 3) = z.

7 240 CHAPTER 4 Vector Spaces 5. Verf the commutatve law of addton for vectors n R Verf the assocatve law of addton for vectors n R Verf propertes (4.1.5) (4.1.8) for vectors n R Show wth eamples that f s a vector n the frst quadrant of R 2 (.e., both coordnates of are postve) and s a vector n the thrd quadrant of R 2 (.e., both coordnates of are negatve), then the sum + could occur n an of the four quadrants. 4.2 Defnton of a Vector Space In the prevous secton, we showed how the set R n of all ordered n-tuples of real numbers, together wth the addton and scalar multplcaton operatons defned on t, has the same algebrac propertes as the famlar algebra of geometrc vectors. We now push ths abstracton one step further and ntroduce the dea of a vector space. Such an abstracton wll enable us to develop a mathematcal framework for studng a broad class of lnear problems, such as sstems of lnear equatons, lnear dfferental equatons, and sstems of lnear dfferental equatons, whch have far-reachng applcatons n all areas of appled mathematcs, scence, and engneerng. Let V be a nonempt set. For our purposes, t s useful to call the elements of V vectors and use the usual vector notaton u, v,...,to denote these elements. For eample, f V s the set of all 2 2 matrces, then the vectors n V are 2 2 matrces, whereas f V s the set of all postve ntegers, then the vectors n V are postve ntegers. We wll be nterested onl n the case when the set V has an addton operaton and a scalar multplcaton operaton defned on ts elements n the followng senses: Vector Addton: A rule for combnng an two vectors n V. We wll use the usual + sgn to denote an addton operaton, and the result of addng the vectors u and v wll be denoted u + v. Real (or Comple) Scalar Multplcaton: A rule for combnng each vector n V wth an real (or comple) number. We wll use the usual notaton kv to denote the result of scalar multplng the vector v b the real (or comple) number k. To combne the two tpes of scalar multplcaton, we let F denote the set of scalars for whch the operaton s defned. Thus, for us, F s ether the set of all real numbers or the set of all comple numbers. For eample, f V s the set of all 2 2 matrces wth comple elements and F denotes the set of all comple numbers, then the usual operaton of matr addton s an addton operaton on V, and the usual method of multplng a matr b a scalar s a scalar multplcaton operaton on V. Notce that the result of applng ether of these operatons s alwas another vector (2 2 matr) n V. As a further eample, let V be the set of postve ntegers, and let F be the set of all real numbers. Then the usual operatons of addton and multplcaton wthn the real numbers defne addton and scalar multplcaton operatons on V. Note n ths case, however, that the scalar multplcaton operaton, n general, wll not eld another vector n V, snce when we multpl a postve nteger b a real number, the result s not, n general, a postve nteger. We are now n a poston to gve a precse defnton of a vector space.

Formulas for the Determinant

Formulas for the Determinant page 224 224 CHAPTER 3 Determnants e t te t e 2t 38 A = e t 2te t e 2t e t te t 2e 2t 39 If 123 A = 345, 456 compute the matrx product A adj(a) What can you conclude about det(a)? For Problems 40 43, use

More information

CHAPTER-5 INFORMATION MEASURE OF FUZZY MATRIX AND FUZZY BINARY RELATION

CHAPTER-5 INFORMATION MEASURE OF FUZZY MATRIX AND FUZZY BINARY RELATION CAPTER- INFORMATION MEASURE OF FUZZY MATRI AN FUZZY BINARY RELATION Introducton The basc concept of the fuzz matr theor s ver smple and can be appled to socal and natural stuatons A branch of fuzz matr

More information

SCALARS AND VECTORS All physical quantities in engineering mechanics are measured using either scalars or vectors.

SCALARS AND VECTORS All physical quantities in engineering mechanics are measured using either scalars or vectors. SCALARS AND ECTORS All phscal uanttes n engneerng mechancs are measured usng ether scalars or vectors. Scalar. A scalar s an postve or negatve phscal uantt that can be completel specfed b ts magntude.

