Chapter 14. Matrix Representations of Linear Transformations


 Matilda Rhoda Sanders
 2 years ago
 Views:
Transcription
1 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 do mtrix opertions rther esily. Also, mtrices tend to be good wy to store informtion in computer. When writing computer code for liner trnsformtion bsed on the formul cn t times be very tedious, but using mtrix multipliction is much esier. In this chpter, we tlk bout when we re ble to use mtrices s tool for trnsformtions. Let us begin with theorem tht tells us bout ll trnsformtions tht re defined using mtrix multipliction. Theorem Define T : R n R m by T x = Mx, where M is m n mtrix. Then T is liner trnsformtion. Proof. Let M M m n (R nd define T : R n R m s bove. We will show tht T stisfies the linerity condition given in Eqution 2.. Let x, y R n nd let α R be sclr. Then, using properties of multipliction by mtrix, we get the following. T (αx + y = M(αx + y = M(αx + My = αmx + My = αt x + T y. 253
2 254 CHAPTER 4. MATRIX REPRESENTATIONS Thus, T is liner. Theorem 4.0. shows tht trnsformtion defined using mtrix multipliction is liner trnsformtion. This leds us to sk whether it possible to define ny liner trnsformtion using mtrix multipliction. If so, tht would be extremely helpful. The potentil stumbling block is tht we cnnot just multiply vector, in sy P 2 (R, by mtrix. Wht would tht men? In this chpter, we combine our knowledge bout coordinte spces nd liner trnsformtions to write liner trnsformtions using mtrix multipliction. 4. Mtrix Trnsformtions Suppose we hve two vector spces V nd W. Let V be ndimensionl with bsis V nd W be mdimensionl with bsis W. Suppose we re given liner trnsformtion T : V W. We re interested in figuring out how to trnsform vectors from V to W, possibly tking new pth using mtrix multipliction. Recll tht the trnsformtion T : V R n defined by T (v = [v] is liner (see Theorem Let T 2 be the trnsformtion tht tkes coordinte vectors in R m bck to their corresponding vectors in W. We know tht T 2 is liner trnsformtion (see Exercise 6. We know tht we cn multiply vectors in R n by m n mtrices to get vectors in R m. We wnt to find M M m n so tht we cn define nd so tht T 3 : R n R m by T (x = Mx for ll x R n T 3 ([v] V = [T (v] W. Tht is, we wnt T 3 to trnsform [v] V into [w] W in the sme wy T trnsforms v into w. (See Figure 4.. Recll tht Corollry 2.3. tells us tht to find trnsformtion T 2 M T equivlent to T, we need only consider their ctions on bsis for V. Definition 4... Given mtrix M nd trnsformtion T : R n R m defined by T (x = Mx for every x R n, we sy tht M is the mtrix representtion for the trnsformtion T.
3 4.. MATRIX TRANSFORMATIONS 255 T = T 2 M T T M T 2 V R n R m W Figure 4.: Illustrtion of the equivlence of liner trnsformtion T with the composition of two coordinte trnsformtions T nd T 2 nd one mtrix multiply. It is common to indicte the mtrix representtion M of liner trnsformtion T : V W by M = [T ] W V, where V nd W re the chosen bses for V nd W, respectively. If V nd W re the sme vector spces, with bsis, then we typiclly write M = [T ] to indicte M = [T ]. Suppose V = {v, v 2,..., v n } is bsis for V nd W = {w, w 2,..., w m } is bsis for W. Then the ction of T requires tht M must mp the coordinte vector of v k to the coordinte vector of T (v k. Tht is, [T (v k ] W = M[v k ] V for k =, 2,..., n. Notice tht [v k ] V = e k, the k th stndrd bsis vector of R n. So, M[v k ] V = Me k is equl to the k th column of M. Thus, the k th column of M must equl [T (v k ] W. These ides suggest the following procedure for constructing the mtrix M. Procedure: Let V nd W be vector spces with ordered bses V = {v, v 2,..., v n } nd W = {w, w 2,..., w m }, respectively. Also, let T : V W be liner. Then the mtrix representtion M = [T ] W V is given by M = [T (v ] W [T (v 2 ] W... [T (v n ] W, (4. where [T (v k ] W is the k th column of M.
