Review of linear algebra. Nuno Vasconcelos UCSD
|
|
- Hope Jones
- 5 years ago
- Views:
Transcription
1 Revew of lner lgebr Nuno Vsconcelos UCSD
2 Vector spces Defnton: vector spce s set H where ddton nd sclr multplcton re defned nd stsf: ) +( + ) (+ )+ 5) λ H 2) + + H 6) 3) H, + 7) λ(λ ) (λλ ) 4) H, - + 8) λ(+ ) λ + λ 9) (λ+λ ) λ + λ the cnoncl emple s R d wth stndrd vector ddton nd sclr multplcton e d e d α + e e e 2 e 2
3 Vector spces but there re much more nterestng emples e.g., the spce of functons f:x R wth (f+g)() f() + g() R d s vector spce of fnte dmenson f (f,..., f d ) when d goes to nfnt we hve functon f f(t) the spce of functons s n nfnte dmensonl vector spce (λf)() λf()
4 Dt n ths course we wll tlk lot bout dt n ll cses dt wll be represented n vector spce: n emple s rell just pont ( dtpont ) on such spce from bove we know how to perform bsc opertons on dtponts ths s nce, becuse dtponts cn be qute bstrct e.g. mges: n mge s functon on the mge plne t ssgns color f(,) to ech ech mge locton (,) the spce Ψ of mges s vector spce (note: ssumes tht mges cn be negtve) ths mge s pont on Ψ
5 Imges becuse of ths we cn mnpulte mges s we would mnpulte vectors e.g. I wnt to morph (,) nto b(,): ths s the pth long the lne from to b: c(α) + α(b-) (-α) + α b for α we hve for α we hve b for α n (,) we hve pont on the lne between nd b to morph we cn smpl ppl the sme rule to the mges! b b- α(b-)
6 Imges when we mke c(,) (-α) (,) + α b(,) we get mge morphng α α.2 α.4 b b- α(b-) α.6 α.8 α the pont s tht ths s possble onl becuse we re n vector spce
7 Imges note tht mges cn lso be represented s ponts n R d smple mge on fnte grd to get rr of pels (,) (,j) mges re lws stored lke ths on computers stck ll the rows nto vector, e.g. 3 3 mge s converted nto 9 vector n generl n m mge s trnsformed nto nm vector note tht ths s et nother vector spce the pont s tht there re multple vector spces nto whch the dt cn be represented
8 Tet nother common tpe of dt s tet documents re represented b word counts: ssocte counter wth ech word slde wndow through the tet whenever the word occurs ncrement ts counter ths s the w serch engnes represent web pges
9 Tet e.g. word counts for three documents n certn corpus (onl 2 words shown for clrt) note tht: ech document s 2 dmensonl vector f I dd two word count vectors (documents), I get new word count vector (document) f I multpl word count vector (document) b sclr, I get word count vector note: once gn we ssume word counts could be negtve (to mke ths hppen we cn smpl subtrct the verge vlue) ths mens: we re once gn on vector spce (postve subset of R d ) document s pont n ths spce
10 Blner forms one reson to use vector spces s tht the llow us to mesure dstnces between dtponts we wll see tht ths s crucl for clssfcton the mn tool for ths s the dot-product to defne dot-product we frst need to recll the noton of blner form Defnton: blner form on vector spce H s mppng Q: H H R (, ) Q(, ) such tht,, H ) Q[(λ+λ ), ] λq(, ) + λ Q(, ) ) Q[,(λ+λ )] λq(,) + λ Q(, )
11 Dot products Defnton: dot-product on vector spce H s blner form <.,.>: H H R (, ) <, > such tht ) <,>, H ) <,> f nd onl f ) <,> <,> for ll nd note tht ) nd ) mke the dot-product nturl mesure of smlrt nothng cn be more smlr to thn tself ths becomes more precse wth ntroducton of norm
12 Dot products nd norms n dot-product defnes norm v 2 <,> the norm hs the followng propertes non-negtvt:, sclr multplcton: λ λ trngle neqult: + + ths defnes metrc d(,) - whch s mesure of the dstnce between nd lws remember tht the norm chnges wth the dotproduct!