More information

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

COMPLEX NUMBERS AND QUADRATIC EQUATIONS COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not

More information

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix

Lectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

APPENDIX A Some Linear Algebra

APPENDIX A Some Linear Algebra APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,

More information

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be

More information

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0

n α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0 MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector

More information

More metrics on cartesian products

More metrics on cartesian products More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of

More information

Lecture 12: Discrete Laplacian

Lecture 12: Discrete Laplacian Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly

More information

Solutions to Homework 7, Mathematics 1. 1 x. (arccos x) (arccos x) 1

Solutions to Homework 7, Mathematics 1. 1 x. (arccos x) (arccos x) 1 Solutons to Homework 7, Mathematcs 1 Problem 1: a Prove that arccos 1 1 for 1, 1. b* Startng from the defnton of the dervatve, prove that arccos + 1, arccos 1. Hnt: For arccos arccos π + 1, the defnton

More information

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal

Inner Product. Euclidean Space. Orthonormal Basis. Orthogonal Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,

More information

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars

More information

Section 3.6 Complex Zeros

Section 3.6 Complex Zeros 04 Chapter Secton 6 Comple Zeros When fndng the zeros of polynomals, at some pont you're faced wth the problem Whle there are clearly no real numbers that are solutons to ths equaton, leavng thngs there

More information

THE SUMMATION NOTATION Ʃ

THE SUMMATION NOTATION Ʃ Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the

More information

Unit 5: Quadratic Equations & Functions

Unit 5: Quadratic Equations & Functions Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton

More information

One Dimensional Axial Deformations

One Dimensional Axial Deformations One Dmensonal al Deformatons In ths secton, a specfc smple geometr s consdered, that of a long and thn straght component loaded n such a wa that t deforms n the aal drecton onl. The -as s taken as the

More information

9. Complex Numbers. 1. Numbers revisited. 2. Imaginary number i: General form of complex numbers. 3. Manipulation of complex numbers

9. Complex Numbers. 1. Numbers revisited. 2. Imaginary number i: General form of complex numbers. 3. Manipulation of complex numbers 9. Comple Numbers. Numbers revsted. Imagnar number : General form of comple numbers 3. Manpulaton of comple numbers 4. The Argand dagram 5. The polar form for comple numbers 9.. Numbers revsted We saw

More information

Lecture Notes Introduction to Cluster Algebra

Lecture Notes Introduction to Cluster Algebra Lecture Notes Introducton to Cluster Algebra Ivan C.H. Ip Updated: Ma 7, 2017 3 Defnton and Examples of Cluster algebra 3.1 Quvers We frst revst the noton of a quver. Defnton 3.1. A quver s a fnte orented

More information

Physics 201, Lecture 4. Vectors and Scalars. Chapters Covered q Chapter 1: Physics and Measurement.

Physics 201, Lecture 4. Vectors and Scalars. Chapters Covered q Chapter 1: Physics and Measurement. Phscs 01, Lecture 4 Toda s Topcs n Vectors chap 3) n Scalars and Vectors n Vector ddton ule n Vector n a Coordnator Sstem n Decomposton of a Vector n Epected from prevew: n Scalars and Vectors, Vector

More information

Affine transformations and convexity

Affine transformations and convexity Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/

More information

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X

3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number

More information

Chapter Twelve. Integration. We now turn our attention to the idea of an integral in dimensions higher than one. Consider a real-valued function f : D

Chapter Twelve. Integration. We now turn our attention to the idea of an integral in dimensions higher than one. Consider a real-valued function f : D Chapter Twelve Integraton 12.1 Introducton We now turn our attenton to the dea of an ntegral n dmensons hgher than one. Consder a real-valued functon f : R, where the doman s a nce closed subset of Eucldean

More information

MEM 255 Introduction to Control Systems Review: Basics of Linear Algebra

MEM 255 Introduction to Control Systems Review: Basics of Linear Algebra MEM 255 Introducton to Control Systems Revew: Bascs of Lnear Algebra Harry G. Kwatny Department of Mechancal Engneerng & Mechancs Drexel Unversty Outlne Vectors Matrces MATLAB Advanced Topcs Vectors A

More information

2.3 Nilpotent endomorphisms

2.3 Nilpotent endomorphisms s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms

More information

Mathematics Intersection of Lines

Mathematics Intersection of Lines a place of mnd F A C U L T Y O F E D U C A T I O N Department of Currculum and Pedagog Mathematcs Intersecton of Lnes Scence and Mathematcs Educaton Research Group Supported b UBC Teachng and Learnng Enhancement

More information

Physics 2A Chapter 3 HW Solutions

Physics 2A Chapter 3 HW Solutions Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C

More information

1 GSW Iterative Techniques for y = Ax

1 GSW Iterative Techniques for y = Ax 1 for y = A I m gong to cheat here. here are a lot of teratve technques that can be used to solve the general case of a set of smultaneous equatons (wrtten n the matr form as y = A), but ths chapter sn

More information

Group Theory Worksheet

Group Theory Worksheet Jonathan Loss Group Theory Worsheet Goals: To ntroduce the student to the bascs of group theory. To provde a hstorcal framewor n whch to learn. To understand the usefulness of Cayley tables. To specfcally

More information

Difference Equations

Difference Equations Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1

More information

12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product

12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product 12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton

More information

Week 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product

Week 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product The Vector Product Week 11: Chapter 11 Angular Momentum There are nstances where the product of two vectors s another vector Earler we saw where the product of two vectors was a scalar Ths was called the

More information

Problem Do any of the following determine homomorphisms from GL n (C) to GL n (C)?

