Fundamentals of Linear Algebra

Size: px
Start display at page:

Download "Fundamentals of Linear Algebra"

Transcription

1 -7/8-797 Mchine Lerning for Signl rocessing Fundmentls of Liner Alger Administrivi Registrtion: Anone on witlist still? Homework : Will e hnded out with clss Liner lger Clss - Sep Instructor: Bhiksh Rj Overview Vectors nd mtrices Bsic vector/mtri opertions Vector products Mtri products Vrious mtri tpes Mtri inversion Mtri interprettion Eigennlsis Singulr vlue decomposition Book Fundmentls of Liner Alger, Gilert Strng Importnt to e ver comfortle with liner lger Appers repetedl in the form of Eigen nlsis, SVD, Fctor nlsis Appers through h vrious properties of mtrices tht t re used in mchine lerning, prticulrl when pplied to imges nd sound Tod s lecture: Definitions Ver smll suset of ll tht s used Importnt suset, intended to help ou recollect Incentive to use liner lger rett nottion! T A j j i i ij And other things ou cn do Frequenc From Bch s Fugue in Gm Esier intuition Rell convenient geometric interprettions Opertions es to descrie verll Es code trnsltion! for i=:n for j=:m c(i)=c(i)+(j)*(i)*(i,j) end end C=*A* Rottion + rojection + Scling Mnipulte Imges Mnipulte Sounds Time Decomposition (MF)

2 Sclrs, vectors, mtrices, A sclr is numer =, =, = -, etc A vector is liner rrngement of collection of sclrs, A mtri A is rectngulr rrngement of collection of vectors A MATLAB snt: =[ ], A=[ ; ] 7 Vector/Mtri tpes nd shpes Vectors re either column or row vectors c, r c c, s A sound cn e vector, series of dil tempertures cn e vector, etc Mtrices cn e squre or rectngulr S c d, R c d e f, M Imges cn e mtri, collections of sounds cn e mtri, etc 8 Dimensions of mtri The mtri sie is specified the numer of rows nd columns c, r c c c = mtri: rows nd column r = mtri: ti row nd d columns S c, R d d c e f S = mtri R = mtri cmn = 99 mtri v Representing n imge s mtri Vlues onl; nd re implicit pcmen A 99 mtri Row nd Column = position A 879 mtri Triples of, nd vlue A 879 vector Unrveling the mtri ote: All of these cn e recst s the mtri tht forms the imge Representtions nd re equivlent The position is not represented 9 Emple of vector Vectors usull hold sets of numericl ttriutes,, vlue [,, ] Ernings, losses, suicides [$ $ ] Etc Consider reltive Mnhttn vector rovides reltive position giving numer of venues nd streets to cross, eg [v st] [-v st] [v st] [v 8st] Vectors Ordered collection of numers Emples: [ ], [ c d], [ ]!= [ ] Order is importnt Tpicll viewed s identifing (the pth from origin to) loction in n -dimensionl spce (,,) (,,)

3 Vectors vs Mtrices (,,) A vector is geometric nottion for how to get from (,) to some loction in the spce A mtri is simpl collection of destintions! roperties of mtrices re verge properties of the trveller s pth to these destintions Bsic rithmetic opertions Addition nd sutrction Element-wise opertions A B MATLAB snt: + nd - Vector Opertions Opertions emple (,,) (,-,-) - - (,,) Opertions tell us how to get from ({}) to the result of the vector opertions (,,) + (,-,-) = (,,) Adding rndom vlues to different representtions of the imge + + Rndom(,columns(M)) Vector norm Mesure of how ig vector is: otted s In Mnhttn vectors mesure of distnce 7 7 MATLAB snt: norm() [-v 7st] [-v st] 7 Vector orm Length = sqrt( + + ) (,,) Geometricll the shortest distnce to trvel from the origin to the destintion As the crow flies Assuming Eucliden Geometr 8