13 Dot products bck to our emples: n R d the stndrd dot-product s whch leds to the stndrd Euclden norm the dstnce between two vectors s the stndrd Euclden dstnce d T, d T 2 d T d 2 ) ( ) ( ) ( ), (
14 Dot-products nd norms note tht ths mmedtel gves mesure of smlrt between web pges compute word count vector from pge, for ll dstnce between pge nd pge j s smpl j T d(, j ) j ( j ) ( j ) ths llows us to fnd, n the web, the most smlr pge to n gven pge ou would be mzed b how close to ths s the mesure of smlrt used b most serch engnes! wht bout functons, e.g. mges?
15 Dot products recll tht the spce of functons s n nfnte dmensonl vector spce the stndrd dot-product s the nturl etenson of tht n R d (just replce summtons b ntegrls) the norm becomes the energ of the functon f ( ), g( ) f ( ) g( ) d f ( ) 2 f ( ) d nd the dstnce between functons the energ of the dfference between them d( f ( ), g( )) 2 f ( ) g( ) [ f ( ) g( )] d
16 Bss we know how to mesure dstnces n vector spce nother nterestng propert s tht we cn full chrcterze the spce b one of ts bses set of vectors,, k re bss of vector spce H f the re lnerl ndependent c c, the spn H : for n v n H, v cn be wrtten s v c
17 Bss note tht b mkng the the columns of mtr X ths cn be compctl wrtten s lner ndependent: Xc c spn H v, c v Xc lso, ll bss of H hve the sme number of vectors, whch s clled the dmenson of H ths s vld for n vector spce!
18 Bss emple the bss of the vector spce of mges of fces the fgure onl show the frst 6 bss vectors but there ctull more these vectors re orthonorml
19 Orthogonlt two vectors re orthogonl f nd onl f ther dot product s zero 2π 2 2π e.g. sn sn( )cos( ) d 2 n the spce of functons defned on [,2π], cos() nd sn() re orthogonl two subspces V nd W re orthogonl f ever vector n V s orthogonl to ever vector n W set of vectors,, k s clled orthogonl f ll prs of vectors re orthogonl orthonorml f the vectors lso hve unt norm, j,, f f j j
20 tr m n mtr s n opertor tht mps vector from R m nto vector n R n e.g. the equton A, or sends n R n to n R m m m O n mn n e n e m A e e e 2 note tht there s nothng mgcl bout ths, t follows rther mechncll from the defnton of mtr-vector multplcton
21 tr vector multplcton consder A,.e. jn j j but we cn thnk of ths s where ( ) mens the th row of A hence the th component of s the dot product of ( ) nd s the projecton of on the spce spnned b the rows of A ( ) L n j j j n n n e e 2 e m A - m m - 2-2
22 tr vector multplcton but there s more. Let A,.e. jn j j whch cn be seen s where wth bove nd below mens the th column of A hence s the th component of n the spce spnned b the columns of A s lner combnton of the columns of A n n n mn m n n n j j j m L L L e e 2 e m A n n R n
23 tr vector multplcton two lterntve pctures coordntes of n row spce of A (belongs to R m ) coordntes of n column spce of A (belongs to R n ) e e 2 e m A - m m n n ( ) n n m + + L
24 A cool trck the mtr multplcton formul C AB lso pples to block mtrces when these re defned properl for emple, f A,B,C,D,E,F,G,H re mtrces c j k k b kj A C BE D G F H AE + BG CE + DG AF CF + + BD DH onl but mportnt cvet: the szes of A,B,C,D,E,F,G,H hve to be such tht the ntermedte opertons mke sense!