Problem Do any of the following determine homomorphisms from GL n (C) to GL n (C)? Homework 8 solutons. Problem 16.1. Whch of the followng defne homomomorphsms from C\{0} to C\{0}? Answer. a) f 1 : z z Yes, f 1 s a homomorphsm. We have that z s the complex conjugate of z. If z 1,z 2

More information

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017

U.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017 U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that

More information

Subset Topological Spaces and Kakutani s Theorem

Subset Topological Spaces and Kakutani s Theorem MOD Natural Neutrosophc Subset Topologcal Spaces and Kakutan s Theorem W. B. Vasantha Kandasamy lanthenral K Florentn Smarandache 1 Copyrght 1 by EuropaNova ASBL and the Authors Ths book can be ordered

More information

332600_08_1.qxp 4/17/08 11:29 AM Page 481

332600_08_1.qxp 4/17/08 11:29 AM Page 481 336_8_.qxp 4/7/8 :9 AM Page 48 8 Complex Vector Spaces 8. Complex Numbers 8. Conjugates and Dvson of Complex Numbers 8.3 Polar Form and DeMovre s Theorem 8.4 Complex Vector Spaces and Inner Products 8.5

More information

of Nebraska - Lincoln

of Nebraska - Lincoln Unversty of Nebraska - Lncoln DgtalCommons@Unversty of Nebraska - Lncoln MAT Exam Expostory Papers Math n the Mddle Insttute Partnershp 008 The Square Root of Tffany Lothrop Unversty of Nebraska-Lncoln

More information

Correlation and Regression. Correlation 9.1. Correlation. Chapter 9

Correlation and Regression. Correlation 9.1. Correlation. Chapter 9 Chapter 9 Correlaton and Regresson 9. Correlaton Correlaton A correlaton s a relatonshp between two varables. The data can be represented b the ordered pars (, ) where s the ndependent (or eplanator) varable,

More information

PHYS 705: Classical Mechanics. Calculus of Variations II

PHYS 705: Classical Mechanics. Calculus of Variations II 1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary

More information

DEMO #8 - GAUSSIAN ELIMINATION USING MATHEMATICA. 1. Matrices in Mathematica

DEMO #8 - GAUSSIAN ELIMINATION USING MATHEMATICA. 1. Matrices in Mathematica demo8.nb 1 DEMO #8 - GAUSSIAN ELIMINATION USING MATHEMATICA Obectves: - defne matrces n Mathematca - format the output of matrces - appl lnear algebra to solve a real problem - Use Mathematca to perform

More information

χ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body

χ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown

More information

So far: simple (planar) geometries

So far: simple (planar) geometries Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector

More information

The internal structure of natural numbers and one method for the definition of large prime numbers

The internal structure of natural numbers and one method for the definition of large prime numbers The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract

More information

Physics 5153 Classical Mechanics. Principle of Virtual Work-1

Physics 5153 Classical Mechanics. Principle of Virtual Work-1 P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal

More information

Ballot Paths Avoiding Depth Zero Patterns

Ballot Paths Avoiding Depth Zero Patterns Ballot Paths Avodng Depth Zero Patterns Henrch Nederhausen and Shaun Sullvan Florda Atlantc Unversty, Boca Raton, Florda nederha@fauedu, ssull21@fauedu 1 Introducton In a paper by Sapounaks, Tasoulas,

More information

Chapter 8. Potential Energy and Conservation of Energy

Chapter 8. Potential Energy and Conservation of Energy Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and non-conservatve forces Mechancal Energy Conservaton of Mechancal

More information

e i is a random error

e i is a random error Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where + β + β e for,..., and are observable varables e s a random error How can an estmaton rule be constructed for the unknown