4 256 CHAPTER 4. MATRIX REPRESENTATIONS This result is verified rigorously s Theorem Exmple 4... Let V = {x 2 +bx+(+b, b R} nd let W = M 2 2. Consider the trnsformtion T : V W defined by ( b T (x 2 + bx + ( + b = + b + 2b We cn show tht T is liner (be sure you know how to do this. So, we cn find mtrix representtion, M, of T. First, we must find bses for V nd W so tht we cn consider the coordinte spces nd determine the size of M.. V = {x 2 + bx + ( + b, b R} = spn { x 2 +, x + }. So bsis for V is V = {x 2 +, x + }. We will use the stndrd bsis for M 2 2. Since V is 2dimensionl spce, the cooresponding coordinte spce is R 2. W, being 4dimensionl spce, cooresponds to the coordinte spce R 4. We will find M tht cn multiply by vector in R 2 to get vector in R 4. This mens tht M M 4 2. We lso wnt M to ct like T. Tht is, we wnt [T (v] W = M[v] V. We need to determine where the bsis elements of V get mpped. ( T (x 2 + = ( 0 T (x + = 2 Writing these outputs s coordinte vectors in R 4 gives [T (x 2 x + ] W = [T (x + ] W = [( [( 0 2 ] ] W = W = 0 2
5 4.. MATRIX TRANSFORMATIONS 257 According to the procedure bove, the coordinte vectors re the columns of M. Tht is, 0 M =. 2 We cn (nd should check tht the trnsformtion tht T : R 2 R 4 defined by T (x = Mx trnsforms the coordinte vectors in the sme wy T trnsforms vectors. Let v = 2x 2 + 4x + 6. We know tht v V becuse it corresponds to the choice = 2, b = 4. Now, ccording to the definition for T, we get ( T (v = T (2x 2 + 4x + (2 + 4 = (4 = ( Next, we check T (x = Mx. Notice tht v = 2(x (x +. So Now, we compute [T (v] W = M[v] V = 0 2 [v] V = ( 2 4 ( 2 4. = Notice tht this is exctly wht we expect becuse [( ] = W = We cn check this more rigorously by using n rbitrry vector in V. Let v = x 2 + bx + + b. Then ( b T (v =. + b + 2b
6 258 CHAPTER 4. MATRIX REPRESENTATIONS The coordinte vectors of these re [v] V = ( b nd [( b + b + 2b ] S = + b b + 2b. Finlly, we compute T ([v] V. T ([v] V = M[v] V = Thus, [T (v] S = T ([v] V. 0 2 ( b = + b b + 2b In Chpter, we wrote the rdiogrphic trnsformtion s mtrix. However, we did not hve brin imge objects vectors in R N nor were the rdiogrphs vectors in R M. We will use the bove informtion to explore, through n exmple, how the mtrix we found ws the mtrix representtion of the rdiogrphic trnsformtion. Let V = I 2 2, the spce of 2 2 objects. Let T be the rdiogrphic trnsformtion with 6 views hving 2 pixels ech. This mens tht the codomin is the set of rdiogrphs with 2 pixels. To figure out the mtrix M representing this rdiogrphic trnsformtion, we first chnge the objects in V to coordinte vectors in R 4 vi the trnsformtion T. So T is defined s T. V x x 2 x 3 x 4 where we hve used the stndrd bsis for I 2 2. After multiplying by the mtrix representtion, we will chnge from coordinte vectors in R 2 bck to rdiogrphs vi T 2 which is defined by: R 4 x x 2 x 3 x 4
7 4.2. CHANGE OF ASIS MATRIX 259 T 2 R2 W b b b 2 b 2 b 3 b 4 b 3 b 5 b 6 b 4 b 7 b 8 b 5 b b 6 9 b 0 b b 7 b 2 b 8 b 9 b 0 b b 2 where, gin, we hve used the stndrd bsis for the rdiogrph spce. Our rdiogrphic trnsformtion is then represented by the mtrix M (which we clled T in Chpter. M will be 2 4 mtrix determined by the rdiogrphic set up nd the chosen bses. 4.2 Chnge of sis Mtrix Situtions rise in mny pplictions so tht it will be useful to chnge our coordinte representtions from the use of one bsis to nother. Consider brin imges represented in coordinte spce R N reltive to bsis 0 = {u, u 2,..., u N }. Perhps this bsis is the stndrd bsis for brin imges. Now suppose tht we hve nother bsis = {v, v 2,..., v N } for R N for which v 43 is brin imge strongly correlted with disese X. If brin imge x is represented s coordinte vector [x] 0, it my be simpler to perform necessry clcultions, but it my be more involved to dignose if disese X is present. However, the 43 st coordinte of [x] tells us directly the reltive contribution of v 43 to the brin imge. Ides such s this inspire the benefits of being ble to quickly chnge our coordinte system. Let T : R n R n be the chnge of coordintes trnsformtion from ordered bsis = {b, b 2,..., b n } to ordered bsis = { b, b 2,..., b n }. We represent the trnsformtion s mtrix M = [T ]. The key ide is tht chnge of coordintes does not chnge the vectors themselves, only their representtion. Thus, T must be the identity trnsformtion. We hve
8 260 CHAPTER 4. MATRIX REPRESENTATIONS [T ] = [I] = [I(b ] [I(b 2 ]... [I(b n ] = [b ] [b 2 ]... [b n ]. Definition Let nd be two ordered bses for vector spce V. The mtrix representtion [I] for the trnsformtion chnging coordinte spces is clled chnge of bsis mtrix. Note: The k th column of the chnge of bsis mtrix is the coordinte representtion of the k th bsis vector of reltive to the bsis. Exmple Consider n ordered bsis for R 3 given by = v =, v 2 = 0, v 3 =. 0 Find the chnge of bsis mtrix M from the stndrd bsis 0 for R 3 to. We hve M = [e ] [e 2 ] [e 2 ]. We cn find [e ] by finding sclrs, b, c so tht e = v + bv 2 + cv 3.