4 Trnsposition A trnsposed row vector ecomes column (nd vice vers), c c c, T c A trnsposed mtri gets ll its row (or column) vectors trnsposed in order c d e f, T c MATLAB snt: d e f M, M T 9 Vector multipliction Multipliction is not element-wise! Dot product, or inner product Vectors must hve the sme numer of elements Row vector times column vector = sclr d c e d e c f f f Cross product, outer product or vector direct product Column vector times row vector = mtri d e f d e f d e f c c d c e c f MATLAB snt: * Vector dot product in Mnhttn Multipling the rd vectors Insted of venue/street we ll use rds = [ ], = [77 ] The dot product of the two vectors reltes to the length of projection How much of the first vector hve we covered following the second one? The nswer comes ck s unit of the first vector so we divide its length 77 T 9d [d d] norm norm 9d [77d d] norm 8 Vector dot product Sqrt(energ) D S D frequenc frequenc frequenc 9 Vectors re spectr Energ t discrete set of frequencies Actull 9 is is the inde of the numer in the vector Represents frequenc is is the vlue of the numer in the vector Represents mgnitude Vector dot product Sqrt(energ) D S D Vector cross product frequenc frequenc frequenc 9 How much of D is lso in S How much cn ou fke D pling n S DS / D S = ot ver much How much of D is in D? DD / D / D = ot d, ou cn fke it To do this, D, S, nd D must e the sme sie The column vector is the spectrum The row vector is n mplitude modultion The crossproduct is spectrogrm Shows how the energ in ech frequenc vries with time The pttern in ech column is scled version of the spectrum Ech row is scled version of the modultion

5 Mtri multipliction Generlition of vector multipliction Dot product of ech vector pir A B Dimensions must mtch!! Columns of first mtri = rows of second Result inherits the numer of rows from the first mtri nd the numer of columns from the second mtri MATLAB snt: * Multipling Vector Mtri (,:) 9 (,:) Multipliction of vector mtri epresses the vector in terms of projections of on the row vectors of the mtri It scles nd rottes the vector Alterntel viewed, it scles nd rottes the spce the underling plne Mtri Multipliction Mtri Multipliction 7 The mtri rottes nd scles the spce Including its own vectors 7 The normls to the row vectors in the mtri ecome the new es is = norml to the second row vector Scled the inverse of the length of the first row vector 8 Mtri Multipliction is projection The k-th is scorresponds dstot the norml tot the hperplne pep e represented the k-,k+-th row vectors in the mtri An set of K- vectors represent hperplne of dimension K- or less The distnce long the new is equls the length of the projection on the k-th row vector Epressed in inverse-lengths of the vector Mtri Multipliction: Column spce d e c c f d e f So much for spces wht does multipling mtri vector rell do? It mies the column vectors of the mtri using the numers in the vector The column spce of the Mtri is the complete set of ll vectors tht cn e formed miing its columns 9

6 Mtri Multipliction: Row spce d e c f c d e f Left multipliction mies the row vectors of the mtri The row spce of the Mtri is the complete set of ll vectors tht cn e formed miing its rows Mtri multipliction: Miing vectors 9 7 = A phsicl emple The three column vectors of the mtri re the spectr of three notes The multipling column vector is just miing vector The result is sound tht is the miture of the three notes Mtri multipliction: Miing vectors 7 Miing two imges The imges re rrnged s columns position vlue not included The result of the multipliction is rerrnged s n imge M Mtri multipliction: nother view A B M M Wht does this men? M K K k k k Mk k k k k kkk kk Mk K K K K M M M K The outer product of the first column of A nd the first row of B + outer product of the second column of A nd the second row of B + Wh is tht useful? Mtri multipliction: Miing modulted spectr 9 Sounds: Three notes modulted independentl Sounds: Three notes modulted independentl

7 Mtri multipliction: Miing modulted spectr Sounds: Three notes modulted independentl Mtri multipliction: Miing modulted spectr Sounds: Three notes modulted independentl 7 8 Mtri multipliction: Miing modulted spectr Sounds: Three notes modulted independentl Mtri multipliction: Miing modulted spectr Sounds: Three notes modulted independentl 9 Mtri multipliction: Imge trnsition Mtri multipliction: Imge trnsition i j i j Imge fdes out linerl Imge fdes in linerl i j i j i 9i 8i i 9 i 8 i i 9 i 8 i Ech column is one imge The columns represent sequence of imges of decresing intensit Imge fdes out linerl 7

