2. Elementary Linear Algebra Problems

Size: px
Start display at page:

Download "2. Elementary Linear Algebra Problems"

Transcription

1 . Eleety e lge Pole. BS: B e lge Suoute (Pog pge wth PCK) Su of veto opoet:. Coputto y f- poe: () () () (3) N Full y tee Depth te tep log()n

2 Veto updte the f- poe wth N : ) ( ) ( ) ( ) ( ) ( ) ( ) ( N N N N ( ) ( ) [ ] ( ) ( ) [ ] ed ed fo N fo N N ; ) ; ; ( ) ; ; ( log()n te tep pllel fo veto of legth

3 evel- BS BS oute wth () pole (veto oly y ε IR ). Ft eple: DT-podut y f-: y T y y y 3 y 3 4 y 4 5 y 5 6 y y 7 8 y 8 3

4 Te tep pllel of the DT-podut ot ette th log(). Evey oputto volvg f- wll te log() te tep pllel! ddedu: Coputg the DT-podut o pel htetue: Dtute dt o -deol y of poeo wth / poeo 3 (-) P P P 3 P / P - P Ptto veto d y lo of veto wth legth. Eh poeo h to opute ptl DT-podut: ( ) y( ) y... 4

5 Te tep pllel oputto ed o th odel: ( dd ult ) te fo ddto d ultplto pllel poeo p fo ptl DT-podut fte oputg the ptl DT-podut eh poeo pllel teedte eult hve to e olleted d dded up to the fl eult. To th the poeo P (P ) ed t eult to the eghog poeo P (P - ) tht dd th ue to t ow ptl DT-podut. The the eult ed to the et egho tht dd two ptl eult g d o o utl the fl eult oputed the poeo P /. ( ) 3 (-) P P P 3 P / P - P 5

6 Totl te depedg o d : f ( ) ( dd ( dd ult) ( ed dd) dd ed ult) ze the totl te f() :! f '( ) ( dd ult) dd ed Ο ( ) ptu: ( ) te tep dt dtuted o ( ) poeo. f ( ) / Ο ( ) Cope f-: poeo log() te tep! 6

7 Futhe evel- BS Pole BS-Notto: defe SXPY: S --- gle peo (D fo doule C fo ople) --- α l X --- veto P --- plu opeto Y --- veto y α y Y α X Y Vetozto of SXPY y ppelg: α α y : α ultplto ddto α y y 7

8 8 SXPY Pllelzto y Pttog { } R I I I > < R R Y Y y y y y X X wth hot veto X d Y of legth /R. Eh poeo P get ptl veto X d Y d opute Y α X Y R. Reult: SXPY vey good vetozle d pllelzle X X X X X

9 Futhe evel-bs Route SCPY: y o y Cope SXPY No: T Cope DT-podut 9

10 evel- BS t-veto opeto wth ( ) opeto (equetlly) BS-Notto: S --- gle peo G E geel t V --- veto defe SGEV t-veto podut: y α y the evel- BS: Solvg tgul yte wth tgul t

11 evel-3 BS t-t opeto wth ( 3 ) opeto (equetlly) BS-Notto: S --- gle peo G E geel t --- t defe SGE t-t podut: C α B C PCK uoute fo olvg le equto let que pole QR-deopoto egevlue gul vlue ed o BS

12 . ly of t-veto- Podut ( ) IR IR IR Vetozto DT-podut of legth SXPY of legth (GXPY)

13 Peudoode: ()-fo: ; fo fo ed ed DT podut Dot podut of -th ow of wth veto 3

14 Peudoode: ()-fo: ; fo fo ed ed SXPY GXPY SXPY updtg veto wth -th olu of GXPY: Sequee of SXPY elted to the e veto dvtge: veto tht updted e ept ft eoy! No ddtol dt tfe. 4

15 .. Pllelzto y uldg lo Ide: Redue t-veto podut o lle t-veto podut. < > < > {... } I I I {... } J J J S R I dut: J I J fo fo Ue -deol y of poeo P. P get t lo :(I J ) :(J ) :(I ). S S ( ) I J I J 5

16 Peudoode fo R fo S () ; ed ed fo R ; fo S () ; ed ed Sll depedet t-veto podut. No outo eey! Blowe olleto d ddto of veto. Rowwe outo! F-. 6

17 7 Spel e: S P P.. No outo eey etwee poeo P P R The oputto of vetozle y GXPY. R : ( ) e depedet. The olleto of ptl eult fo poeo P P. F-. Fl u oe poeo: vetozle y GXPY.

18 Rule () Ie loop of pog hould e ple vetozle () ute loop of pog hould e uttl depedet pllelzle fo uttl d pllelzle fo. ed ple vetozle ed (3) Reue of dt (Che l dt tfe log) 8

19 9..3 fo Bded t Bdwdth (yet) dgol: dg. udg. upedg. : tdgol

20 N N N Stog ete dgolwe () t ted of. ow fo... d d ow dgol d d [ ] { } { } [ ] l [ ] { } { } [ ] l

21 Coputto of the t-veto podut ed o th toge hee o veto poeo: Fo : l l lgoth: Fo - : Fo {-} : {-} ed ed Geel tde No SXPY o Fo : Fo {--} : {-} ed ed Ptl Dot podut Spty le opeto ut lo lo of effey thee opeto

22 Pllel Pttog: R < > U fo ed I I dut l ; Poeo P get ow to de et I :[ ] ode to opute t pt of the fl veto. Wht pt of veto doe poeo P eed ode to opute t pt of?