9 4.3. PROPERTIES OF MATRIX REPRESENTATIONS 26 Solving the corresponding system of equtions, we get =, b =, c =. 0 So, [e ] =. Similrly, we find tht [e 2 ] = nd [e 3 ] = 0. Thus M = 0 0 Now, given ny coordinte vector (with respect to the stndrd bsis v = b, we cn write s coordinte vector in terms of by c [v] = M[v] 0 = 0 0 b c. = c b + b + c 4.3 Properties of Mtrix Representtions Consider the following theorems which help us mke sense of mtrix representtions of multiple liner trnsformtions. The proof of ech follow from the properties of mtrix multipliction nd the definition of the mtrix representtion. The first theorem shows tht mtrix representtions of liner trnsformtions stisfy linerity properties themselves.. Theorem Let T, U : V W be liner, α sclr, nd V nd W be finite dimensionl vector spces with ordered bses nd, respectively. Then ( [T + U] = [T ] + [U] (b [αt ] = [T ]. Proof. See Exercise 7.
10 262 CHAPTER 4. MATRIX REPRESENTATIONS The second theorem shows tht mtrix representtions of compositions of liner trnsformtions behve s mtrix multipliction opertions (in pproprite bses representtions. Theorem Let T : V W nd U : W X be liner, u V, V, W, nd X be finite dimensionl vector spces with ordered bses,, nd, respectively. Then [U T ] = [U] [T ]. Proof. See Exercise 8. The third theorem verifies our mtrix representtion construction of Section 4. Theorem Let T : V W be liner, v V, nd V, W be finite dimensionl vector spces with ordered bses nd, respectively. Then [T (v] = [T ] [v]. Proof. See Exercise Exercises Find the mtrix representtion, M = [T ] W V with the given bses V nd W. of ech trnsformtion below. T : V O V R, where V O is the spce of objects with 4 voxels nd V R is the spce of rdiogrphs with 4 pixels nd T x x 3 x 2 x 4 = 2 Where VO nd VR re the stndrd bses. x + x 2 x 3 + x 4 3 x + x x. 4 3 x + x x 4
11 4.4. EXERCISES T : V O R 4, where V O is the spce of objects with 4 voxels nd V R is the spce of rdiogrphs with 4 pixels nd T x x 3 x 2 x 4 = x x 3 x 2 x 4 where is the stndrd bsis for V O. Where VO nd R 4 re the stndrd bses. ( b 3. T : M 2 2 R 4 defined by T = b c d + b {( ( ( c d ( } 0 Where M2 2 =,,, nd R 4 is the stndrd bsis. 4. T : P 2 (R P 2 (R defined by T (x 2 + bx + c = cx 2 + x + b, where P2 = {x 2 +, x, }. 5. T : P 2 (R P 2 (R defined by T (x 2 + bx + c = ( + bx 2 b + c, where P2 is the stndrd bsis. 6. T : H 4 (R R 4 defined by T (v = [v] Y, where H4 (R = Y the bsis given in Exmple nd R 4 is the stndrd bsis. 7. T : D(Z 2 D(Z 2 defined by T (x = x + x, where D(Z is the bsis given in Exmple The trnsformtion of Exercise 8 in Chpter 2 on het sttes, where H4 (R = Y the bsis given in Exmple For Exercises 93, choose common vector spces V nd W nd liner trnsformtion T : V W for which M is the mtrix representtion of T when using the stndrd bses for V nd W.Check your nswers with t lest two exmples. ( M = 2,
12 264 CHAPTER 4. MATRIX REPRESENTATIONS ( 0. M = M = ( M = M = For Exercises 46, find the mtrix representtion of the trnsformtion T : V W. 4. V = P 3 with the bsis = {x 3, x 2 +, x +, } nd W = P 2 with the stndrd bsis. T (x 3 + bx 2 + cx + d = 3x 2 + 2bx + c. 5. V = R 3 with the stndrd bsis nd W = R 3 with bsis 0 x = 0,, 0 nd T y = 0 z x + y y z 0 6. V = M 3 2 with the stndrd bsis nd W = M 2 2 with the bsis {( ( ( ( } =,,, ( T 2 22 = ,2 Additionl Exercises. 7. Prove Theorem Prove Theorem Prove Theorem
Chapter 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 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 informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 25pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More 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 informationChapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY
Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
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 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 information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationBases for Vector Spaces
Bses for Vector Spces 22625 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly welldefined, is too restrictive for mny purposes; there re functions which
More informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationRecitation 3: Applications of the Derivative. 1 HigherOrder Derivatives and their Applications
Mth 1c TA: Pdric Brtlett Recittion 3: Applictions of the Derivtive Week 3 Cltech 013 1 HigherOrder Derivtives nd their Applictions Another thing we could wnt to do with the derivtive, motivted by wht
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationLecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationContinuous Quantum Systems
Chpter 8 Continuous Quntum Systems 8.1 The wvefunction So fr, we hve been tlking bout finite dimensionl Hilbert spces: if our system hs k qubits, then our Hilbert spce hs n dimensions, nd is equivlent
More informationPHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS
PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS To strt on tensor clculus, we need to define differentition on mnifold.a good question to sk is if the prtil derivtive of tensor tensor on mnifold?
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 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 informationModule 6: LINEAR TRANSFORMATIONS
Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for
More informationMatrix Solution to Linear Equations and Markov Chains
Trding Systems nd Methods, Fifth Edition By Perry J. Kufmn Copyright 2005, 2013 by Perry J. Kufmn APPENDIX 2 Mtrix Solution to Liner Equtions nd Mrkov Chins DIRECT SOLUTION AND CONVERGENCE METHOD Before
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 informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationPractice final exam solutions
University of Pennsylvni Deprtment of Mthemtics Mth 26 Honors Clculus II Spring Semester 29 Prof. Grssi, T.A. Asher Auel Prctice finl exm solutions 1. Let F : 2 2 be defined by F (x, y (x + y, x y. If
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
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 informationLecture 3: Equivalence Relations
Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts
More information308K. 1 Section 3.2. Zelaya Eufemia. 1. Example 1: Multiplication of Matrices: X Y Z R S R S X Y Z. By associativity we have to choices:
8K Zely Eufemi Section 2 Exmple : Multipliction of Mtrices: X Y Z T c e d f 2 R S X Y Z 2 c e d f 2 R S 2 By ssocitivity we hve to choices: OR: X Y Z R S cr ds er fs X cy ez X dy fz 2 R S 2 Suggestion
More information1 1D heat and wave equations on a finite interval
1 1D het nd wve equtions on finite intervl In this section we consider generl method of seprtion of vribles nd its pplictions to solving het eqution nd wve eqution on finite intervl ( 1, 2. Since by trnsltion
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd GussJordn elimintion to solve systems of liner
More informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationNumerical Linear Algebra Assignment 008
Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com
More informationdx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.
Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd
More 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 informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
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 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 informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationLinearly Similar Polynomials
Linerly Similr Polynomils rthur Holshouser 3600 Bullrd St. Chrlotte, NC, US Hrold Reiter Deprtment of Mthemticl Sciences University of North Crolin Chrlotte, Chrlotte, NC 28223, US hbreiter@uncc.edu stndrd
More informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
More informationLECTURE 3. Orthogonal Functions. n X. It should be noted, however, that the vectors f i need not be orthogonal nor need they have unit length for
ECTURE 3 Orthogonl Functions 1. Orthogonl Bses The pproprite setting for our iscussion of orthogonl functions is tht of liner lgebr. So let me recll some relevnt fcts bout nite imensionl vector spces.
More informationp(t) dt + i 1 re it ireit dt =
Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)
More informationMath 33A Discussion Example Austin Christian October 23, Example 1. Consider tiling the plane by equilateral triangles, as below.