8 Mtri multipliction: Imge trnsition Mtri multipliction: Imge trnsition i j i j Imge fdes in linerl i j i j Imge fdes out linerl Imge fdes in linerl The Identit Mtri Digonl Mtri An identit mtri is squre mtri where All digonl elements re All off-digonl elements re Multipliction n identit mtri does not chnge vectors All off-digonl elements re ero Digonl elements re non-ero Scles the es M flip es Digonl mtri to trnsform imges Stretching How? Loction-sed representtion Scling mtri onl scles the is The is nd piel vlue re scled identit ot good w of scling 7 8 8

9 9 Stretching D = -7/8-797 Better w ) ( ewpic EA A Sep 9 Modifing color B G R -7/8-797 Scle onl Green ewpic Sep ermuttion Mtri (,,) Z (old ) (old Z) Z (old ) -7/8-797 A permuttion mtri simpl rerrnges the es The row entries re is vectors in different order The result is comintion of rottions nd reflections The permuttion mtri effectivel permutes the rrngement of the elements in vector ( ) Sep ermuttion Mtri -7/8-797 Reflections nd 9 degree rottions of imges nd ojects Sep ermuttion Mtri -7/8-797 Reflections nd 9 degree rottions of imges nd ojects Oject represented s mtri of -Dimensionl position vectors ositions identif ech point on the surfce Sep Rottion Mtri ' ' cos sin sin cos new R (,) new R (,) (, ) cos sin ' sin cos ' -7/8-797 A rottion mtri rottes the vector some ngle Alterntel viewed, it rottes the es The new es re t n ngle to the old one Sep

10 Rotting picture cos R sin sin cos ote the representtion: -row mtri Rottion onl pplies on the coordinte rows The vlue does not chnge Wh is pcmn grin? -D Rottion new new Znew Z degrees of freedom seprte ngles Wht will the rottion mtri e? rojections rojection Mtri 9degrees W W projection Wht would we see if the cone to the left were trnsprent if we looked t it long the norml to the plne The plne goes through the origin Answer: the figure to the right How do we get this? rojection Consider n plne specified set of vectors W, W Or mtri [W W ] An vector cn e projected onto this plne The mtri A tht rottes nd scles the vector so tht it ecomes its projection is projection mtri 7 8 rojection Mtri 9degrees rojections W W projection Given set of vectors W, W, which form mtri W = [W W ] The projection mtri tht trnsforms n vector to its projection on the plne is = W (W T W) - W T We will visit mtri inversion shortl Mgic n set of vectors from the sme plne tht re epressed s mtri will give ou the sme projection mtri = V (V T V) - V T 9 HOW?

11 rojections rojections Drw n two vectors W nd W tht lie on the plne A two so long s the hve different ngles Compose mtri W = [W W] Compose the projection mtri = W (W T W) - W T Multipl ever point on the cone to get its projection View it I m missing step here wht is it? The projection ctull projects it onto the plne, ut ou re still seeing the plne in D The result of the projection is -D vector = W (W T W) - W T =, *Vector = The imge must e rotted till the plne is in the plne of the pper The Z is in this cse will lws e ero nd cn e ignored How will ou rotte it? (rememer ou know W nd W) rojection mtri properties The projection of n vector tht is lred on the plne is the vector itself = if is on the plne If the oject is lred on the plne, there is no further projection to e performed The projection of projection is the projection () = Tht is ecuse is lred on the plne rojection mtrices re idempotent = Sep Follows from the ove -7/8-797

Fundamentals of Linear Algebra

Fundamentals of Linear Algebra -7/8-797 Mchine Lerning for Signl Processing Fundmentls of Liner Alger Administrivi Registrtion: Anone on witlist still? Homewor : Will pper over weeend Liner lger Clss Aug Instructor: Bhish Rj -7/8-797

More information

11-755/ Machine Learning for Signal Processing. Algebra. Class Sep Instructor: Bhiksha Raj

11-755/ Machine Learning for Signal Processing. Algebra. Class Sep Instructor: Bhiksha Raj -755/8-797 Mchine Lerning for Signl Processing Fundmentls of Liner Alger Clss -3 Sep Instructor: Bhiksh Rj Sep -755/8-797 Administrivi Registrtion: Anyone on witlist still? Homework : Will e hnded out

More information

11-755/ Machine Learning for Signal Processing. Algebra. Class August Instructor: Bhiksha Raj