More information

1 Matrix representations of canonical matrices

1 Matrix representations of canonical matrices 1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:

More information

5.76 Lecture #21 2/28/94 Page 1. Lecture #21: Rotation of Polyatomic Molecules I

5.76 Lecture #21 2/28/94 Page 1. Lecture #21: Rotation of Polyatomic Molecules I 5.76 Lecture # /8/94 Page Lecture #: Rotaton of Polatomc Molecules I A datomc molecule s ver lmted n how t can rotate and vbrate. * R s to nternuclear as * onl one knd of vbraton A polatomc molecule can

More information

SL n (F ) Equals its Own Derived Group

SL n (F ) Equals its Own Derived Group Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu

More information

The Fundamental Theorem of Algebra. Objective To use the Fundamental Theorem of Algebra to solve polynomial equations with complex solutions

The Fundamental Theorem of Algebra. Objective To use the Fundamental Theorem of Algebra to solve polynomial equations with complex solutions 5-6 The Fundamental Theorem of Algebra Content Standards N.CN.7 Solve quadratc equatons wth real coeffcents that have comple solutons. N.CN.8 Etend polnomal denttes to the comple numbers. Also N.CN.9,

More information

For all questions, answer choice E) NOTA" means none of the above answers is correct.

For all questions, answer choice E) NOTA means none of the above answers is correct. 0 MA Natonal Conventon For all questons, answer choce " means none of the above answers s correct.. In calculus, one learns of functon representatons that are nfnte seres called power 3 4 5 seres. For

More information

Structure and Drive Paul A. Jensen Copyright July 20, 2003

Structure and Drive Paul A. Jensen Copyright July 20, 2003 Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.

More information

arxiv: v1 [math.ho] 18 May 2008

arxiv: v1 [math.ho] 18 May 2008 Recurrence Formulas for Fbonacc Sums Adlson J. V. Brandão, João L. Martns 2 arxv:0805.2707v [math.ho] 8 May 2008 Abstract. In ths artcle we present a new recurrence formula for a fnte sum nvolvng the Fbonacc

More information

From Biot-Savart Law to Divergence of B (1)

From Biot-Savart Law to Divergence of B (1) From Bot-Savart Law to Dvergence of B (1) Let s prove that Bot-Savart gves us B (r ) = 0 for an arbtrary current densty. Frst take the dvergence of both sdes of Bot-Savart. The dervatve s wth respect to

More information

The Geometry of Logit and Probit

The Geometry of Logit and Probit The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.

More information

Foundations of Arithmetic

Foundations of Arithmetic Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an

More information

CALCULUS CLASSROOM CAPSULES

CALCULUS CLASSROOM CAPSULES CALCULUS CLASSROOM CAPSULES SESSION S86 Dr. Sham Alfred Rartan Valley Communty College salfred@rartanval.edu 38th AMATYC Annual Conference Jacksonvlle, Florda November 8-, 202 2 Calculus Classroom Capsules

More information

2 More examples with details

2 More examples with details Physcs 129b Lecture 3 Caltech, 01/15/19 2 More examples wth detals 2.3 The permutaton group n = 4 S 4 contans 4! = 24 elements. One s the dentty e. Sx of them are exchange of two objects (, j) ( to j and

More information

Grid Generation around a Cylinder by Complex Potential Functions

Grid Generation around a Cylinder by Complex Potential Functions Research Journal of Appled Scences, Engneerng and Technolog 4(): 53-535, 0 ISSN: 040-7467 Mawell Scentfc Organzaton, 0 Submtted: December 0, 0 Accepted: Januar, 0 Publshed: June 0, 0 Grd Generaton around

More information

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look

More information

8.6 The Complex Number System

8.6 The Complex Number System 8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want

More information

Random Walks on Digraphs

Random Walks on Digraphs Random Walks on Dgraphs J. J. P. Veerman October 23, 27 Introducton Let V = {, n} be a vertex set and S a non-negatve row-stochastc matrx (.e. rows sum to ). V and S defne a dgraph G = G(V, S) and a drected

More information

Pre-Calculus Summer Assignment

Pre-Calculus Summer Assignment Pre-Calculus Summer Assgnment Dear Future Pre-Calculus Student, Congratulatons on our successful completon of Algebra! Below ou wll fnd the summer assgnment questons. It s assumed that these concepts,

More information

FREQUENCY DISTRIBUTIONS Page 1 of The idea of a frequency distribution for sets of observations will be introduced,