23 Neey fo I : : l { } { } { } { } Poeo P wth de et I eed fo the de [ { } { } ] 3

24 4.3 ly of t-t Podut ( ) ( ) ( ) q q B C B ed ed q Fo Fo : :

25 ()-Fo: lgoth Fo : Fo : q Fo : ; ed ed ed Dot podut of legth B fo ll ll ete e fully oputed oe fte othe. e to d C owwe to B oluwe. 5

26 6 Dffeet vew o the t-t podut: ( ) ( ) ( ) T T e e e e t odeed oto of olu o ow ( ) T T e e e e B u of full te y oute podut of the -th olu of d the -th ow of B Full q - te

27 ()-Fo lgoth Fo : q Fo : Fo : ; ed ed ed SXPY Veto updte. GXPY Sequee of SXPY fo the e veto. C oputed oluwe; e to oluwe. e to B oluwe ut delyed. 7

28 ()-Fo lgoth 3 Fo : Fo : q Fo : ; ed ed ed SXPY No GXPY Sequee of SXPY fo dffeet veto. Veto updte. e to oluwe. e to B delyed. C oputed wth teedte vlue () whh e oputed oluwe. 8

29 vevew ove dffeet Fo e to y e to B y Coput -to of C ow ow olu olu olu ow ow olu ow ow ow olu olu olu Coput det delyed delyed det delyed delyed -to Veto opeto Veto leght DT GXPY SXPY DT GXPY SXPY q q Bette: GXPY loge veto legth. e to te odg to toge hee (owwe o oluwe) 9

30 t-t Pllel R < > U I < > U K < q > U J S T t t. Dtute the lo eltve to de et I K d J t y P t : to poeo K J t J t K. I () t I B t () Poeo P t opute ll t-t podut ll Poeo pllel. ( ) t B t () Copute u y f- : S ( t t ) 3

31 Spel Ce S J t J t. I t I B t I th e eh poeo P t opute t pt of t depedetly wthout y outo. Eh poeo eed full lo of ow of eltve to de et I d full lo of olu of B eltve to de et J t ode to opute t eltve to ow I d olu J t. Epelly wth q poeo eh poeo h to opute oe DT podut t t wth () pllel te tep. F- y q ddtol poeo fo ll thee Dot podut edue the ue of pllel te tep to (log()). 3

Level-2 BLAS. Matrix-Vector operations with O(n 2 ) operations (sequentially) BLAS-Notation: S --- single precision G E general matrix M V --- vector

Level-2 BLAS. Matrix-Vector operations with O(n 2 ) operations (sequentially) BLAS-Notation: S --- single precision G E general matrix M V --- vector evel-2 BS trx-vector opertos wth 2 opertos sequetlly BS-Notto: S --- sgle precso G E geerl mtrx V --- vector defes SGEV, mtrx-vector product: r y r α x β r y ther evel-2 BS: Solvg trgulr system x wth trgulr

More information

2.Decision Theory of Dependence

2.Decision Theory of Dependence .Deciio Theoy of Depedece Theoy :I et of vecto if thee i uet which i liely depedet the whole et i liely depedet too. Coolly :If the et i liely idepedet y oepty uet of it i liely idepedet. Theoy : Give

More information

Current Programmed Control (i.e. Peak Current-Mode Control) Lecture slides part 2 More Accurate Models

Current Programmed Control (i.e. Peak Current-Mode Control) Lecture slides part 2 More Accurate Models Curret Progred Cotrol.e. Pek Curret-Mode Cotrol eture lde prt More Aurte Model ECEN 5807 Drg Mkovć Sple Frt-Order CPM Model: Sury Aupto: CPM otroller operte delly, Ueful reult t low frequee, well uted

More information

SOLVING SYSTEMS OF EQUATIONS, DIRECT METHODS

SOLVING SYSTEMS OF EQUATIONS, DIRECT METHODS ELM Numecl Alyss D Muhem Mecmek SOLVING SYSTEMS OF EQUATIONS DIRECT METHODS ELM Numecl Alyss Some of the cotets e dopted fom Luee V. Fusett Appled Numecl Alyss usg MATLAB. Petce Hll Ic. 999 ELM Numecl

More information

The formulae in this booklet have been arranged according to the unit in which they are first

The formulae in this booklet have been arranged according to the unit in which they are first Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge og to the ut whh the e fst toue. Thus te sttg ut m e eque to use the fomule tht wee toue peeg ut e.g. tes sttg C mght e epete to use fomule

More information

Moments of Generalized Order Statistics from a General Class of Distributions

Moments of Generalized Order Statistics from a General Class of Distributions ISSN 684-843 Jol of Sttt Vole 5 28. 36-43 Moet of Geelzed Ode Sttt fo Geel l of Dtto Att Mhd Fz d Hee Ath Ode ttt eod le d eel othe odel of odeed do le e ewed el e of geelzed ode ttt go K 995. I th e exlt

More information

5 - Determinants. r r. r r. r r. r s r = + det det det

5 - Determinants. r r. r r. r r. r s r = + det det det 5 - Detemts Assote wth y sque mtx A thee s ume lle the etemt of A eote A o et A. Oe wy to efe the etemt, ths futo fom the set of ll mtes to the set of el umes, s y the followg thee popetes. All mtes elow

More information

X-Ray Notes, Part III

X-Ray Notes, Part III oll 6 X-y oe 3: Pe X-Ry oe, P III oe Deeo Coe oupu o x-y ye h look lke h: We efe ue of que lhly ffee efo h ue y ovk: Co: C ΔS S Sl o oe Ro: SR S Co o oe Ro: CR ΔS C SR Pevouly, we ee he SR fo ye hv pxel

More information

On Almost Increasing Sequences For Generalized Absolute Summability

On Almost Increasing Sequences For Generalized Absolute Summability Joul of Applied Mthetic & Bioifotic, ol., o., 0, 43-50 ISSN: 79-660 (pit), 79-6939 (olie) Itetiol Scietific Pe, 0 O Alot Iceig Sequece Fo Geelized Abolute Subility W.. Suli Abtct A geel eult coceig bolute

More information

Chapter #2 EEE State Space Analysis and Controller Design

Chapter #2 EEE State Space Analysis and Controller Design Chpte EEE8- Chpte # EEE8- Stte Spce Al d Cotolle Deg Itodcto to tte pce Obevblt/Cotollblt Modle ede: D D Go - d.go@cl.c.k /4 Chpte EEE8-. Itodcto Ae tht we hve th ode te: f, ', '',.... Ve dffclt to td

More information

3/20/2013. Splines There are cases where polynomial interpolation is bad overshoot oscillations. Examplef x. Interpolation at -4,-3,-2,-1,0,1,2,3,4

3/20/2013. Splines There are cases where polynomial interpolation is bad overshoot oscillations. Examplef x. Interpolation at -4,-3,-2,-1,0,1,2,3,4 // Sples There re ses where polyoml terpolto s d overshoot oslltos Emple l s Iterpolto t -,-,-,-,,,,,.... - - - Ide ehd sples use lower order polyomls to oet susets o dt pots mke oetos etwee djet sples

More information

Matrix. Definition 1... a1 ... (i) where a. are real numbers. for i 1, 2,, m and j = 1, 2,, n (iii) A is called a square matrix if m n.