11-755/ Machine Learning for Signal Processing. Algebra. Class August Instructor: Bhiksha Raj -755/8-797 Mchine Lerning for Signl Processing Fundmentls of Liner Alger Clss 6 August 9 Instructor: Bhiksh Rj 6 Aug -755/8-797 Administrivi Registrtion: Anyone on witlist still? Our TA is here Sourish

More information

Machine Learning for Signal Processing Fundamentals of Linear Algebra

Machine Learning for Signal Processing Fundamentals of Linear Algebra Mchine Lerning for Signl Processing Fundmentls of Liner Alger Clss 6 Sep 6 Instructor: Bhiksh Rj -755/8-797 Overview Vectors nd mtrices Bsic vector/mtrix opertions Vrious mtrix types Projections -755/8-797

More information

Lecture 8 Wrap-up Part1, Matlab

Lecture 8 Wrap-up Part1, Matlab Lecture 8 Wrp-up Prt1, Mtlb Homework Polic Plese stple our homework (ou will lose 10% credit if not stpled or secured) Submit ll problems in order. This mens to plce ever item relting to problem 3 (our

More information

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

Things 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

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

Analytically, vectors will be represented by lowercase bold-face 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 information

Chapter 2. Vectors. 2.1 Vectors Scalars and Vectors

Chapter 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 information

Operations with Matrices

Operations with Matrices Section. Equlit of Mtrices Opertions with Mtrices There re three ws to represent mtri.. A mtri cn be denoted b n uppercse letter, such s A, B, or C.. A mtri cn be denoted b representtive element enclosed

More information

Matrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24

Matrix 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 information

8.6 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

8.6 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 information

Introduction to Algebra - Part 2

Introduction to Algebra - Part 2 Alger Module A Introduction to Alger - Prt Copright This puliction The Northern Alert Institute of Technolog 00. All Rights Reserved. LAST REVISED Oct., 008 Introduction to Alger - Prt Sttement of Prerequisite

More information

Computer Graphics (CS 4731) Lecture 7: Linear Algebra for Graphics (Points, Scalars, Vectors)

Computer 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 information

Uses of transformations. 3D transformations. Review of vectors. Vectors in 3D. Points vs. vectors. Homogeneous coordinates S S [ H [ S \ H \ S ] H ]

Uses of transformations. 3D transformations. Review of vectors. Vectors in 3D. Points vs. vectors. Homogeneous coordinates S S [ H [ S \ H \ S ] H ] Uses of trnsformtions 3D trnsformtions Modeling: position nd resize prts of complex model; Viewing: define nd position the virtul cmer Animtion: define how objects move/chnge with time y y Sclr (dot) product

More information

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: Volumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge

More information

Lecture Solution of a System of Linear Equation

Lecture 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 information

Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY

Chapter 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 information

On the diagram below the displacement is represented by the directed line segment OA.

On 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 information

September 13 Homework Solutions

September 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 information

INTRODUCTION TO LINEAR ALGEBRA

INTRODUCTION 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 information

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 2

CS434a/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 information

Geometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.

Geometric 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 information

M344 - ADVANCED ENGINEERING MATHEMATICS

M344 - ADVANCED ENGINEERING MATHEMATICS M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If

More information

MATH1131 Mathematics 1A Algebra

MATH1131 Mathematics 1A Algebra MATH1131 Mthemtics 1A Alger UNSW Sydney Semester 1, 017 Mike Mssierer Mike is pronounced like Mich mike@unsweduu Plese emil me if you hve ny questions or comments Office hours TBA (week ), or emil me to

More information

1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and.

1.2 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 fill-in-lnks-prolems (print.pdf from www.motiongenesis.com Textooks Resources). 1.1 Solving prolems wht engineers

More information

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?