FREQUENCY DISTRIBUTIONS Page 1 of The idea of a frequency distribution for sets of observations will be introduced, FREQUENCY DISTRIBUTIONS Page 1 of 6 I. Introducton 1. The dea of a frequency dstrbuton for sets of observatons wll be ntroduced, together wth some of the mechancs for constructng dstrbutons of data. Then

More information

j) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1

j) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1 Random varables Measure of central tendences and varablty (means and varances) Jont densty functons and ndependence Measures of assocaton (covarance and correlaton) Interestng result Condtonal dstrbutons

More information

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle

More information

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of

More information

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems

Chapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons

More information

x yi In chapter 14, we want to perform inference (i.e. calculate confidence intervals and perform tests of significance) in this setting.

x yi In chapter 14, we want to perform inference (i.e. calculate confidence intervals and perform tests of significance) in this setting. The Practce of Statstcs, nd ed. Chapter 14 Inference for Regresson Introducton In chapter 3 we used a least-squares regresson lne (LSRL) to represent a lnear relatonshp etween two quanttatve explanator

More information

Quantum Mechanics I - Session 4

Quantum Mechanics I - Session 4 Quantum Mechancs I - Sesson 4 Aprl 3, 05 Contents Operators Change of Bass 4 3 Egenvectors and Egenvalues 5 3. Denton....................................... 5 3. Rotaton n D....................................

More information

PHYS 1443 Section 003 Lecture #17

PHYS 1443 Section 003 Lecture #17 PHYS 144 Secton 00 ecture #17 Wednesda, Oct. 9, 00 1. Rollng oton of a Rgd od. Torque. oment of Inerta 4. Rotatonal Knetc Energ 5. Torque and Vector Products Remember the nd term eam (ch 6 11), onda, Nov.!

More information

Kinematics in 2-Dimensions. Projectile Motion

Kinematics in 2-Dimensions. Projectile Motion Knematcs n -Dmensons Projectle Moton A medeval trebuchet b Kolderer, c1507 http://members.net.net.au/~rmne/ht/ht0.html#5 Readng Assgnment: Chapter 4, Sectons -6 Introducton: In medeval das, people had

More information

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2

Salmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2 Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to

More information

New Method for Solving Poisson Equation. on Irregular Domains

New Method for Solving Poisson Equation. on Irregular Domains Appled Mathematcal Scences Vol. 6 01 no. 8 369 380 New Method for Solvng Posson Equaton on Irregular Domans J. Izadan and N. Karamooz Department of Mathematcs Facult of Scences Mashhad BranchIslamc Azad

More information

SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION

SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION talan journal of pure appled mathematcs n. 33 2014 (63 70) 63 SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION M.R. Farhangdoost Department of Mathematcs College of Scences Shraz Unversty Shraz, 71457-44776

More information

Math1110 (Spring 2009) Prelim 3 - Solutions

Math1110 (Spring 2009) Prelim 3 - Solutions Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.

More information

Interactive Bi-Level Multi-Objective Integer. Non-linear Programming Problem

Interactive Bi-Level Multi-Objective Integer. Non-linear Programming Problem Appled Mathematcal Scences Vol 5 0 no 65 3 33 Interactve B-Level Mult-Objectve Integer Non-lnear Programmng Problem O E Emam Department of Informaton Systems aculty of Computer Scence and nformaton Helwan

More information

p 1 c 2 + p 2 c 2 + p 3 c p m c 2

p 1 c 2 + p 2 c 2 + p 3 c p m c 2 Where to put a faclty? Gven locatons p 1,..., p m n R n of m houses, want to choose a locaton c n R n for the fre staton. Want c to be as close as possble to all the house. We know how to measure dstance

More information

An Introduction to Morita Theory

An Introduction to Morita Theory An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac K-Theory

More information

MTH 263 Practice Test #1 Spring 1999

MTH 263 Practice Test #1 Spring 1999 Pat Ross MTH 6 Practce Test # Sprng 999 Name. Fnd the area of the regon bounded by the graph r =acos (θ). Observe: Ths s a crcle of radus a, for r =acos (θ) r =a ³ x r r =ax x + y =ax x ax + y =0 x ax

More information

Chapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.

Chapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise. Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the

More information

Beyond Zudilin s Conjectured q-analog of Schmidt s problem

Beyond Zudilin s Conjectured q-analog of Schmidt s problem Beyond Zudln s Conectured q-analog of Schmdt s problem Thotsaporn Ae Thanatpanonda thotsaporn@gmalcom Mathematcs Subect Classfcaton: 11B65 33B99 Abstract Usng the methodology of (rgorous expermental mathematcs

More information

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system.