Matrix. Definition 1... a1 ... (i) where a. are real numbers. for i 1, 2,, m and j = 1, 2,, n (iii) A is called a square matrix if m n. Mtrx Defto () s lled order of m mtrx, umer of rows ( 橫行 ) umer of olums ( 直列 ) m m m where j re rel umers () B j j for,,, m d j =,,, () s lled squre mtrx f m (v) s lled zero mtrx f (v) s lled detty mtrx

More information

The linear system. The problem: solve

The linear system. The problem: solve The ler syste The prole: solve Suppose A s vertle, the there ests uue soluto How to effetly opute the soluto uerlly??? A A A evew of dret ethods Guss elto wth pvotg Meory ost: O^ Coputtol ost: O^ C oly

More information

CHAPTER 5 Vectors and Vector Space

CHAPTER 5 Vectors and Vector Space HAPTE 5 Vetors d Vetor Spe 5. Alger d eometry of Vetors. Vetor A ordered trple,,, where,, re rel umers. Symol:, B,, A mgtude d dreto.. Norm of vetor,, Norm =,, = = mgtude. Slr multplto Produt of slr d

More information

Chapter Linear Regression

Chapter Linear Regression Chpte 6.3 Le Regesso Afte edg ths chpte, ou should be ble to. defe egesso,. use sevel mmzg of esdul cte to choose the ght cteo, 3. deve the costts of le egesso model bsed o lest sques method cteo,. use

More information

= y and Normed Linear Spaces

= y and Normed Linear Spaces 304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads

More information

Certain Expansion Formulae Involving a Basic Analogue of Fox s H-Function

Certain Expansion Formulae Involving a Basic Analogue of Fox s H-Function vlle t htt:vu.edu l. l. Mth. ISSN: 93-9466 Vol. 3 Iue Jue 8. 8 36 Pevouly Vol. 3 No. lcto d led Mthetc: Itetol Joul M Cet Exo Foule Ivolvg c logue o Fox -Fucto S.. Puoht etet o c-scece Mthetc College o

More information

ME 501A Seminar in Engineering Analysis Page 1

ME 501A Seminar in Engineering Analysis Page 1 Mtr Trsformtos usg Egevectors September 8, Mtr Trsformtos Usg Egevectors Lrry Cretto Mechcl Egeerg A Semr Egeerg Alyss September 8, Outle Revew lst lecture Trsformtos wth mtr of egevectors: = - A ermt

More information

Rendering Equation. Linear equation Spatial homogeneous Both ray tracing and radiosity can be considered special case of this general eq.

Rendering Equation. Linear equation Spatial homogeneous Both ray tracing and radiosity can be considered special case of this general eq. Rederg quto Ler equto Sptl homogeeous oth ry trcg d rdosty c be cosdered specl cse of ths geerl eq. Relty ctul photogrph Rdosty Mus Rdosty Rederg quls the dfferece or error mge http://www.grphcs.corell.edu/ole/box/compre.html

More information

Review of Linear Algebra

Review of Linear Algebra PGE 30: Forulto d Soluto Geosstes Egeerg Dr. Blhoff Sprg 0 Revew of Ler Alger Chpter 7 of Nuercl Methods wth MATLAB, Gerld Recktewld Vector s ordered set of rel (or cople) uers rrged s row or colu sclr

More information

3. REVIEW OF PROPERTIES OF EIGENVALUES AND EIGENVECTORS

3. REVIEW OF PROPERTIES OF EIGENVALUES AND EIGENVECTORS . REVIEW OF PROPERTIES OF EIGENVLUES ND EIGENVECTORS. EIGENVLUES ND EIGENVECTORS We hll ow revew ome bc fct from mtr theory. Let be mtr. clr clled egevlue of f there et ozero vector uch tht Emle: Let 9

More information

CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find

CBSE SAMPLE PAPER SOLUTIONS CLASS-XII MATHS SET-2 CBSE , ˆj. cos. SECTION A 1. Given that a 2iˆ ˆj. We need to find BSE SMLE ER SOLUTONS LSS-X MTHS SET- BSE SETON Gv tht d W d to fd 7 7 Hc, 7 7 7 Lt, W ow tht Thus, osd th vcto quto of th pl z - + z = - + z = Thus th ts quto of th pl s - + z = Lt d th dstc tw th pot,,

More information

1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1.

1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1. SPECIAL MATRICES SYMMETRIC MATRICES Def: A mtr A s symmetr f d oly f A A, e,, Emple A s symmetr Def: A mtr A s skew symmetr f d oly f A A, e,, Emple A s skew symmetr Remrks: If A s symmetr or skew symmetr,

More information

AN ALGEBRAIC APPROACH TO M-BAND WAVELETS CONSTRUCTION

AN ALGEBRAIC APPROACH TO M-BAND WAVELETS CONSTRUCTION AN ALGEBRAIC APPROACH TO -BAN WAELETS CONSTRUCTION Toy L Qy S Pewe Ho Ntol Lotoy o e Peeto Pe Uety Be 8 P. R. C Att T e eet le o to ott - otool welet e. A yte of ott eto ote fo - otool flte te olto e o

More information

Chapter 17. Least Square Regression

Chapter 17. Least Square Regression The Islmc Uvest of Gz Fcult of Egeeg Cvl Egeeg Deptmet Numecl Alss ECIV 336 Chpte 7 Lest que Regesso Assocte Pof. Mze Abultef Cvl Egeeg Deptmet, The Islmc Uvest of Gz Pt 5 - CURVE FITTING Descbes techques

More information

A convex hull characterization

A convex hull characterization Pue d ppled Mthets Joul 4; (: 4-48 Pulshed ole My 4 (http://www.seepulshggoup.o//p do:.648/.p.4. ove hull htezto Fo Fesh Gov Qut Deptet DISG Uvesty of Se Itly El ddess: fesh@us.t (F. Fesh qut@us.t (G.