Mth 33A Discussion Exmple Austin Christin October 3 6 Exmple Consider tiling the plne by equilterl tringles s below Let v nd w be the ornge nd green vectors in this figure respectively nd let {v w} be
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
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 informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
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 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 informationChapter 3 Polynomials
Dr M DRAIEF As described in the introduction of Chpter 1, pplictions of solving liner equtions rise in number of different settings In prticulr, we will in this chpter focus on the problem of modelling
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
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 informationMATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35
MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationNatural examples of rings are the ring of integers, a ring of polynomials in one variable, the ring
More generlly, we define ring to be nonempty set R hving two binry opertions (we ll think of these s ddition nd multipliction) which is n Abelin group under + (we ll denote the dditive identity by 0),
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationREPRESENTATION THEORY OF PSL 2 (q)
REPRESENTATION THEORY OF PSL (q) YAQIAO LI Following re notes from book [1]. The im is to show the qusirndomness of PSL (q), i.e., the group hs no low dimensionl representtion. 1. Representtion Theory
More informationInfinite Geometric Series
Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to
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 informationAnalytical Methods Exam: Preparatory Exercises
Anlyticl Methods Exm: Preprtory Exercises Question. Wht does it men tht (X, F, µ) is mesure spce? Show tht µ is monotone, tht is: if E F re mesurble sets then µ(e) µ(f). Question. Discuss if ech of the
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
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 information1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a nonconstant can be solved with the same idea as above.
1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationGeneralized Fano and nonfano networks
Generlized Fno nd nonfno networks Nildri Ds nd Brijesh Kumr Ri Deprtment of Electronics nd Electricl Engineering Indin Institute of Technology Guwhti, Guwhti, Assm, Indi Emil: {d.nildri, bkri}@iitg.ernet.in
More information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
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 informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
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 informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be relvlues nd smooth The pproximtion of n integrl by numericl
More informationarxiv: v1 [math.ra] 1 Nov 2014
CLASSIFICATION OF COMPLEX CYCLIC LEIBNIZ ALGEBRAS DANIEL SCOFIELD AND S MCKAY SULLIVAN rxiv:14110170v1 [mthra] 1 Nov 2014 Abstrct Since Leibniz lgebrs were introduced by Lody s generliztion of Lie lgebrs,
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors PreChpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
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 informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationNumerical Methods I Orthogonal Polynomials
Numericl Methods I Orthogonl Polynomils Aleksndr Donev Cournt Institute, NYU 1 donev@cournt.nyu.edu 1 MATHGA 2011.003 / CSCIGA 2945.003, Fll 2014 Nov 6th, 2014 A. Donev (Cournt Institute) Lecture IX
More informationWe will see what is meant by standard form very shortly
THEOREM: For fesible liner progrm in its stndrd form, the optimum vlue of the objective over its nonempty fesible region is () either unbounded or (b) is chievble t lest t one extreme point of the fesible
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO  Ares Under Functions............................................ 3.2 VIDEO  Applictions
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationLecture 2: Fields, Formally
Mth 08 Lecture 2: Fields, Formlly Professor: Pdric Brtlett Week UCSB 203 In our first lecture, we studied R, the rel numbers. In prticulr, we exmined how the rel numbers intercted with the opertions of
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationHere we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.
Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous
More informationpadic Egyptian Fractions
padic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Setup 3 4 pgreedy Algorithm 5 5 pegyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationExponentials  Grade 10 [CAPS] *
OpenStxCNX module: m859 Exponentils  Grde 0 [CAPS] * Free High School Science Texts Project Bsed on Exponentils by Rory Adms Free High School Science Texts Project Mrk Horner Hether Willims This work
More information4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve
Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions
More informationVyacheslav Telnin. Search for New Numbers.
Vycheslv Telnin Serch for New Numbers. 1 CHAPTER I 2 I.1 Introduction. In 1984, in the first issue for tht yer of the Science nd Life mgzine, I red the rticle "NonStndrd Anlysis" by V. Uspensky, in which
More informationLecture Note 9: Orthogonal Reduction
MATH : Computtionl Methods of Liner Algebr 1 The Row Echelon Form Lecture Note 9: Orthogonl Reduction Our trget is to solve the norml eution: Xinyi Zeng Deprtment of Mthemticl Sciences, UTEP A t Ax = A
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationMath Lecture 23
Mth 8  Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationLinearity, linear operators, and self adjoint eigenvalue problems
Linerity, liner opertors, nd self djoint eigenvlue problems 1 Elements of liner lgebr The study of liner prtil differentil equtions utilizes, unsurprisingly, mny concepts from liner lgebr nd liner ordinry
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
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 information