25 tr vector multplcton ths mkes t es to derve the two lterntve pctures the row spce pcture s just lke sclr multplcton, wth blocks ( -) nd the column spce pcture s just dot product, wth blocks nd the columns of A ( ) ( ) L n n n n n { { ( ) ( ) n m m n n n n L L
26 Orthogonl mtrces mtr s clled orthogonl f t s squre nd hs orthonorml columns mportnt propertes: ) the nverse of n orthogonl mtr s ts trnspose ths cn be esl shown wth the block mtr trck 2) n orthogonl mtr s rotton mtr ths follows from the fct tht t does not chnge the norms of the vectors on whch t opertes ( ) K O K K K n j n T T A A ) ( ) ( A A A A A T T T T
27 Rotton mtrces the combnton of. opertor nterpretton 2. block mtr trck s useful n mn stutons poll: wht s the mtr R tht rottes R 2 b θ degrees? e 2 e 2 e θ e
28 Rotton mtrces the ke s to consder how the mtr opertes on the vectors of the cnoncl bss note tht R sends e to e e 2 e' r r 2 usng the column spce pcture e' r r r r 2 22 r + r r r from whch we hve the frst column of the mtr 2 e 2 sn θ θ cos θ e e R e' r r 2 22 cosθ snθ r r 2 22
29 Rotton mtrces nd we do the sme for e 2 R sends e 2 to e 2 e' r r r r r r + r r r r 2 22 from whch R cosθ snθ snθ [ e e' ] ' 2 cosθ check cos θ e cosθ snθ cosθ snθ R T R I snθ cosθ snθ cosθ e 2 -sn θ e 2 cos θ sn θ θ θ e
30 Anlss/snthess one nterestng cse s tht of mtrces wth orthogonl columns note tht, n ths cse, the columns of A re bss of the column spce of A bss of the row spce of A T ths leds to n nterestng nterpretton of the two pctures the projecton of nto the column spce of A s equl to tht nto the row spce of A T A T cn then be sntheszed b usng the column spce pcture A Q: s the sntheszed equl to?
31 Projectons A: not necessrl! Recll A T nd A f nd onl f AA T I! ths mens tht A hs to be orthonorml. wht hppens when ths s not the cse? we get the projecton of on the column spce of A e.g. let then nd A ' e e 2 e 3 column spce of A row spce of A T
32 Null spce wht hppens to the prt tht s lost? ths s the null spce of A T N ( T ) { T A A } n the emple ll vectors of the tpe snce A T α α note tht N(A) s (lws!) orthogonl to row spce of A: s n null spce f t s orthogonl to ll rows of A here ths mens tht N(A T ) s orthogonl to the column spce of A α e 3 e e 2 column spce of A row spce of A T null spce of A T
33 Orthonorml mtrces Q: wh s the orthonorml cse specl? becuse here there s no null spce of A T recll tht for ll n N(A T ) A T A the onl vector n the null spce s ths mkes sense: A hs n orthonorml columns, e.g. A these spn ll of R n there s no etr room for n orthogonl spce the null spce of A T hs to be empt the projecton nto row spce of A T (column spce of A) s the vector tself n ths cse, we s tht the mtr hs full rnk
34 The three fundmentl spces there re ctull four, but here we re onl concerned wth three these hold for n mtr column spce: spce spnned b the columns row spce: spce spnned b the rows null spce: spce of vectors orthogonl to ll rows (lso known s the orthogonl complement of the row spce) squre smmetrc mtrces (A A T ): column spce s the sme s row spce null spce orthogonl to column spce nn, orthonorml (A T A I) mtrces column spce row spce R n null spce {}
35 Squre mtrces n ths cse the two spces re n R n - n - n e m A e e 2 n n 2 2
36
Linear Algebra Concepts
Ler Algebr Cocepts Ke Kreutz-Delgdo (Nuo Vscocelos) ECE 75A Wter 22 UCSD Vector spces Defto: vector spce s set H where ddto d sclr multplcto re defed d stsf: ) +( + ) (+ )+ 5) l H 2) + + H 6) 3) H, + 7)
More informationLinear Algebra Concepts
Ler Algebr Cocepts Nuo Vscocelos (Ke Kreutz-Delgdo) UCSD Vector spces Defto: vector spce s set H where ddto d sclr multplcto re defed d stsf: ) +( + ) = (+ )+ 5) H 2) + = + H 6) = 3) H, + = 7) ( ) = (
More informationPrinciple Component Analysis
Prncple Component Anlyss Jng Go SUNY Bufflo Why Dmensonlty Reducton? We hve too mny dmensons o reson bout or obtn nsghts from o vsulze oo much nose n the dt Need to reduce them to smller set of fctors
More informationRank One Update And the Google Matrix by Al Bernstein Signal Science, LLC
Introducton Rnk One Updte And the Google Mtrx y Al Bernsten Sgnl Scence, LLC www.sgnlscence.net here re two dfferent wys to perform mtrx multplctons. he frst uses dot product formulton nd the second uses
More informationMath 497C Sep 17, Curves and Surfaces Fall 2004, PSU
Mth 497C Sep 17, 004 1 Curves nd Surfces Fll 004, PSU Lecture Notes 3 1.8 The generl defnton of curvture; Fox-Mlnor s Theorem Let α: [, b] R n be curve nd P = {t 0,...,t n } be prtton of [, b], then the
More informationThe Number of Rows which Equal Certain Row
Interntonl Journl of Algebr, Vol 5, 011, no 30, 1481-1488 he Number of Rows whch Equl Certn Row Ahmd Hbl Deprtment of mthemtcs Fcult of Scences Dmscus unverst Dmscus, Sr hblhmd1@gmlcom Abstrct Let be X
More informationAnnouncements. Image Formation: Outline. The course. Image Formation and Cameras (cont.)