More information

10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n

10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n 0. Sere I th ecto, we wll troduce ere tht wll be dcug for the ret of th chpter. Wht ere? If we dd ll term of equece, we get whch clled fte ere ( or jut ere) d deoted, for hort, by the ymbol or Doe t mke

More information

Lecture 9-3/8/10-14 Spatial Description and Transformation

Lecture 9-3/8/10-14 Spatial Description and Transformation Letue 9-8- tl Deton nd nfomton Homewo No. Due 9. Fme ngement onl. Do not lulte...8..7.8 Otonl et edt hot oof tht = - Homewo No. egned due 9 tud eton.-.. olve oblem:.....7.8. ee lde 6 7. e Mtlb on. f oble.

More information

COMP 465: Data Mining More on PageRank

COMP 465: Data Mining More on PageRank COMP 465: Dt Mnng Moe on PgeRnk Sldes Adpted Fo: www.ds.og (Mnng Mssve Dtsets) Powe Iteton: Set = 1/ 1: = 2: = Goto 1 Exple: d 1/3 1/3 5/12 9/24 6/15 = 1/3 3/6 1/3 11/24 6/15 1/3 1/6 3/12 1/6 3/15 Iteton

More information

ECONOMETRIC ANALYSIS ON EFFICIENCY OF ESTIMATOR ABSTRACT

ECONOMETRIC ANALYSIS ON EFFICIENCY OF ESTIMATOR ABSTRACT ECOOMETRIC LYSIS O EFFICIECY OF ESTIMTOR M. Khohev, Lectue, Gffth Uvet, School of ccoutg d Fce, utl F. K, tt Pofeo, Mchuett Ittute of Techolog, Deptet of Mechcl Egeeg, US; cuetl t Shf Uvet, I. Houl P.

More information

Chapter Gauss-Seidel Method

Chapter Gauss-Seidel Method Chpter 04.08 Guss-Sedel Method After redg ths hpter, you should be ble to:. solve set of equtos usg the Guss-Sedel method,. reogze the dvtges d ptflls of the Guss-Sedel method, d. determe uder wht odtos

More information

Week 10: DTMC Applications Ranking Web Pages & Slotted ALOHA. Network Performance 10-1

Week 10: DTMC Applications Ranking Web Pages & Slotted ALOHA. Network Performance 10-1 Week : DTMC Alictions Rnking Web ges & Slotted ALOHA etwok efonce - Outline Aly the theoy of discete tie Mkov chins: Google s nking of web-ges Wht ge is the use ost likely seching fo? Foulte web-gh s Mkov

More information

Difference Sets of Null Density Subsets of

Difference Sets of Null Density Subsets of dvces Pue Mthetcs 95-99 http://ddoog/436/p37 Pulshed Ole M (http://wwwscrpog/oul/p) Dffeece Sets of Null Dest Susets of Dwoud hd Dsted M Hosse Deptet of Mthetcs Uvest of Gul Rsht I El: hd@gulc h@googlelco

More information

CURVE FITTING LEAST SQUARES METHOD

CURVE FITTING LEAST SQUARES METHOD Nuercl Alss for Egeers Ger Jord Uverst CURVE FITTING Although, the for of fucto represetg phscl sste s kow, the fucto tself ot be kow. Therefore, t s frequetl desred to ft curve to set of dt pots the ssued

More information

Analele Universităţii din Oradea, Fascicula: Protecţia Mediului, Vol. XIII, 2008

Analele Universităţii din Oradea, Fascicula: Protecţia Mediului, Vol. XIII, 2008 Alele Uverstăţ d Orde Fsul: Proteţ Medulu Vol. XIII 00 THEORETICAL AND COMPARATIVE STUDY REGARDING THE MECHANICS DISPLASCEMENTS UNDER THE STATIC LOADINGS FOR THE SQUARE PLATE MADE BY WOOD REFUSE AND MASSIF

More information

African Journal of Science and Technology (AJST) Science and Engineering Series Vol. 4, No. 2, pp GENERALISED DELETION DESIGNS

African Journal of Science and Technology (AJST) Science and Engineering Series Vol. 4, No. 2, pp GENERALISED DELETION DESIGNS Af Joul of See Tehology (AJST) See Egeeg See Vol. 4, No.,. 7-79 GENERALISED DELETION DESIGNS Mhel Ku Gh Joh Wylff Ohbo Dee of Mhe, Uvey of Nob, P. O. Bo 3097, Nob, Key ABSTRACT:- I h e yel gle ele fol

More information

MATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER

MATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER MATHEMATICS II PUC VECTOR ALGEBRA QUESTIONS & ANSWER I One M Queston Fnd the unt veto n the deton of Let ˆ ˆ 9 Let & If Ae the vetos & equl? But vetos e not equl sne the oespondng omponents e dstnt e detons

More information

Advanced Higher Maths: Formulae

Advanced Higher Maths: Formulae Advced Highe Mths: Fomule Advced Highe Mthemtics Gee (G): Fomule you solutely must memoise i ode to pss Advced Highe mths. Rememe you get o fomul sheet t ll i the em! Ame (A): You do t hve to memoise these

More information

148 CIVIL ENGINEERING

148 CIVIL ENGINEERING STRUTUR NYSS fluee es fo Bems d Tusses fluee le sows te vto of effet (eto, se d momet ems, foe tuss) used movg ut lod oss te stutue. fluee le s used to deteme te posto of movele set of lods tt uses te

More information

Introduction to Modern Control Theory

Introduction to Modern Control Theory Itroductio to Moder Cotrol Theory MM : Itroductio to Stte-Spce Method MM : Cotrol Deig for Full Stte Feedck MM 3: Etitor Deig MM 4: Itroductio of the Referece Iput MM 5: Itegrl Cotrol d Rout Trckig //4

More information

Exercise # 2.1 3, 7, , 3, , -9, 1, Solution: natural numbers are 3, , -9, 1, 2.5, 3, , , -9, 1, 2 2.5, 3, , -9, 1, , -9, 1, 2.