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 information

Chapter 1 VECTOR ALGEBRA

Chapter 1 VECTOR ALGEBRA Chpter 1 VECTOR LGEBR INTRODUCTION: Electromgnetics (EM) m be regrded s the stud of the interctions between electric chrges t rest nd in motion. Electromgnetics is brnch of phsics or electricl engineering

More information

Reference. Vector Analysis Chapter 2

Reference. Vector Analysis Chapter 2 Reference Vector nlsis Chpter Sttic Electric Fields (3 Weeks) Chpter 3.3 Coulomb s Lw Chpter 3.4 Guss s Lw nd pplictions Chpter 3.5 Electric Potentil Chpter 3.6 Mteril Medi in Sttic Electric Field Chpter

More information

Introduction To Matrices MCV 4UI Assignment #1

Introduction 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 information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 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 information

What we should know about Linear Algebra

What we should know about Linear Algebra Wht we should know out iner Alger D Coordinte geometr Vectors in spce nd spce Dot product nd cross product definitions nd uses Vector nd mtri nottion nd lger Properties (mtri ssocitiit ut NOT mtri commuttiit)

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

MATRICES AND VECTORS SPACE

MATRICES 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 information

Matrices and Determinants

Matrices 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 information

MATHEMATICS FOR MANAGEMENT BBMP1103

MATHEMATICS FOR MANAGEMENT BBMP1103 T o p i c M T R I X MTHEMTICS FOR MNGEMENT BBMP Ojectives: TOPIC : MTRIX. Define mtri. ssess the clssifictions of mtrices s well s know how to perform its opertions. Clculte the determinnt for squre mtri

More information

LINEAR ALGEBRA APPLIED

LINEAR 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 information

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution

Multiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: olumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge

More information

The Algebra (al-jabr) of Matrices

The 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 information

P 1 (x 1, y 1 ) is given by,.

P 1 (x 1, y 1 ) is given by,. MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous 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 information

MATRIX DEFINITION A matrix is any doubly subscripted array of elements arranged in rows and columns.

MATRIX 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 information

Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Partial 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 right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

Review of Gaussian Quadrature method

Review 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 information

Maths in Motion. Theo de Haan. Order now: 29,95 euro

Maths in Motion. Theo de Haan. Order now:   29,95 euro Mths in Motion Theo de Hn Order now: www.mthsinmotion.org 9,95 euro Cover Design: Drwings: Photogrph: Printing: Niko Spelbrink Lr Wgterveld Mrijke Spelbrink Rddrier, Amsterdm Preview: Prts of Chpter 6,

More information

Chapter 1: Logarithmic functions and indices

Chapter 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 information

Chapter 7: Applications of Integrals

Chapter 7: Applications of Integrals Chpter 7: Applictions of Integrls 78 Chpter 7 Overview: Applictions of Integrls Clculus, like most mthemticl fields, egn with tring to solve everd prolems. The theor nd opertions were formlized lter. As

More information

S56 (5.3) Vectors.notebook January 29, 2016

S56 (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 information

To Do. Vectors. Motivation and Outline. Vector Addition. Cartesian Coordinates. Foundations of Computer Graphics (Spring 2010) x y

To 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 information

Matrices. Elementary Matrix Theory. Definition of a Matrix. Matrix Elements:

Matrices. 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 information

Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Math 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, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of

More information

ES.182A Topic 32 Notes Jeremy Orloff

ES.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 information

Quadratic Forms. Quadratic Forms

Quadratic 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 information

THE DISCRIMINANT & ITS APPLICATIONS

THE DISCRIMINANT & ITS APPLICATIONS THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND 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 information

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?

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 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 information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.

Lecture 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 information

Algebra Of Matrices & Determinants

Algebra 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 information

Numerical Linear Algebra Assignment 008

Numerical 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 information

ESCI 241 Meteorology Lesson 0 Math and Physics Review

ESCI 241 Meteorology Lesson 0 Math and Physics Review UNITS ESCI 41 Meteorolog Lesson 0 Mth nd Phsics Review A numer is meningless unless it is ccompnied unit telling wht the numer represents. The stndrd unit sstem used interntionll scientists is known s

More information

Signal Flow Graphs. Consider a complex 3-port microwave network, constructed of 5 simpler microwave devices:

Signal Flow Graphs. Consider a complex 3-port microwave network, constructed of 5 simpler microwave devices: 3/3/009 ignl Flow Grphs / ignl Flow Grphs Consider comple 3-port microwve network, constructed of 5 simpler microwve devices: 3 4 5 where n is the scttering mtri of ech device, nd is the overll scttering

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT SCHOOL OF ENGINEERING & BUIL ENVIRONMEN MARICES FOR ENGINEERING Dr Clum Mcdonld Contents Introduction Definitions Wht is mtri? Rows nd columns of mtri Order of mtri Element of mtri Equlity of mtrices Opertions

More information

Here we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.