nnouncements Imge Formton nd Cmers (cont.) ssgnment : Cmer & Lenses, gd Trnsformtons, nd Homogrph wll be posted lter tod. CSE 5 Lecture 5 CS5, Fll CS5, Fll CS5, Fll The course rt : The phscs of mgng rt
More informationESCI 342 Atmospheric Dynamics I Lesson 1 Vectors and Vector Calculus
ESI 34 tmospherc Dnmcs I Lesson 1 Vectors nd Vector lculus Reference: Schum s Outlne Seres: Mthemtcl Hndbook of Formuls nd Tbles Suggested Redng: Mrtn Secton 1 OORDINTE SYSTEMS n orthonorml coordnte sstem
More informationINTRODUCTION TO COMPLEX NUMBERS
INTRODUCTION TO COMPLEX NUMBERS The numers -4, -3, -, -1, 0, 1,, 3, 4 represent the negtve nd postve rel numers termed ntegers. As one frst lerns n mddle school they cn e thought of s unt dstnce spced
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 VECTORS VECTORS VECTORS. 2. Vector Representation. 1. Definition. 3. Types of Vectors. 5. Vector Operations I. 4. Equal and Opposite Vectors
1. Defnton A vetor s n entt tht m represent phsl quntt tht hs mgntude nd dreton s opposed to slr tht ls dreton.. Vetor Representton A vetor n e represented grphll n rrow. The length of the rrow s the mgntude
More informationDCDM BUSINESS SCHOOL NUMERICAL METHODS (COS 233-8) Solutions to Assignment 3. x f(x)
DCDM BUSINESS SCHOOL NUMEICAL METHODS (COS -8) Solutons to Assgnment Queston Consder the followng dt: 5 f() 8 7 5 () Set up dfference tble through fourth dfferences. (b) Wht s the mnmum degree tht n nterpoltng
More information4. Eccentric axial loading, cross-section core
. Eccentrc xl lodng, cross-secton core Introducton We re strtng to consder more generl cse when the xl force nd bxl bendng ct smultneousl n the cross-secton of the br. B vrtue of Snt-Vennt s prncple we
More informationDefinition of Tracking
Trckng Defnton of Trckng Trckng: Generte some conclusons bout the moton of the scene, objects, or the cmer, gven sequence of mges. Knowng ths moton, predct where thngs re gong to project n the net mge,
More informationChapter 2 Introduction to Algebra. Dr. Chih-Peng Li ( 李 )
Chpter Introducton to Algebr Dr. Chh-Peng L 李 Outlne Groups Felds Bnry Feld Arthetc Constructon of Glos Feld Bsc Propertes of Glos Feld Coputtons Usng Glos Feld Arthetc Vector Spces Groups 3 Let G be set
More informationCENTROID (AĞIRLIK MERKEZİ )
CENTOD (ĞLK MEKEZİ ) centrod s geometrcl concept rsng from prllel forces. Tus, onl prllel forces possess centrod. Centrod s tougt of s te pont were te wole wegt of pscl od or sstem of prtcles s lumped.
