Math 270A: Numerical Linear Algebra

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Math 270A: Numerical Linear Algebra"

Transcription

1 Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner Systems 1. Recll tht the trnspose A T of n n n mtrix A is defined s A T ij = A ji. Another chrcteriztion of the trnspose mtrix A T is tht it is the unique djoint opertor stisfying where (, is the usul Eucliden inner-product, (Au, v = (u, A T v, u, v R n, (u, v = n u i v i. Show tht these two chrcteriztions re equivlent.. Given n SPD (symmetric positive definite n n mtrix A, show tht it defines new inner-product (u, v A = (Au, v = i=1 n (Au i v i, u, v R n, i=1 clled the A-inner-product. I.e., show tht (u, v A is true inner-product, in tht it stisfies the three inner-product properties. 3. The A-djoint of mtrix M, denoted M, is defined s the djoint in the A-inner-product; i.e., the unique mtrix stisfying: (AMu, v = (Au, M v, u, v R n. Show tht tht n equivlent definition of M is M = A 1 M T A. 4. Now tht we hve new inner-product (u, v A, give definition of the norm it induces, which we will denote u A. Give n rgument tht it stisfies the three norm properties. 5. Use the vector norm u A to define the opertor (mtrix norm M A which it induces for ny n n mtrix M. Give n rgument tht this definition gives us ccess to the following inequlity: Mu A M A u A, u R n. 6. Use the A-norm to define the A-condition number of ny squre n n mtrix M, which we denote s κ A (M. 7. Prove the following: If M R n n is A-self-djoint for n SPD A R n n, then ρ(m = M A, κ A (M = λ mx(m λ min (M.

2 8. Let A, B R n n be SPD, nd let α R with α 0. Our gol is to solve the liner system Au = f for the unknown u R n, given f R n, nd given B A 1. Derive the following identity nd rgue tht it hs unique solution u R n tht grees with the solution to Au = f: u = (I αbau + αbf. 9. Derive the prmeterized form of the Bsic Liner Method for the problem Au = f: u n+1 = (I αbau n + αbf. 10. Use the Bnch Fixed Point Theorem to prove tht the BLM converges if α is restricted to subset of R, nd give tht subset. 11. Prove tht when the optiml vlue is chosen for α, the following error bound holds for the Bsic Liner Method: e n+1 ( n+1 1 e 0, 1 + κ A (BA where e n+1 = u u n+1, nd where κ A (BA denotes the A-condition number of BA. 1. Drive more generl error bound for the Bsic Liner Method, similr to the result bove, but one tht holds for ny vlue of α. (Hint: The norm of the error propgtor E = I BA will pper more nturlly thn the condition number of BA. 13. Assume we hve the optiml vlue for α. Assuming tht we would like to chieve the following ccurcy in our itertion fter some number of steps n: e n+1 A e 0 A < ɛ, use the pproximtion: ln ( ( ( 1/ = ln = ( 1/ [ ( ( ( ] <, to show tht we cn chieve this error tolernce with the Bsic Liner Method if n stisfies: n = O (κ A (BA ln ɛ. 14. Derive similr itertion bound result tht holds for ny α. 15. Mny types of mtrices hve O(1 non-zeros per row (the finite difference nd finite element discretiztions of ordinry nd prtil differentil equtions we looked t briefly in 70A, nd will look t more in 70C, lwys generte such mtrices. Assume tht the cost to store nd pply B is O(n. Prove tht the cost of one itertion of BLM will be O(n, where n is the dimension of the problem, i.e., A is n n n mtrix. 16. Wht is the overll complexity (in terms of n nd κ A (BA to solve the problem with the BLM to given tolernce ɛ? (Assume you hve the optiml α. If κ A (BA cn be bounded by constnt, independent of the problem size n, wht is the complexity? Is this then n optiml method?