Here 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 information

Coordinate geometry and vectors

Coordinate 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 information

HW3, Math 307. CSUF. Spring 2007.

HW3, 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 information

set is not closed under matrix [ multiplication, ] and does not form a group.

set 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 information

1. Extend QR downwards to meet the x-axis at U(6, 0). y

1. Extend QR downwards to meet the x-axis at U(6, 0). y In the digrm, two stright lines re to be drwn through so tht the lines divide the figure OPQRST into pieces of equl re Find the sum of the slopes of the lines R(6, ) S(, ) T(, 0) Determine ll liner functions

More information

Geometrical Transformations

Geometrical Transformations Geometricl Trnsformtions Did Atkinson D.Atkinson@cl.c.k Compttionl Aspects of MRI References Fole, n Dm, Feiner, Hghes. Compter Grphics: Principles nd Prctice. Chpter 5. Wolfrm MthWorld http://mthworld.wolfrm.com/

More information

CSCI 5525 Machine Learning

CSCI 5525 Machine Learning CSCI 555 Mchine Lerning Some Deini*ons Qudrtic Form : nn squre mtri R n n : n vector R n the qudrtic orm: It is sclr vlue. We oten implicitly ssume tht is symmetric since / / I we write it s the elements

More information

Math 017. Materials With Exercises

Math 017. Materials With Exercises Mth 07 Mterils With Eercises Jul 0 TABLE OF CONTENTS Lesson Vriles nd lgeric epressions; Evlution of lgeric epressions... Lesson Algeric epressions nd their evlutions; Order of opertions....... Lesson

More information

Lecture 6. Notes. Notes. Notes. Representations Z A B and A B R. BTE Electronics Fundamentals August Bern University of Applied Sciences

Lecture 6. Notes. Notes. Notes. Representations Z A B and A B R. BTE Electronics Fundamentals August Bern University of Applied Sciences Lecture 6 epresenttions epresenttions TE52 - Electronics Fundmentls ugust 24 ern University of pplied ciences ev. c2d5c88 6. Integers () sign-nd-mgnitude representtion The set of integers contins the Nturl

More information

7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 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 information

Vectors. Introduction. Definition. The Two Components of a Vector. Vectors

Vectors. 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 information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 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 information

The practical version

The practical version Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht

More information

C Precalculus Review. C.1 Real Numbers and the Real Number Line. Real Numbers and the Real Number Line

C Precalculus Review. C.1 Real Numbers and the Real Number Line. Real Numbers and the Real Number Line C. Rel Numers nd the Rel Numer Line C C Preclculus Review C. Rel Numers nd the Rel Numer Line Represent nd clssif rel numers. Order rel numers nd use inequlities. Find the solute vlues of rel numers nd

More information

10.2 The Ellipse and the Hyperbola

10.2 The Ellipse and the Hyperbola CHAPTER 0 Conic Sections Solve. 97. Two surveors need to find the distnce cross lke. The plce reference pole t point A in the digrm. Point B is meters est nd meter north of the reference point A. Point

More information

THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES

THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES THREE-DIMENSIONAL KINEMATICS OF RIGID BODIES 1. TRANSLATION Figure shows rigid body trnslting in three-dimensionl spce. Any two points in the body, such s A nd B, will move long prllel stright lines if

More information

STRAND B: NUMBER THEORY

STRAND B: NUMBER THEORY Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors,

More information

Matching patterns of line segments by eigenvector decomposition

Matching patterns of line segments by eigenvector decomposition Title Mtching ptterns of line segments y eigenvector decomposition Author(s) Chn, BHB; Hung, YS Cittion The 5th IEEE Southwest Symposium on Imge Anlysis nd Interprettion Proceedings, Snte Fe, NM., 7-9

More information

What Is Calculus? 42 CHAPTER 1 Limits and Their Properties

What Is Calculus? 42 CHAPTER 1 Limits and Their Properties 60_00.qd //0 : PM Pge CHAPTER Limits nd Their Properties The Mistress Fellows, Girton College, Cmridge Section. STUDY TIP As ou progress through this course, rememer tht lerning clculus is just one of

More information

Section 6: Area, Volume, and Average Value

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

SCHOOL 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 information

Electromagnetics P5-1. 1) Physical quantities in EM could be scalar (charge, current, energy) or vector (EM fields).