More informationThe Schur-Cohn Algorithm
Modelng, Estmton nd Otml Flterng n Sgnl Processng Mohmed Njm Coyrght 8, ISTE Ltd. Aendx F The Schur-Cohn Algorthm In ths endx, our m s to resent the Schur-Cohn lgorthm [] whch s often used s crteron for
More informationChapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter.4 Newton-Rphson Method of Solvng Nonlner Equton After redng ths chpter, you should be ble to:. derve the Newton-Rphson method formul,. develop the lgorthm of the Newton-Rphson method,. use the Newton-Rphson
More informationMultiple view geometry
EECS 442 Computer vson Multple vew geometry Perspectve Structure from Moton - Perspectve structure from moton prolem - mgutes - lgerc methods - Fctorzton methods - Bundle djustment - Self-clrton Redng:
More informationFINITE NEUTROSOPHIC COMPLEX NUMBERS. W. B. Vasantha Kandasamy Florentin Smarandache
INITE NEUTROSOPHIC COMPLEX NUMBERS W. B. Vsnth Kndsmy lorentn Smrndche ZIP PUBLISHING Oho 11 Ths book cn be ordered from: Zp Publshng 1313 Chespeke Ave. Columbus, Oho 31, USA Toll ree: (61) 85-71 E-ml:
More informationLeast squares. Václav Hlaváč. Czech Technical University in Prague
Lest squres Václv Hlváč Czech echncl Unversty n Prgue hlvc@fel.cvut.cz http://cmp.felk.cvut.cz/~hlvc Courtesy: Fred Pghn nd J.P. Lews, SIGGRAPH 2007 Course; Outlne 2 Lner regresson Geometry of lest-squres
More informationGAUSS ELIMINATION. Consider the following system of algebraic linear equations
Numercl Anlyss for Engneers Germn Jordnn Unversty GAUSS ELIMINATION Consder the followng system of lgebrc lner equtons To solve the bove system usng clsscl methods, equton () s subtrcted from equton ()
More informationJens Siebel (University of Applied Sciences Kaiserslautern) An Interactive Introduction to Complex Numbers
Jens Sebel (Unversty of Appled Scences Kserslutern) An Interctve Introducton to Complex Numbers 1. Introducton We know tht some polynoml equtons do not hve ny solutons on R/. Exmple 1.1: Solve x + 1= for
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 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 informationLogarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.
Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the
More informationp (i.e., the set of all nonnegative real numbers). Similarly, Z will denote the set of all
th Prelmnry E 689 Lecture Notes by B. Yo 0. Prelmnry Notton themtcl Prelmnres It s ssumed tht the reder s fmlr wth the noton of set nd ts elementry oertons, nd wth some bsc logc oertors, e.g. x A : x s
More informationProof that if Voting is Perfect in One Dimension, then the First. Eigenvector Extracted from the Double-Centered Transformed
Proof tht f Votng s Perfect n One Dmenson, then the Frst Egenvector Extrcted from the Doule-Centered Trnsformed Agreement Score Mtrx hs the Sme Rn Orderng s the True Dt Keth T Poole Unversty of Houston
More informationVectors and Tensors. R. Shankar Subramanian. R. Aris, Vectors, Tensors, and the Equations of Fluid Mechanics, Prentice Hall (1962).
005 Vectors nd Tensors R. Shnkr Subrmnn Good Sources R. rs, Vectors, Tensors, nd the Equtons of Flud Mechncs, Prentce Hll (96). nd ppendces n () R. B. Brd, W. E. Stewrt, nd E. N. Lghtfoot, Trnsport Phenomen,
More informationFall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. with respect to λ. 1. χ λ χ λ ( ) λ, and thus:
More on χ nd errors : uppose tht we re fttng for sngle -prmeter, mnmzng: If we epnd The vlue χ ( ( ( ; ( wth respect to. χ n Tlor seres n the vcnt of ts mnmum vlue χ ( mn χ χ χ χ + + + mn mnmzes χ, nd
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 informationMATH 174A: PROBLEM SET 5. Suggested Solution
MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion
More informationCS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 2
CS434/54: Pttern Recognition Prof. Olg Veksler Lecture Outline Review of Liner Algebr vectors nd mtrices products nd norms vector spces nd liner trnsformtions eigenvlues nd eigenvectors Introduction to
More informationLecture 4: Piecewise Cubic Interpolation
Lecture notes on Vrtonl nd Approxmte Methods n Appled Mthemtcs - A Perce UBC Lecture 4: Pecewse Cubc Interpolton Compled 6 August 7 In ths lecture we consder pecewse cubc nterpolton n whch cubc polynoml