Exercise # 2.1 3, 7, , 3, , -9, 1, Solution: natural numbers are 3, , -9, 1, 2.5, 3, , , -9, 1, 2 2.5, 3, , -9, 1, , -9, 1, 2. Chter Chter Syste of Rel uers Tertg Del frto: The del frto whh Gve fte uers of dgts ts del rt s lled tertg del frto. Reurrg ( o-tertg )Del frto: The del frto (No tertg) whh soe dgts re reeted g d g the

More information

xl yl m n m n r m r m r r! The inner sum in the last term simplifies because it is a binomial expansion of ( x + y) r : e +.

xl yl m n m n r m r m r r! The inner sum in the last term simplifies because it is a binomial expansion of ( x + y) r : e +. Ler Trsfortos d Group Represettos Hoework #3 (06-07, Aswers Q-Q re further exerses oer dots, self-dot trsfortos, d utry trsfortos Q3-6 volve roup represettos Of these, Q3 d Q4 should e quk Q5 s espelly

More information

Spherical refracting surface. Here, the outgoing rays are on the opposite side of the surface from the Incoming rays.

Spherical refracting surface. Here, the outgoing rays are on the opposite side of the surface from the Incoming rays. Sphericl refrctig urfce Here, the outgoig ry re o the oppoite ide of the urfce from the Icomig ry. The oject i t P. Icomig ry PB d PV form imge t P. All prxil ry from P which trike the phericl urfce will

More information

E-Companion: Mathematical Proofs

E-Companion: Mathematical Proofs E-omnon: Mthemtcl Poo Poo o emm : Pt DS Sytem y denton o t ey to vey tht t ncee n wth d ncee n We dene } ] : [ { M whee / We let the ttegy et o ech etle n DS e ]} [ ] [ : { M w whee M lge otve nume oth

More information

G8-11 Congruence Rules

G8-11 Congruence Rules G8-11 ogruee Rules If two polgos re ogruet, ou ple the oe o top of the other so tht the th etl. The verties tht th re lled orrespodig verties. The gles tht th re lled orrespodig gles. The sides tht th

More information

SYSTEMS OF NON-LINEAR EQUATIONS. Introduction Graphical Methods Close Methods Open Methods Polynomial Roots System of Multivariable Equations

SYSTEMS OF NON-LINEAR EQUATIONS. Introduction Graphical Methods Close Methods Open Methods Polynomial Roots System of Multivariable Equations SYSTEMS OF NON-LINEAR EQUATIONS Itoduto Gaphal Method Cloe Method Ope Method Polomal Root Stem o Multvaale Equato Chapte Stem o No-Lea Equato /. Itoduto Polem volvg o-lea equato egeeg lude optmato olvg

More information

The formulae in this booklet have been arranged according to the unit in which they are first

The formulae in this booklet have been arranged according to the unit in which they are first Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge ccog to the ut whch the e fst touce. Thus cte sttg ut m e eque to use the fomule tht wee touce peceg ut e.g. ctes sttg C mght e epecte to use

More information

Chapter Unary Matrix Operations

Chapter Unary Matrix Operations Chpter 04.04 Ury trx Opertos After redg ths chpter, you should be ble to:. kow wht ury opertos mes,. fd the trspose of squre mtrx d t s reltoshp to symmetrc mtrces,. fd the trce of mtrx, d 4. fd the ermt

More information

Parametric Methods. Autoregressive (AR) Moving Average (MA) Autoregressive - Moving Average (ARMA) LO-2.5, P-13.3 to 13.4 (skip

Parametric Methods. Autoregressive (AR) Moving Average (MA) Autoregressive - Moving Average (ARMA) LO-2.5, P-13.3 to 13.4 (skip Pmeti Methods Autoegessive AR) Movig Avege MA) Autoegessive - Movig Avege ARMA) LO-.5, P-3.3 to 3.4 si 3.4.3 3.4.5) / Time Seies Modes Time Seies DT Rdom Sig / Motivtio fo Time Seies Modes Re the esut

More information

Project 3: Using Identities to Rewrite Expressions

Project 3: Using Identities to Rewrite Expressions MAT 5 Projet 3: Usig Idetities to Rewrite Expressios Wldis I lger, equtios tht desrie properties or ptters re ofte lled idetities. Idetities desrie expressio e repled with equl or equivlet expressio tht

More information

Describes techniques to fit curves (curve fitting) to discrete data to obtain intermediate estimates.

Describes techniques to fit curves (curve fitting) to discrete data to obtain intermediate estimates. CURVE FITTING Descbes techques to ft cuves (cuve fttg) to dscete dt to obt temedte estmtes. Thee e two geel ppoches fo cuve fttg: Regesso: Dt ehbt sgfct degee of sctte. The stteg s to deve sgle cuve tht

More information

Linear Open Loop Systems

Linear Open Loop Systems Colordo School of Me CHEN43 Trfer Fucto Ler Ope Loop Sytem Ler Ope Loop Sytem... Trfer Fucto for Smple Proce... Exmple Trfer Fucto Mercury Thermometer... 2 Derblty of Devto Vrble... 3 Trfer Fucto for Proce

More information

DETAIL MEASURE EVALUATE

DETAIL MEASURE EVALUATE MEASURE EVALUATE B I M E q u i t y BIM Workflow Guide MEASURE EVALUATE Introduction We o e to ook 2 i t e BIM Workflow Guide i uide wi tr i you i re ti ore det i ed ode d do u e t tio u i r i d riou dd

More information

minimize c'x subject to subject to subject to

minimize c'x subject to subject to subject to z ' sut to ' M ' M N uostrd N z ' sut to ' z ' sut to ' sl vrls vtor of : vrls surplus vtor of : uostrd s s s s s s z sut to whr : ut ost of :out of : out of ( ' gr of h food ( utrt : rqurt for h utrt

More information

Noise in electronic components.