Electromagnetics P5-1. 1) Physical quantities in EM could be scalar (charge, current, energy) or vector (EM fields). Electromgnetics 5- Lesson 5 Vector nlsis Introduction ) hsicl quntities in EM could be sclr (chrge current energ) or ector (EM fields) ) Specifing ector in -D spce requires three numbers depending on the

More information

Chapter 14. Matrix Representations of Linear Transformations

Chapter 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 information

Prerequisites CHAPTER P

Prerequisites CHAPTER P CHAPTER P Prerequisites P. Rel Numers P.2 Crtesin Coordinte System P.3 Liner Equtions nd Inequlities P.4 Lines in the Plne P.5 Solving Equtions Grphiclly, Numericlly, nd Algericlly P.6 Comple Numers P.7

More information

GRAND PLAN. Visualizing Quaternions. I: Fundamentals of Quaternions. Andrew J. Hanson. II: Visualizing Quaternion Geometry. III: Quaternion Frames

GRAND PLAN. Visualizing Quaternions. I: Fundamentals of Quaternions. Andrew J. Hanson. II: Visualizing Quaternion Geometry. III: Quaternion Frames Visuliing Quternions Andrew J. Hnson Computer Siene Deprtment Indin Universit Siggrph Tutoril GRAND PLAN I: Fundmentls of Quternions II: Visuliing Quternion Geometr III: Quternion Frmes IV: Clifford Algers

More information

50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS

50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS 68 CHAPTE MULTIPLE INTEGALS 46. e da, 49. Evlute tn 3 4 da, where,. [Hint: Eploit the fct tht is the disk with center the origin nd rdius is smmetric with respect to both es.] 5. Use smmetr to evlute 3

More information

Lecture 2e Orthogonal Complement (pages )

Lecture 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 information

dx 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.

dx 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 information

DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS

DEFINITION 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 information

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description

More information

MATH 13 FINAL STUDY GUIDE, WINTER 2012

MATH 13 FINAL STUDY GUIDE, WINTER 2012 MATH 13 FINAL TUY GUI, WINTR 2012 This is ment to be quick reference guide for the topics you might wnt to know for the finl. It probbly isn t comprehensive, but should cover most of wht we studied in

More information

Vectors 3-1 VECTORS AND THEIR COMPONENTS. What Is Physics? Vectors and Scalars. Learning Objectives

Vectors 3-1 VECTORS AND THEIR COMPONENTS. What Is Physics? Vectors and Scalars. Learning Objectives C H A P T E R 3 Vectors 3-1 VECTORS AND THEIR COMPONENTS Lerning Ojectives After reding this module, ou should e le to... 3.01 Add vectors drwing them in hed-to-til rrngements, ppling the commuttive nd

More information

Logarithms LOGARITHMS.

Logarithms LOGARITHMS. Logrithms LOGARITHMS www.mthletis.om.u Logrithms LOGARITHMS Logrithms re nother method to lulte nd work with eponents. Answer these questions, efore working through this unit. I used to think: In the

More information

CHAPTER 1 PROGRAM OF MATRICES

CHAPTER 1 PROGRAM OF MATRICES CHPTER PROGRM OF MTRICES -- INTRODUCTION definition of engineering is the science y which the properties of mtter nd sources of energy in nture re mde useful to mn. Thus n engineer will hve to study the

More information

Introduction to Electrical & Electronic Engineering ENGG1203

Introduction to Electrical & Electronic Engineering ENGG1203 Introduction to Electricl & Electronic Engineering ENGG23 2 nd Semester, 27-8 Dr. Hden Kwok-H So Deprtment of Electricl nd Electronic Engineering Astrction DIGITAL LOGIC 2 Digitl Astrction n Astrct ll

More information

Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus

Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = -x + 8x )Use

More information

Chapter 3 Single Random Variables and Probability Distributions (Part 2)

Chapter 3 Single Random Variables and Probability Distributions (Part 2) Chpter 3 Single Rndom Vriles nd Proilit Distriutions (Prt ) Contents Wht is Rndom Vrile? Proilit Distriution Functions Cumultive Distriution Function Proilit Densit Function Common Rndom Vriles nd their

More information

Matrix Eigenvalues and Eigenvectors September 13, 2017

Matrix 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 information