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 informationUNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS. M.Sc. in Economics MICROECONOMIC THEORY I. Problem Set II
Mcroeconomc Theory I UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS MSc n Economcs MICROECONOMIC THEORY I Techng: A Lptns (Note: The number of ndctes exercse s dffculty level) ()True or flse? If V( y )
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 informationCOMPLEX NUMBERS INDEX
COMPLEX NUMBERS INDEX. The hstory of the complex numers;. The mgnry unt I ;. The Algerc form;. The Guss plne; 5. The trgonometrc form;. The exponentl form; 7. The pplctons of the complex numers. School
More informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More informationPhysics for Scientists and Engineers I
Phscs for Scentsts nd Engneers I PHY 48, Secton 4 Dr. Betr Roldán Cuen Unverst of Centrl Flord, Phscs Deprtment, Orlndo, FL Chpter - Introducton I. Generl II. Interntonl Sstem of Unts III. Converson of
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 Newton-Raphson Method of Solving a Nonlinear Equation
Chpter 0.04 Newton-Rphson Method o Solvng Nonlner Equton Ater redng ths chpter, you should be ble to:. derve the Newton-Rphson method ormul,. develop the lgorthm o the Newton-Rphson method,. use the Newton-Rphson
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More informationAPPENDIX 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 information5.2 Exponent Properties Involving Quotients
5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
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 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 nth-order
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
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 information( ) ( )()4 x 10-6 C) ( ) = 3.6 N ( ) = "0.9 N. ( )ˆ i ' ( ) 2 ( ) 2. q 1 = 4 µc q 2 = -4 µc q 3 = 4 µc. q 1 q 2 q 3
3 Emple : Three chrges re fed long strght lne s shown n the fgure boe wth 4 µc, -4 µc, nd 3 4 µc. The dstnce between nd s. m nd the dstnce between nd 3 s lso. m. Fnd the net force on ech chrge due to the
More informationThe Algebra (al-jabr) of Matrices
Section : Mtri lgebr nd Clculus Wshkewicz College of Engineering he lgebr (l-jbr) of Mtrices lgebr s brnch of mthemtics is much broder thn elementry lgebr ll of us studied in our high school dys. In sense
More informationES.182A Topic 32 Notes Jeremy Orloff
ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In
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 informationLectures - 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 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 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 informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More 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 informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More information7.2 Volume. A cross section is the shape we get when cutting straight through an object.
7. Volume Let s revew the volume of smple sold, cylnder frst. Cylnder s volume=se re heght. As llustrted n Fgure (). Fgure ( nd (c) re specl cylnders. Fgure () s rght crculr cylnder. Fgure (c) s ox. A
More informationψ ij has the eigenvalue
Moller Plesset Perturbton Theory In Moller-Plesset (MP) perturbton theory one tes the unperturbed Hmltonn for n tom or molecule s the sum of the one prtcle Foc opertors H F() where the egenfunctons of
More informationQuiz: Experimental Physics Lab-I
Mxmum Mrks: 18 Totl tme llowed: 35 mn Quz: Expermentl Physcs Lb-I Nme: Roll no: Attempt ll questons. 1. In n experment, bll of mss 100 g s dropped from heght of 65 cm nto the snd contner, the mpct s clled
More informationMTH 5102 Linear Algebra Practice Exam 1 - Solutions Feb. 9, 2016
Nme (Lst nme, First nme): MTH 502 Liner Algebr Prctice Exm - Solutions Feb 9, 206 Exm Instructions: You hve hour & 0 minutes to complete the exm There re totl of 6 problems You must show your work Prtil
More informationA Family of Multivariate Abel Series Distributions. of Order k
Appled Mthemtcl Scences, Vol. 2, 2008, no. 45, 2239-2246 A Fmly of Multvrte Abel Seres Dstrbutons of Order k Rupk Gupt & Kshore K. Ds 2 Fculty of Scence & Technology, The Icf Unversty, Agrtl, Trpur, Ind
More informationapproaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below
. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.