Noise in electronic components. No lto opot5098, JDS No lto opot Th PN juto Th ut thouh a PN juto ha fou opot t: two ffuo ut (hol fo th paa to th aa a lto th oppot to) a thal at oty ha a (hol fo th aa to th paa a lto th oppot to, laka

More information

D. Bertsekas and R. Gallager, "Data networks." Q: What are the labels for the x-axis and y-axis of Fig. 4.2?

D. Bertsekas and R. Gallager, Data networks. Q: What are the labels for the x-axis and y-axis of Fig. 4.2? pd by J. Succ ECE 543 Octob 22 2002 Outl Slottd Aloh Dft Stblzd Slottd Aloh Uslottd Aloh Splttg Algoths Rfc D. Btsks d R. llg "Dt twoks." Rvw (Slottd Aloh): : Wht th lbls fo th x-xs d y-xs of Fg. 4.2?

More information

Lectures # He-like systems. October 31 November 4,6

Lectures # He-like systems. October 31 November 4,6 Lectue #5-7 7 Octoe 3 oveme 4,6 Self-conitent field Htee-Foc eqution: He-lie ytem Htee-Foc eqution: cloed-hell hell ytem Chpte 3, pge 6-77, Lectue on Atomic Phyic He-lie ytem H (, h ( + h ( + h ( Z Z:

More information

STUDY PACKAGE. Subject : Mathematics Topic : DETERMINANTS & MATRICES Available Online :

STUDY PACKAGE. Subject : Mathematics Topic : DETERMINANTS & MATRICES Available Online : o/u opkj Hkh# tu] ugh vkjehks dke] oi s[k NksM+s qj e/;e eu dj ';kea iq#"k lg ldyi dj] lgs oi vusd] ^uk^ u NksM+s /;s; dks] j?kqj jk[ks VsdAA jp% ekuo /kez iz.ksk l~q# Jh j.knksm+klth egkjkt STUDY PAKAGE

More information

Some Different Perspectives on Linear Least Squares

Some Different Perspectives on Linear Least Squares Soe Dfferet Perspectves o Lear Least Squares A stadard proble statstcs s to easure a respose or depedet varable, y, at fed values of oe or ore depedet varables. Soetes there ests a deterstc odel y f (,,

More information

Lecture 3 summary. C4 Lecture 3 - Jim Libby 1

Lecture 3 summary. C4 Lecture 3 - Jim Libby 1 Lecue su Fes of efeece Ivce ude sfoos oo of H wve fuco: d-fucos Eple: e e - µ µ - Agul oeu s oo geeo Eule gles Geec slos cosevo lws d Noehe s heoe C4 Lecue - Lbb Fes of efeece Cosde fe of efeece O whch

More information

XII. Addition of many identical spins

XII. Addition of many identical spins XII. Addto of may detcal sps XII.. ymmetc goup ymmetc goup s the goup of all possble pemutatos of obects. I total! elemets cludg detty opeato. Each pemutato s a poduct of a ceta fte umbe of pawse taspostos.

More information

Numerical Analysis Topic 4: Least Squares Curve Fitting

Numerical Analysis Topic 4: Least Squares Curve Fitting Numerl Alss Top 4: Lest Squres Curve Fttg Red Chpter 7 of the tetook Alss_Numerk Motvto Gve set of epermetl dt: 3 5. 5.9 6.3 The reltoshp etwee d m ot e ler. Fd futo f tht est ft the dt 3 Alss_Numerk Motvto

More information

Multi-Electron Atoms-Helium

Multi-Electron Atoms-Helium Multi-lecto Atos-Heliu He - se s H but with Z He - electos. No exct solutio of.. but c use H wve fuctios d eegy levels s sttig poit ucleus sceeed d so Zeffective is < sceeig is ~se s e-e epulsio fo He,

More information

ELECTROPHORESIS IN STRUCTURED COLLOIDS

ELECTROPHORESIS IN STRUCTURED COLLOIDS ELKIN 4 ELECTROPHORESIS IN STRUCTURE COLLOIS José M. Médez A. Cvestv Mexo I ollboto wt O. Aló-Wess ULA d J. J. Beel-Mstett Cvestv. V µ E; µ 6πη ε ζ ; ζ 3 ε ζ ζ 4 THE GENERATION OF ONE PARTICLE EFFECTIVE

More information

M5. LTI Systems Described by Linear Constant Coefficient Difference Equations

M5. LTI Systems Described by Linear Constant Coefficient Difference Equations 5. LTI Systes Descied y Lie Costt Coefficiet Diffeece Equtios Redig teil: p.34-4, 245-253 3/22/2 I. Discete-Tie Sigls d Systes Up til ow we itoduced the Fouie d -tsfos d thei popeties with oly ief peview

More information

Born-Oppenheimer Approximation. Kaito Takahashi

Born-Oppenheimer Approximation. Kaito Takahashi o-oppehee ppoato Kato Takahah toc Ut Fo quatu yte uch a ecto ad olecule t eae to ue ut that ft the=tomc UNT Ue a of ecto (ot kg) Ue chage of ecto (ot coulob) Ue hba fo agula oetu (ot kg - ) Ue 4pe 0 fo

More information

( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x

( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x SIO 22B, Rudnick dpted fom Dvis III. Single vile sttistics The next few lectues e intended s eview of fundmentl sttistics. The gol is to hve us ll speking the sme lnguge s we move to moe dvnced topics.

