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


 Adelia Hawkins
 4 years ago
 Views:
Transcription
1 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 round s long s it doesn t grow/shrink or get tilted. If you multiply vector y positive sclr, the direction stys the sme, nd the length is scled. If you multiply vector y negtive sclr, the direction ecomes opposite (ut still prllel), nd the length is scled. We cll two vectors prllel if they re sclr multiples of ech other. We geometriclly dd two vectors together y plcing the hed of one t the til of the other, nd drwing the sum vector from the til of the first to the hed of the second. This is equivlent to tking prticulr digonl of the prllelogrm formed y the two vectors. Using the ove definition, we see tht the difference of two vectors cn e drwn geometriclly y tking the two vectors with their tils together, nd drwing vector etween their heds. R n refers to the spce of vectors with n coordintes. So, R 2 is ll vectors with two coordintes tht is, ll points in the xyplne. Similrly, R 3 is ll vectors with three coordintes tht is, ll points in xyzspce. A unit vector is vector with length 1. In R 2, i = [1, 0] nd j = [1, 0]. So, they re the unit vectors in the directions of the positive xes. Similrly, in R 3, i = [1, 0, 0], j = [0, 1, 0], nd k = [0, 0, 1]. The length of vector = [ 1, 2,..., n ] is Dot Product = n The dot product comines two vectors (with the sme numer of coordintes) nd gives numer. (By contrst, the cross product comines two vectors in R 3 nd gives third vector in R 3.) To clculte the dot product of two vectors, you multiply together their corresponding coordintes, nd dd the products. For exmple: = [ (8)( 1) + (3)(2) + (10)(5) + (3)(6) = 66 ] 3 6 1
2 Given two vectors nd, when we tke their dot product, it tells us something out the ngle θ etween the two of them: = cos θ It follows from the previous ullet point tht two nonzero vectors re orthogonl (or perpendiculr) if nd only if their dot product is zero. This is very common use of the dot product determining orthogonlity. 2 = Projections The projection of onto is the component of tht is prllel to. The picture is this: We drw line from the tip of to tht meets t right ngle. The point where it meets is the hed of proj, nd the til is the til of. So, the lue vector is the projection. We usully show these s cute ngles, ut they work for otuse lso. We relly only cre out the line in the direction of. If the ngle is otuse, the picture is like this: We cn clculte projection s follows: proj = ( ) 2 The dot product ehves lgericlly together with sclr multipliction nd vector ddition much the wy you would like it to. For exmple, ( + c) = + c. Determinnts of Smll Mtrices The determinnt of squre mtrix is numer. So the determinnt function hs squre mtrix s its input, nd numer s its output.
3 To clculte the determinnt of 2y2 mtrix: [ ] det = d c c d If two nonzero vectors in R 2 re prllel (tht is, sclr multiples of ech other) then when we put them s rows in mtrix, the determinnt of tht mtrix will e 0. To clculte the determinnt of 3y3 mtrix: c det d e f g h i [ e f = det h i ] det [ d f g i ] + c det [ ] d e g h Tht is: first, we tke the top row, nd lternte signs +,, +. Then, to decide which sumtrix we ll tke the determinnt of, we delete the row nd column of the entry we chose. If nd re two vectors in R 2, then we cn clculte the re of the prllelogrm spnned y them (s in the picture elow) y mking them the rows of 2y2 mtrix nd tking the solute vlue of the determinnt. [ Are of prllelogrm = ] det Equivlently, the solute vlue of the determinnt is sin θ, where θ is the ngle etween nd. Cross Product The cross product comines two vectors in R 3 to give third vector in R 3. Unlike dot products nd determinnts, we won t define cross product for vectors except those with three coordintes. = ( ); other properties of the cross product re given on Pge 31 of your text i j k = det gives the re of the prllelogrm spnned y nd. Tht is, = sin θ, where θ is the ngle etween nd. So, the length of the cross product is the re of the prllelogrm spnned y the two vectors.
4 Are of prllelogrm = Note: contrst this to the re of the prllelogrm formed y two vectors in R 2. The vector is orthogonl to oth nd. However, we need to e more precise thn this, ecuse there re two directions tht re orthogonl two oth vectors. Its direction (up or down) is given y the righthnd rule. If,, nd c re three vectors in R 3, then we cn clculte the volume of the prllelepiped spnned y them y mking them the rows of 3y3 mtrix nd tking the solute vlue of the determinnt. c Volume of prllelepiped = det c The determinnt ove is lso the triple product of,, nd c, which cn lso e clculted s ( c) or ( ) c. Lines nd Plnes The prmetric eqution of line (in ny dimension: R 2, R 3, etc) is x = q + s Tht is, the line consists of ll points x tht cn e written s some sclr multiple of plus the constnt vector q. The vector gives the direction of the line, nd the vector q is point on the line. The component eqution of line in R 2 is x + y = c, where,, nd c re constnt sclrs. The vector [, ] is orthogonl to the line.