More informationEffects of polarization on the reflected wave
Lecture Notes. L Ros PPLIED OPTICS Effects of polrzton on the reflected wve Ref: The Feynmn Lectures on Physcs, Vol-I, Secton 33-6 Plne of ncdence Z Plne of nterfce Fg. 1 Y Y r 1 Glss r 1 Glss Fg. Reflecton
More informationInner-product spaces
Inner-product spces Definition: Let V be rel or complex liner spce over F (here R or C). An inner product is n opertion between two elements of V which results in sclr. It is denoted by u, v nd stisfies:
More informationDennis Bricker, 2001 Dept of Industrial Engineering The University of Iowa. MDP: Taxi page 1
Denns Brcker, 2001 Dept of Industrl Engneerng The Unversty of Iow MDP: Tx pge 1 A tx serves three djcent towns: A, B, nd C. Ech tme the tx dschrges pssenger, the drver must choose from three possble ctons:
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationVariable time amplitude amplification and quantum algorithms for linear algebra. Andris Ambainis University of Latvia
Vrble tme mpltude mplfcton nd quntum lgorthms for lner lgebr Andrs Ambns Unversty of Ltv Tlk outlne. ew verson of mpltude mplfcton;. Quntum lgorthm for testng f A s sngulr; 3. Quntum lgorthm for solvng
More informationChapter 2. Vectors. 2.1 Vectors Scalars and Vectors
Chpter 2 Vectors 2.1 Vectors 2.1.1 Sclrs nd Vectors A vector is quntity hving both mgnitude nd direction. Emples of vector quntities re velocity, force nd position. One cn represent vector in n-dimensionl
More informationA Matrix 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 informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
More information( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
More informationCISE 301: Numerical Methods Lecture 5, Topic 4 Least Squares, Curve Fitting
CISE 3: umercl Methods Lecture 5 Topc 4 Lest Squres Curve Fttng Dr. Amr Khouh Term Red Chpter 7 of the tetoo c Khouh CISE3_Topc4_Lest Squre Motvton Gven set of epermentl dt 3 5. 5.9 6.3 The reltonshp etween
More informationVECTORS AND TENSORS IV.1.1. INTRODUCTION
Chpter IV Vector nd Tensor Anlyss IV. Vectors nd Tensors Septemer 5, 08 05 IV. VECTORS AND TENSORS IV... INTRODUCTION In mthemtcs nd mechncs, we he to operte wth qunttes whch requre dfferent mthemtcl ojects
More informationBest Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationChapter I Vector Analysis
. Chpte I Vecto nlss . Vecto lgeb j It s well-nown tht n vecto cn be wtten s Vectos obe the followng lgebc ules: scl s ) ( j v v cos ) ( e Commuttv ) ( ssoctve C C ) ( ) ( v j ) ( ) ( ) ( ) ( (v) he lw
More informationBernoulli Numbers Jeff Morton
Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f
More informationProblem Set 4: Solutions Math 201A: Fall 2016
Problem Set 4: s Mth 20A: Fll 206 Problem. Let f : X Y be one-to-one, onto mp between metric spces X, Y. () If f is continuous nd X is compct, prove tht f is homeomorphism. Does this result remin true
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationfractions Let s Learn to
5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More information8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1
8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.
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 informationApplied Statistics Qualifier Examination
Appled Sttstcs Qulfer Exmnton Qul_june_8 Fll 8 Instructons: () The exmnton contns 4 Questons. You re to nswer 3 out of 4 of them. () You my use ny books nd clss notes tht you mght fnd helpful n solvng
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 informationPoint Lattices: Bravais Lattices
Physics for Solid Stte Applictions Februry 18, 2004 Lecture 7: Periodic Structures (cont.) Outline Review 2D & 3D Periodic Crystl Structures: Mthemtics X-Ry Diffrction: Observing Reciprocl Spce Point Lttices:
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationFourier series. Preliminary material on inner products. Suppose V is vector space over C and (, )
Fourier series. Preliminry mteril on inner products. Suppose V is vector spce over C nd (, ) is Hermitin inner product on V. This mens, by definition, tht (, ) : V V C nd tht the following four conditions
More informationElectrochemical Thermodynamics. Interfaces and Energy Conversion
CHE465/865, 2006-3, Lecture 6, 18 th Sep., 2006 Electrochemcl Thermodynmcs Interfces nd Energy Converson Where does the energy contrbuton F zϕ dn come from? Frst lw of thermodynmcs (conservton of energy):
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 informationIntroduction to Group Theory
Introduction to Group Theory Let G be n rbitrry set of elements, typiclly denoted s, b, c,, tht is, let G = {, b, c, }. A binry opertion in G is rule tht ssocites with ech ordered pir (,b) of elements
More informationChapter 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 informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More information