3 Exercise 3.. (Derivtion of the Conjugte Grdient Method. In this problem, we will derive the conjugte grdient (CG method, n extremely importnt itertive method for solving mtrix equtions Au = f when A is symmetric (A = A T nd positive-definite (u T Au > 0, u 0. (We will refer to mtrices hving both properties s SPD. This problem brings together severl different topics we hve discussed in lecture in 70A, including itertive methods for mtrix equtions, mnipultions with inner-products nd norms in vector spce, together with some bsic ides from pproximtion nd orthogonl polynomils tht will be exmined more completely in 70B. 1. The Cyley-Hmilton Theorem sttes tht squre n n mtrix M stisfies its own chrcteristic eqution: P n (M = 0. Using this result, prove tht if M is lso nonsingulr, then the mtrix M 1 cn be written s mtrix polynomil of degree n 1 in M, or M 1 = Q n 1 (M.. Consider now the mtrix eqution: Au = f, where A is n n n SPD mtrix, nd u nd f re n-vectors. It is common to precondition such n eqution before ttempting numericl solution, by multiplying by n pproximte inverse opertor B A 1 nd then solving the preconditioned system: BAu = Bf. If A nd B re both SPD, under wht conditions is BA lso SPD? Show tht if A nd B re both SPD, then BA is A-SPD (symmetric nd positive in the A-inner-product. 3. Given n initil guess u 0 to the solution of BAu = Bf, we cn form the initil residuls: r 0 = f Au 0, s 0 = Br 0 = Bf BAu 0. Do simple mnipultion to show tht the solution u cn be written s: u = u 0 + Q n 1 (BAs 0, where Q( is the mtrix polynomil representing (BA 1. In other words, you hve estblished tht the solution u lies in trnslted Krylov spce: u u 0 + V n 1 (BA, s 0, where V n 1 (BA, s 0 = spn{s 0, BAs 0, (BA s 0,..., (BA n 1 s 0 }. Note tht we cn view the Krylov spces s sequence of expnding subspces V 0 (BA, s 0 V 1 (BA, s 0 V n 1 (BA, s We will now try to construct n itertive method (the CG method for finding u. The lgorithm determines the best pproximtion u k to u in subspce V k (BA, s 0 t ech step k of the lgorithm, by forming u k+1 = u k + α k p k, where p k is such tht p k V k (BA, s 0 t step k, but p k V j (BA, s 0 for j < k. In ddition, we wnt to enforce minimiztion of the error in the A-norm, e k+1 A = u u k+1 A, t step k of the lgorithm. The next itertion expnds the subspce to V k+1 (BA, s 0, finds the best pproximtion in the expnded spce, nd so on, until the exct solution in V n 1 (BA, s 0 is reched. To relize this lgorithm, let s consider how to construct the required vectors p k in n efficient wy. Let p 0 = s 0, nd consider the construction of n A-orthogonl bsis for V n 1 (BA, s 0 using the stndrd Grm-Schmidt procedure: p k+1 = BAp k k i=0 (BAp k, p i A (p i, p i A p i, k = 0,..., n. 3

4 At ech step of the procedure, we will hve generted n A-orthogonl (orthogonl in the A-inner-product bsis {p 0,..., p k } for V k (BA, s 0. Now, note tht by construction, (p k, v A = 0, v V j (BA, s 0, j < k. Using this fct, nd the fct you estblished previously tht BA is A-self-djoint, show tht the Grm-Schmidt procedure hs only three non-zero terms in the sum, nmely p k+1 = BAp k (BApk, p k A (p k, p k A p k (BApk, p k 1 A (p k 1, p k 1 A p k 1, k = 0,..., n 1. Thus, there exists n efficient three-term recursion for generting the A-orthogonl bsis for the solution spce. Note tht this three-term recursion is possible due to the fct tht we re working with orthogonl (mtrix polynomils. 5. We cn nerly write down the CG method now, by ttempting to expnd the solution in terms of our cheply generted A-orthogonl bsis. However, we need to determine how fr to move in ech conjugte direction p k t step k fter we generte p k from the recursion. As remrked erlier, we would like to enforce minimiztion of the quntity e k+1 A = u u k+1 A t step k of the itertive lgorithm. Show tht this is equivlent to enforcing (e k+1, p k A = 0. (I.e., show tht minimizing the error in finding n pproximtion from subspce is mthemticlly equivlent to enforcing tht the pproximtion error be orthogonl to tht subspce. 6. Let s ssume we hve somehow enforced (e k, p i A = 0, i < k, t the previous step of the lgorithm. We hve t our disposl p k V k (BA, s 0, nd let s tke our new pproximtion t step k + 1 s: u k+1 = u k + α k p k, for some step-length α k R in the direction p k. Thus, the error in the new pproximtion is simply: e k+1 = e k + α k p k. Show tht in order to enforce (e k+1, p k A = 0, we must choose α k to be α k = (rk, p k (p k, p k A. 7. The Conjugte Grdient Algorithm you hve derive bove is: (The Conjugte Grdient Algorithm Let u 0 H be given. r 0 = f Au 0, s 0 = Br 0, p 0 = s 0. Do k = 0, 1,... until convergence: α k = (r k, p k /(p k, p k A u k+1 = u k + α k p k r k+1 = r k α k Ap k s k+1 = Br k+1 β k+1 = (BAp k, p k A /(p k, p k A γ k+1 = (BAp k, p k 1 A /(p k 1, p k 1 A p k+1 = BAp k + β k+1 p k + γ k+1 p k 1 End do. Show tht equivlent expressions for some of the prmeters in CG re: ( α k = (r k, s k /(p k, p k A (b δ k+1 = (r k+1, s k+1 /(r k, s k (c p k+1 = s k+1 + δ k+1 p k Remrk: The CG lgorithm tht ppers in textbooks is usully formulted to employ these equivlent expressions due to the reduction in computtionl work of ech itertion. 4