5 The prmetric eqution of plne (in ny dimension: R 3, R 4, etc) is x = q + s + t where nd re not prllel. Tht is, the line consists of ll points x tht cn e written s liner comintion of nd plus the constnt vector q. The vectors nd lie in the plne; tht is, if we position them to their til is point on the plne, then their hed is point on the plne s well. (We lso cll vector tht lies in the plne prllel to the plne.) The constnt vector q is point on the plne. The component eqution of plne in R 3 is x + y + cz = d, where the vector [,, c] is norml to the plne. Tht is, ny vector prllel to the plne is orthogonl to [,, c]. If d = 0, then the plne psses through the origin. The constnt d will determine the position of the plne in spce, while the coefficients,, c, determine the plne s tilt. Note tht for every point (x, y, z) in the plne, [x, y, z] [,, c] = d. The component eqution of line in R 3 is s the intersection of two plnes: 1 x + 2 y + 3 z = 4 1 z + 2 y + 3 z = 4 The vectors [ 1, 2, 3 ] nd [ 1, 2, 3 ] re orthogonl to the line. To get direction vector of the line, you cn clculte the cross product of these two vectors. Liner Systems Let 1 x + 2 y = 3 nd 1 x + 2 y = 3 e lines tht re not prllel. Then their intersection is point. So, the solution to the following system of liner equtions is single point. 1 x + 2 y = 3 1 z + 2 y = 3 Let 1 x + 2 y + 3 z = 4, 1 x + 2 y + 3 z = 4, nd c 2 x + c 2 y + c 3 z = c 4 e plnes in R 3 whose norml vectors do not ll lie on the sme plne. (Tht is, their norml vectors re linerly independent.) Then their intersection is point. So, the solution to the following system of liner equtions is single point. 1 x + 2 y + 3 z = 4 1 z + 2 y + 3 z = 4 c 1 z + c 2 y + c 3 z = c 4 If three vectors lie on the sme plne, then the prllelepiped they define is smooshed ll the wy flt, nd so hs volume zero, nd hence determinnt zero. So, the system of liner equtions in the ullet point ove hs precisely one solution if nd only if det c 1 c 2 c 3 In the cse tht the determinnt is zero, the solutions to the system of liner equtions could form line (tht is, they cn e descried using one prmeter), they could form plne (tht is, it tkes two prmeters to descrie them), they could e ll of R 3 (in the silly cse tht every eqution is 0x + 0y + 0z = 0), or there could e no solutions t ll.
6 Liner Independence A liner comintion of vectors is vector formed y dding sclr multiples of the given vectors. For exmple, the vector πc is liner comintion of the vectors,, nd c. The set of ll liner comintions of collection of vectors is clled the spn of the vectors. A collection of vectors is linerly independent if the only liner comintion of them to equl the zero vector is the liner comintion where ll sclrs re zero. Tht is: consider collection of vectors 1, s,..., n. Set up the system of liner equtions s s s n n = 0. If the only solution is s 1 = s 2 = = s n = 0, then the vectors re linerly independent. If there re other solutions, then the vectors re linerly dependent. The ove is the forml definition of liner independence. For collection of t lest two vectors, it is equivlent to the following definition: collection of vectors is linerly independent if no vector in the collection cn e written s liner comintion of the others. A collection of n linerly independent vectors from R n is clled sis of R n. Suppose you hve collection of vectors tht form sis of R n. Then ny vector in R n cn e written s liner comintion of your vectors, nd there is unique wy to do it. Chpter 3 Solving Liner Systems You should know how to solve liner system using sustitution. We cn use three row opertions on liner system without chnging its solutions: 1. Multiplying n eqution (row) y nonzero sclr 2. Adding multiple of one row to nother row 3. Swpping the verticl order of rows To sve time nd spce, we use shorthnd for systems of liner equtions. We use n ugmented mtrix, y which we men we tke mtrix (representing the coefficients of the vriles) nd ugment it y dding n extr column seprted y verticl line (the column represents the constnts in the equtions). You should know how to use Gussin Elimintion (pivoting) to solve system of liner equtions. You should recognize when mtrix is in row echelon form nd reduced row echelon form. (See the pictures on pge 79 in the text.) The rnk of mtrix (without its ugmenttion) is the numer of nonzero rows tht it hs fter eing reduced. You should know how to express the solutions to system of liner equtions using prmeters. The numer of prmeters needed to descrie the solutions to system of liner equtions with rnk r nd n vriles is n r, provided there re ny solutions t ll.