= = = (a) Use the MATLAB command rref to solve the system. (b) Let A be the coefficient matrix and B be the right-hand side of the system. Chapter Matlab Exercses Chapter Matlab Exercses. Consder the lnear system of Example n Secton.. x x x y z y y z (a) Use the MATLAB command rref to solve the system. (b) Let A be the coeffcent matrx and

More information

Discussion 11 Summary 11/20/2018

Discussion 11 Summary 11/20/2018 Dscusson 11 Summary 11/20/2018 1 Quz 8 1. Prove for any sets A, B that A = A B ff B A. Soluton: There are two drectons we need to prove: (a) A = A B B A, (b) B A A = A B. (a) Frst, we prove A = A B B A.

More information

MA 323 Geometric Modelling Course Notes: Day 13 Bezier Curves & Bernstein Polynomials

MA 323 Geometric Modelling Course Notes: Day 13 Bezier Curves & Bernstein Polynomials MA 323 Geometrc Modellng Course Notes: Day 13 Bezer Curves & Bernsten Polynomals Davd L. Fnn Over the past few days, we have looked at de Casteljau s algorthm for generatng a polynomal curve, and we have

More information

Generalized Linear Methods

Generalized Linear Methods Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set

More information

Numerical Heat and Mass Transfer

Numerical Heat and Mass Transfer Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and

More information

Solutions to exam in SF1811 Optimization, Jan 14, 2015

Solutions to exam in SF1811 Optimization, Jan 14, 2015 Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O -5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable

More information

Linear Regression Analysis: Terminology and Notation

Linear Regression Analysis: Terminology and Notation ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented

More information

Fundamental loop-current method using virtual voltage sources technique for special cases

Fundamental loop-current method using virtual voltage sources technique for special cases Fundamental loop-current method usng vrtual voltage sources technque for specal cases George E. Chatzaraks, 1 Marna D. Tortorel 1 and Anastasos D. Tzolas 1 Electrcal and Electroncs Engneerng Departments,

More information

The Order Relation and Trace Inequalities for. Hermitian Operators

The Order Relation and Trace Inequalities for. Hermitian Operators Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence

More information

Mathematical Preparations

Mathematical Preparations 1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the

More information

h-analogue of Fibonacci Numbers

h-analogue of Fibonacci Numbers h-analogue of Fbonacc Numbers arxv:090.0038v [math-ph 30 Sep 009 H.B. Benaoum Prnce Mohammad Unversty, Al-Khobar 395, Saud Araba Abstract In ths paper, we ntroduce the h-analogue of Fbonacc numbers for

More information

Remarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence

Remarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence Remarks on the Propertes of a Quas-Fbonacc-lke Polynomal Sequence Brce Merwne LIU Brooklyn Ilan Wenschelbaum Wesleyan Unversty Abstract Consder the Quas-Fbonacc-lke Polynomal Sequence gven by F 0 = 1,

More information

Week 6, Chapter 7 Sect 1-5

Week 6, Chapter 7 Sect 1-5 Week 6, Chapter 7 Sect 1-5 Work and Knetc Energy Lecture Quz The frctonal force of the floor on a large sutcase s least when the sutcase s A.pushed by a force parallel to the floor. B.dragged by a force

More information

The optimal delay of the second test is therefore approximately 210 hours earlier than =2.

The optimal delay of the second test is therefore approximately 210 hours earlier than =2. THE IEC 61508 FORMULAS 223 The optmal delay of the second test s therefore approxmately 210 hours earler than =2. 8.4 The IEC 61508 Formulas IEC 61508-6 provdes approxmaton formulas for the PF for smple

More information

arxiv: v1 [math.co] 12 Sep 2014

arxiv: v1 [math.co] 12 Sep 2014 arxv:1409.3707v1 [math.co] 12 Sep 2014 On the bnomal sums of Horadam sequence Nazmye Ylmaz and Necat Taskara Department of Mathematcs, Scence Faculty, Selcuk Unversty, 42075, Campus, Konya, Turkey March

More information

DISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization

DISCRIMINANTS AND RAMIFIED PRIMES. 1. Introduction A prime number p is said to be ramified in a number field K if the prime ideal factorization DISCRIMINANTS AND RAMIFIED PRIMES KEITH CONRAD 1. Introducton A prme number p s sad to be ramfed n a number feld K f the prme deal factorzaton (1.1) (p) = po K = p e 1 1 peg g has some e greater than 1.

More information