More information

Maximize: x (1.1) Where s is slack variable vector of size m 1. This is a maximization problem. Or (1.2)

Maximize: x (1.1) Where s is slack variable vector of size m 1. This is a maximization problem. Or (1.2) A ew Algorth for er Progrg Dhy P. ehedle Deprtet of Eletro See, Sr Prhurhu College, lk Rod, Pue-00, d dhy.p.ehedle@gl.o Atrt- th pper we propoe ew lgorth for ler progrg. h ew lgorth ed o tretg the oetve

More information

Note 7 Root-Locus Techniques

Note 7 Root-Locus Techniques Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 Note 7 Root-Locu Techique Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd Lecture Note

More information

Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:

Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right: Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the

More information

SPH3UW Unit 7.5 Snell s Law Page 1 of Total Internal Reflection occurs when the incoming refraction angle is

SPH3UW Unit 7.5 Snell s Law Page 1 of Total Internal Reflection occurs when the incoming refraction angle is SPH3UW Uit 7.5 Sell s Lw Pge 1 of 7 Notes Physis Tool ox Refrtio is the hge i diretio of wve due to hge i its speed. This is most ommoly see whe wve psses from oe medium to other. Idex of refrtio lso lled

More information

Summary: Binomial Expansion...! r. where

Summary: Binomial Expansion...! r. where Summy: Biomil Epsio 009 M Teo www.techmejcmth-sg.wes.com ) Re-cp of Additiol Mthemtics Biomil Theoem... whee )!!(! () The fomul is ville i MF so studets do ot eed to memoise it. () The fomul pplies oly

More information

ROOT-LOCUS ANALYSIS. Lecture 11: Root Locus Plot. Consider a general feedback control system with a variable gain K. Y ( s ) ( ) K

ROOT-LOCUS ANALYSIS. Lecture 11: Root Locus Plot. Consider a general feedback control system with a variable gain K. Y ( s ) ( ) K ROOT-LOCUS ANALYSIS Coder a geeral feedback cotrol yte wth a varable ga. R( Y( G( + H( Root-Locu a plot of the loc of the pole of the cloed-loop trafer fucto whe oe of the yte paraeter ( vared. Root locu

More information

The z-transform. LTI System description. Prof. Siripong Potisuk

The z-transform. LTI System description. Prof. Siripong Potisuk The -Trsform Prof. Srpog Potsuk LTI System descrpto Prevous bss fucto: ut smple or DT mpulse The put sequece s represeted s ler combto of shfted DT mpulses. The respose s gve by covoluto sum of the put

More information

20.2. The Transform and its Inverse. Introduction. Prerequisites. Learning Outcomes

20.2. The Transform and its Inverse. Introduction. Prerequisites. Learning Outcomes The Trnform nd it Invere 2.2 Introduction In thi Section we formlly introduce the Lplce trnform. The trnform i only pplied to cul function which were introduced in Section 2.1. We find the Lplce trnform

More information

Induction. Induction and Recursion. Induction is a very useful proof technique

Induction. Induction and Recursion. Induction is a very useful proof technique Iductio Iductio is vey useul poo techique Iductio d Recusio CSC-59 Discete Stuctues I compute sciece, iductio is used to pove popeties o lgoithms Iductio d ecusio e closely elted Recusio is desciptio method

More information

Section 11.5 Notes Page Partial Fraction Decomposition. . You will get: +. Therefore we come to the following: x x

Section 11.5 Notes Page Partial Fraction Decomposition. . You will get: +. Therefore we come to the following: x x Setio Notes Pge Prtil Frtio Deompositio Suppose we were sked to write the followig s sigle frtio: We would eed to get ommo deomitors: You will get: Distributig o top will give you: 8 This simplifies to:

More information

Empirical equations for electrical parameters of asymmetrical coupled microstrip lines

Empirical equations for electrical parameters of asymmetrical coupled microstrip lines Epl equons fo elel petes of syel ouple osp lnes I.M. Bsee Eletons eseh Instute El-h steet, Dokk, o, Egypt Abstt: Epl equons e eve fo the self n utul nutne n ptne fo two syel ouple osp lnes. he obne ptne

More information

( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi

( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi Assgmet /MATH 47/Wte Due: Thusday Jauay The poblems to solve ae umbeed [] to [] below Fst some explaatoy otes Fdg a bass of the colum-space of a max ad povg that the colum ak (dmeso of the colum space)

More information

Fredholm Type Integral Equations with Aleph-Function. and General Polynomials

Fredholm Type Integral Equations with Aleph-Function. and General Polynomials Iteto Mthetc Fou Vo. 8 3 o. 989-999 HIKI Ltd.-h.co Fedho Te Iteg uto th eh-fucto d Gee Poo u J K.J. o Ittute o Mgeet tude & eech Mu Id u5@g.co Kt e K.J. o Ittute o Mgeet tude & eech Mu Id dehuh_3@hoo.co

More information

BINOMIAL THEOREM SOLUTION. 1. (D) n. = (C 0 + C 1 x +C 2 x C n x n ) (1+ x+ x 2 +.)

BINOMIAL THEOREM SOLUTION. 1. (D) n. = (C 0 + C 1 x +C 2 x C n x n ) (1+ x+ x 2 +.) BINOMIAL THEOREM SOLUTION. (D) ( + + +... + ) (+ + +.) The coefficiet of + + + +... + fo. Moeove coefficiet of is + + + +... + if >. So. (B)... e!!!! The equied coefficiet coefficiet of i e -.!...!. (A),

More information

CBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane.

CBSE , ˆj. cos CBSE_2015_SET-1. SECTION A 1. Given that a 2iˆ ˆj. We need to find. 3. Consider the vector equation of the plane. CBSE CBSE SET- SECTION. Gv tht d W d to fd 7 7 Hc, 7 7 7. Lt,. W ow tht.. Thus,. Cosd th vcto quto of th pl.. z. - + z = - + z = Thus th Cts quto of th pl s - + z = Lt d th dstc tw th pot,, - to th pl.

More information

Mathematical Notation Math Calculus & Analytic Geometry I

Mathematical Notation Math Calculus & Analytic Geometry I Mthemticl Nottio Mth - Clculus & Alytic Geometry I Use Wor or WorPerect to recrete the ollowig ocumets. Ech rticle is worth poits shoul e emile to the istructor t jmes@richl.eu. Type your me t the top

More information

MTH 146 Class 7 Notes

MTH 146 Class 7 Notes 7.7- Approxmte Itegrto Motvto: MTH 46 Clss 7 Notes I secto 7.5 we lered tht some defte tegrls, lke x e dx, cot e wrtte terms of elemetry fuctos. So, good questo to sk would e: How c oe clculte somethg

More information

Artificial Intelligence Markov Decision Problems

Artificial Intelligence Markov Decision Problems rtificil Intelligence Mrkov eciion Problem ilon - briefly mentioned in hpter Ruell nd orvig - hpter 7 Mrkov eciion Problem; pge of Mrkov eciion Problem; pge of exmple: probbilitic blockworld ction outcome