7 If mtrix hs n rows nd columns, nd rnk n, then its reduced row echelon form will hve 1s long the min digonl nd 0s everywhere else A system of liner equtions tht looks like this will hve precisely one solution, regrdless of wht its constnts re. If mtrix hs n columns, rnk n, nd t lest n rows, then its reduced row echelon form will hve 1s long the min digonl nd 0s everywhere else A system of liner equtions tht looks like this will hve either precisely one solution or no solutions t ll. Homogeneous Systems A system of liner equtions is homogeneous if the constnts re ll zero. For instnce, the system on the left is homogeneous, while the system on the right is not. 1 x + 2 y = 0 1 z + 2 y = 0 1 x + 2 y = 0 1 z + 2 y = 1 A homogeneous system lwys hs t lest one solution (where x = y = z = = 0) So, the solutions of liner eqution cn lwys e written s s s s k k (for some pproprite k, nd possily the vectors re ll 0). Any liner comintion of solutions of A 0 is itself solution of A 0. If you hve system of liner equtions, you cn form the ssocited homogeneous system y simply chnging ll the constnts to 0. This is different system, with different solutions, ut the solutions re relted in predictle mnner. Suppose A is system of liner equtions, nd A 0 is its ssocited homogeneous system. Let s s s k k e the complete set of solutions to A 0. It s possile tht A hs no solutions. If it hs t lest one solution q, then the complete set of solutions to A cn e written s q+s 1 1 +s s k k. There re lots of relted fcts: for instnce, the difference of ny two solutions of A is solution of A 0, nd the sum of ny solution of A with ny solution of A 0 is solution to A.
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 informationAnalytically, vectors will be represented by lowercase boldface 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 information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk 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 informationVECTORS, TENSORS, AND MATRICES. 2 + Az. A vector A can be defined by its length A and the direction of a unit
GG33 Lecture 7 5/17/6 1 VECTORS, TENSORS, ND MTRICES I Min Topics C Vector length nd direction Vector Products Tensor nottion vs. mtrix nottion II Vector Products Vector length: x 2 + y 2 + z 2 vector
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationMatrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24
Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the
More informationLecture 2e Orthogonal Complement (pages )
Lecture 2e Orthogonl Complement (pges ) We hve now seen tht n orthonorml sis is nice wy to descrie suspce, ut knowing tht we wnt n orthonorml sis doesn t mke one fll into our lp. In theory, the process
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationChapter 2. Vectors. 2.1 Vectors Scalars and Vectors
Chpter 2 Vectors 2.1 Vectors 2.1.1 Sclrs nd Vectors A vector is quntity hving both mgnitude nd direction. Emples of vector quntities re velocity, force nd position. One cn represent vector in ndimensionl
More informationVectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3dimensional vectors:
Vectors 1232018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2dimensionl vectors: (2, 3), ( )
More informationOn the diagram below the displacement is represented by the directed line segment OA.
Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples
More informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd GussJordn elimintion to solve systems of liner
More informationMATRICES AND VECTORS SPACE
MATRICES AND VECTORS SPACE MATRICES AND MATRIX OPERATIONS SYSTEM OF LINEAR EQUATIONS DETERMINANTS VECTORS IN SPACE AND SPACE GENERAL VECTOR SPACES INNER PRODUCT SPACES EIGENVALUES, EIGENVECTORS LINEAR
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More informationVectors. Introduction. Definition. The Two Components of a Vector. Vectors
Vectors Introduction This pper covers generl description of vectors first (s cn e found in mthemtics ooks) nd will stry into the more prcticl res of grphics nd nimtion. Anyone working in grphics sujects
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 25pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More information308K. 1 Section 3.2. Zelaya Eufemia. 1. Example 1: Multiplication of Matrices: X Y Z R S R S X Y Z. By associativity we have to choices:
8K Zely Eufemi Section 2 Exmple : Multipliction of Mtrices: X Y Z T c e d f 2 R S X Y Z 2 c e d f 2 R S 2 By ssocitivity we hve to choices: OR: X Y Z R S cr ds er fs X cy ez X dy fz 2 R S 2 Suggestion
More informationLecture Solution of a System of Linear Equation
ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville  D. Keffer, 5/9/98 (updted /) Lecture 8  Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner
More information1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and.
Homework 1. Chpters 2. Bsis independent vectors nd their properties Show work except for fillinlnksprolems (print.pdf from www.motiongenesis.com Textooks Resources). 1.1 Solving prolems wht engineers
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationS56 (5.3) Vectors.notebook January 29, 2016
Dily Prctice 15.1.16 Q1. The roots of the eqution (x 1)(x + k) = 4 re equl. Find the vlues of k. Q2. Find the rte of chnge of 剹 x when x = 1 / 8 Tody we will e lerning out vectors. Q3. Find the eqution
More informationLecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More informationLINEAR ALGEBRA APPLIED
5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nthorder
More informationBases for Vector Spaces
Bses for Vector Spces 22625 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More information9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes
The Vector Product 9.4 Introduction In this section we descrie how to find the vector product of two vectors. Like the sclr product its definition my seem strnge when first met ut the definition is chosen
More informationChapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY
Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationRudimentary Matrix Algebra
Rudimentry Mtrix Alger Mrk Sullivn Decemer 4, 217 i Contents 1 Preliminries 1 1.1 Why does this document exist?.................... 1 1.2 Why does nyone cre out mtrices?................ 1 1.3 Wht is mtrix?...........................
More informationElementary Linear Algebra
Elementry Liner Algebr Anton & Rorres, 1 th Edition Lecture Set 5 Chpter 4: Prt II Generl Vector Spces 163 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 163 คณตศาสตรวศวกรรม 3 สาขาวชาวศวกรรมคอมพวเตอร
More informationNumerical Linear Algebra Assignment 008
Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com
More informationParse trees, ambiguity, and Chomsky normal form
Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for contextfree grmmrs
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationHere we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.
Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous
More informationMatrix Eigenvalues and Eigenvectors September 13, 2017
Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues
More informationINTRODUCTION TO LINEAR ALGEBRA
ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR
More informationElementary Linear Algebra
Elementry Liner Algebr Anton & Rorres, 9 th Edition Lectre Set Chpter : Vectors in Spce nd Spce Chpter Content Introdction to Vectors (Geometric Norm of Vector; Vector Arithmetic Dot Prodct; Projections
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationThomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More informationDEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1.
398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 11.1 ORTHOGONAL FUNCTIONS REVIEW MATERIAL The notions of generlized vectors nd vector spces cn e found in ny liner lger text. INTRODUCTION The concepts
More informationMATRIX DEFINITION A matrix is any doubly subscripted array of elements arranged in rows and columns.
4.5 THEORETICL SOIL MECHNICS Vector nd Mtrix lger Review MTRIX DEFINITION mtrix is ny douly suscripted rry of elements rrnged in rows nd columns. m  Column Revised /0 n Row m,,,,,, n n mn ij nd Order
More informationdx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.
Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd
More informationVector differentiation. Chapters 6, 7
Chpter 2 Vectors Courtesy NASA/JPLCltech 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 higherdimensionl counterprts
More informationAlgebra Of Matrices & Determinants
lgebr Of Mtrices & Determinnts Importnt erms Definitions & Formule 0 Mtrix  bsic introduction: mtrix hving m rows nd n columns is clled mtrix of order m n (red s m b n mtrix) nd mtrix of order lso in
More information1 Linear Least Squares
Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationPractice final exam solutions
University of Pennsylvni Deprtment of Mthemtics Mth 26 Honors Clculus II Spring Semester 29 Prof. Grssi, T.A. Asher Auel Prctice finl exm solutions 1. Let F : 2 2 be defined by F (x, y (x + y, x y. If
More informationIntroduction To Matrices MCV 4UI Assignment #1
Introduction To Mtrices MCV UI Assignment # INTRODUCTION: A mtrix plurl: mtrices) is rectngulr rry of numbers rrnged in rows nd columns Exmples: ) b) c) [ ] d) Ech number ppering in the rry is sid to be
More informationECON 331 Lecture Notes: Ch 4 and Ch 5
Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve
More informationDEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS
3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive
More informationalong the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate
L8 VECTOR EQUATIONS OF LINES HL Mth  Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationChapter 1: Logarithmic functions and indices
Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4
More informationGeometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.