More information

6.6 Moments and Centers of Mass

6.6 Moments and Centers of Mass th 8 www.tetodre.co 6.6 oets d Ceters of ss Our ojectve here s to fd the pot P o whch th plte of gve shpe lces horzotll. Ths pot s clled the ceter of ss ( or ceter of grvt ) of the plte.. We frst cosder

More information

Lattice planes. Lattice planes are usually specified by giving their Miller indices in parentheses: (h,k,l)

Lattice planes. Lattice planes are usually specified by giving their Miller indices in parentheses: (h,k,l) Ltte ples Se the epol ltte of smple u ltte s g smple u ltte d the Mlle des e the oodtes of eto oml to the ples, the use s ey smple lttes wth u symmety. Ltte ples e usully spefed y gg the Mlle des petheses:

More information

Chapter #2 EEE Subsea Control and Communication Systems

Chapter #2 EEE Subsea Control and Communication Systems EEE 87 Chpter # EEE 87 Sube Cotrol d Commuictio Sytem Trfer fuctio Pole loctio d -ple Time domi chrcteritic Extr pole d zero Chpter /8 EEE 87 Trfer fuctio Lplce Trform Ued oly o LTI ytem Differetil expreio

More information

Stochastic simulation methods

Stochastic simulation methods tot ulto etod oue oepe: oltá éd U-lu lu Ro ed@p.ubblu.o o: ote-lo etod ttp://www.p.ubblu.o/~ed/.tl obetve o te oue: - letue ote - opute ode - ee poet - ppe to du - pott oueet - teet l o te web o ve PRAIA

More information

12781 Velp Avenue. West County B Rural Residential Development

12781 Velp Avenue. West County B Rural Residential Development U PL & EET E 28 Vel ee eded 2 P.. ) LL EET T E 2) PPVE E ) ET E ) e e e e eded eebe 2 Plg & g eeg b) Bldg Pe e: eebe ) PUBL FU ( -E TE): g be bg bee e Plg & g eel ll be de ll be e. 5) UEETFEEBK: ) be ll

More information

BEM with Linear Boundary Elements for Solving the Problem of the 3D Compressible Fluid Flow around Obstacles

BEM with Linear Boundary Elements for Solving the Problem of the 3D Compressible Fluid Flow around Obstacles EM wth L ou Elts o olvg th Pol o th D opssl Flu Flow ou Ostls Lut Gu o Vlsu stt hs pp psts soluto o th sgul ou tgl quto o th D opssl lu low ou ostl whh uss sopt l ou lts o Lgg tp. h sgul ou tgl quto oult

More information

Course Updates. Reminders: 1) Assignment #8 available. 2) Chapter 28 this week.

Course Updates. Reminders: 1) Assignment #8 available. 2) Chapter 28 this week. Couse Updtes http://www.phys.hwii.edu/~vne/phys7-sp1/physics7.html Remindes: 1) Assignment #8 vilble ) Chpte 8 this week Lectue 3 iot-svt s Lw (Continued) θ d θ P R R θ R d θ d Mgnetic Fields fom long

More information

Asymptotic Dominance Problems. is not constant but for n 0, f ( n) 11. 0, so that for n N f

Asymptotic Dominance Problems. is not constant but for n 0, f ( n) 11. 0, so that for n N f Asymptotc Domce Prolems Dsply ucto : N R tht s Ο( ) ut s ot costt 0 = 0 The ucto ( ) = > 0 s ot costt ut or 0, ( ) Dee the relto " " o uctos rom N to R y g d oly = Ο( g) Prove tht s relexve d trstve (Recll:

More information

Accuplacer Elementary Algebra Study Guide

Accuplacer Elementary Algebra Study Guide Testig Ceter Studet Suess Ceter Aupler Elemetry Alger Study Guide The followig smple questios re similr to the formt d otet of questios o the Aupler Elemetry Alger test. Reviewig these smples will give

More information

The use of linear parametric approximation in numerical solving of nonlinear non-smooth Fuzzy equations

The use of linear parametric approximation in numerical solving of nonlinear non-smooth Fuzzy equations vlle ole t www.choleechl.co chve o ppled Scece Reech 5 6:49-6 http://choleechl.co/chve.htl ISSN 975-58X CODEN US SRC9 The ue o le petc ppoto uecl olvg o ole o-ooth uzz equto Mjd Hllj e Ze d l Vhd Kd Nehu

More information

Physics 604 Problem Set 1 Due Sept 16, 2010

Physics 604 Problem Set 1 Due Sept 16, 2010 Physics 64 Polem et 1 Due ept 16 1 1) ) Inside good conducto the electic field is eo (electons in the conducto ecuse they e fee to move move in wy to cncel ny electic field impessed on the conducto inside

More information

On the energy of complement of regular line graphs

On the energy of complement of regular line graphs MATCH Coucato Matheatcal ad Coputer Chetry MATCH Cou Math Coput Che 60 008) 47-434 ISSN 0340-653 O the eergy of copleet of regular le graph Fateeh Alaghpour a, Baha Ahad b a Uverty of Tehra, Tehra, Ira

More information

The limit comparison test

The limit comparison test Roerto s Notes o Ifiite Series Chpter : Covergece tests Sectio 4 The limit compriso test Wht you eed to kow lredy: Bsics of series d direct compriso test. Wht you c ler here: Aother compriso test tht does

More information

Handout #2. Introduction to Matrix: Matrix operations & Geometric meaning

Handout #2. Introduction to Matrix: Matrix operations & Geometric meaning Hdout # Title: FAE Course: Eco 8/ Sprig/5 Istructor: Dr I-Mig Chiu Itroductio to Mtrix: Mtrix opertios & Geometric meig Mtrix: rectgulr rry of umers eclosed i pretheses or squre rckets It is covetiolly

More information

Chapter 2: Descriptive Statistics

Chapter 2: Descriptive Statistics Chapte : Decptve Stattc Peequte: Chapte. Revew of Uvaate Stattc The cetal teecy of a oe o le yetc tbuto of a et of teval, o hghe, cale coe, ofte uaze by the athetc ea, whch efe a We ca ue the ea to ceate

More information