Geometric Sequences Geometric Sequence sequence whose consecutive terms hve common rtio. Geometric Sequence A sequence is geometric if the rtios of consecutive terms re the sme. 2 3 4... 2 3 The number
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationTHREEDIMENSIONAL KINEMATICS OF RIGID BODIES
THREEDIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in threedimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if
More informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
More information, MATHS H.O.D.: SUHAG R.KARIYA, BHOPAL, CONIC SECTION PART 8 OF
DOWNLOAD FREE FROM www.tekoclsses.com, PH.: 0 903 903 7779, 98930 5888 Some questions (Assertion Reson tpe) re given elow. Ech question contins Sttement (Assertion) nd Sttement (Reson). Ech question hs
More informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities RentHep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationI1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3
2 The Prllel Circuit Electric Circuits: Figure 2 elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rellife exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More information1. Review. t 2. t 1. v w = vw cos( ) where is the angle between v and w. The above leads to the Schwarz inequality: v w vw.
1. Review 1.1. The Geometry of Curves. AprmetriccurveinR 3 is mp R R 3 t (t) = (x(t),y(t),z(t)). We sy tht is di erentile if x, y, z re di erentile. We sy tht it is C 1 if, in ddition, the derivtives re
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose
More informationSTRAND J: TRANSFORMATIONS, VECTORS and MATRICES
Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors
More informationCHAPTER 2d. MATRICES
CHPTER d. MTRICES University of Bhrin Deprtment of Civil nd rch. Engineering CEG Numericl Methods in Civil Engineering Deprtment of Civil Engineering University of Bhrin Every squre mtrix hs ssocited
More informationCoordinate geometry and vectors
MST124 Essentil mthemtics 1 Unit 5 Coordinte geometry nd vectors Contents Contents Introduction 4 1 Distnce 5 1.1 The distnce etween two points in the plne 5 1.2 Midpoints nd perpendiculr isectors 7 2
More informationMATH 260 Final Exam April 30, 2013
MATH 60 Finl Exm April 30, 03 Let Mpn,Rq e the spce of nyn mtrices with rel entries () We know tht (with the opertions of mtrix ddition nd sclr multipliction), M pn, Rq is vector spce Wht is the dimension
More informationDeterminants Chapter 3
Determinnts hpter Specil se : x Mtrix Definition : the determinnt is sclr quntity defined for ny squre n x n mtrix nd denoted y or det(). x se ecll : this expression ppers in the formul for x mtrix inverse!
More informationLinear Systems with Constant Coefficients
Liner Systems with Constnt Coefficients 4305 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system
More informationIMPOSSIBLE NAVIGATION
Sclrs versus Vectors IMPOSSIBLE NAVIGATION The need for mgnitude AND direction Sclr: A quntity tht hs mgnitude (numer with units) ut no direction. Vector: A quntity tht hs oth mgnitude (displcement) nd
More information8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.
8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the
More informationChapter 2. Determinants
Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if dbc0. The expression dbc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is
More informationContinuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom
Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive
More informationA Matrix Algebra Primer
A Mtrix Algebr Primer Mtrices, Vectors nd Sclr Multipliction he mtrix, D, represents dt orgnized into rows nd columns where the rows represent one vrible, e.g. time, nd the columns represent second vrible,
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More information11755/ Machine Learning for Signal Processing. Algebra. Class Sep Instructor: Bhiksha Raj
755/8797 Mchine Lerning for Signl Processing Fundmentls of Liner Alger Clss 3 Sep Instructor: Bhiksh Rj Sep 755/8797 Administrivi Registrtion: Anyone on witlist still? Homework : Will e hnded out
More informationCHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx
CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twicedifferentile function of x, then t
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More informationLesson Notes: Week 40Vectors
Lesson Notes: Week 40Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationComputer Graphics (CS 4731) Lecture 7: Linear Algebra for Graphics (Points, Scalars, Vectors)
Computer Grphics (CS 4731) Lecture 7: Liner Alger for Grphics (Points, Sclrs, Vectors) Prof Emmnuel Agu Computer Science Dept. Worcester Poltechnic Institute (WPI) Annoncements Project 1 due net Tuesd,
More informationBasics of space and vectors. Points and distance. Vectors
Bsics of spce nd vectors Points nd distnce One wy to describe our position in three dimensionl spce is using Crtesin coordintes x, y, z) where we hve fixed three orthogonl directions nd we move x units
More informationTo Do. Vectors. Motivation and Outline. Vector Addition. Cartesian Coordinates. Foundations of Computer Graphics (Spring 2010) x y
Foundtions of Computer Grphics (Spring 2010) CS 184, Lecture 2: Review of Bsic Mth http://inst.eecs.erkeley.edu/~cs184 o Do Complete Assignment 0 Downlod nd compile skeleton for ssignment 1 Red instructions
More informationa a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.
Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting
More informationMatrices. 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 informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More information