5 Exercise 3.3. (Properties of the Conjugte Grdient Method. In this problem, we will estblish some simple properties of the CG method derived in Problem 1. (Although this nlysis is stndrd, you will hve difficulty finding ll of the pieces in one text. 1. Show tht the error in the CG lgorithm propgtes s: e k+1 = [I BAp k (BA]e 0, where p k P k, the spce of polynomils of degree k.. By construction, we know tht the polynomil bove is such tht e k+1 A = min p k P k [I BAp k (BA]e 0 A. Now, since BA is A-SPD, we know tht it hs rel positive eigenvlues λ j σ(ba, nd further, tht the corresponding eigenvectors v j of BA re orthonorml. Using the expnsion of the initil error estblish the inequlity: e 0 = Σ n j=1α j v j, ( [ ] e k+1 A min mx 1 λ jp k (λ j e 0 A. p k P k λ j σ(ba The polynomil which minimizes the mximum norm bove is sid to solve mini-mx problem. 3. It is well-known in pproximtion theory tht the Chebyshev polynomils T k (x = cos(k rccos x solve mini-mx problems of the bove type, in the sense tht they devite lest from zero (in the mx-norm sense in the intervl [ 1, 1], which cn be shown is due to their unique equi-oscilltion property. (These fcts cn be found in ny introductory numericl nlysis text, such s the one used in Mth 170C. If we extend the Chebyshev polynomils outside the intervl [ 1, 1] in the nturl wy (outlined in clss, it be shown tht shifted nd scled forms of the Chebyshev polynomils solve the bove mini-mx problem. In prticulr, the solution is simply ( λ T mx+λ min λ k+1 λ mx λ min 1 λp k (λ = p k+1 (λ = (. λ T mx+λ min k+1 λ mx λ min Use n obvious property of the polynomils T k+1 (x to conclude e k+1 A [ ( ] 1 λmx + λ min T k+1 e 0 A. λ mx λ min 4. Use one of the Chebyshev polynomil results given in the lst problem below to refine this inequlity to: λmx(ba e k+1 λ A 1 min(ba λmx(ba λ + 1 min(ba k+1 e 0 A. 5. Using the fct tht BA is A-self-djoint nd positive definite, show tht the bove error reduction inequlity cn be written more simply s e k+1 A ( k+1 ( κa (BA 1 e κa (BA A = κ A (BA k+1 e 0 A. 6. Assume tht we would like to chieve the following ccurcy in our itertion fter some number of steps n: e n+1 A e 0 A < ɛ. 5

6 Using the pproximtion: ln ( ( ( 1/ = ln = ( 1/ [ ( show tht we cn chieve this error tolernce if n stisfies ( n = O κ 1/ ln A (BA ɛ. ( ( ] <, 7. Mny types of mtrices hve O(1 non-zeros per row (the finite difference nd finite element discretiztions of ordinry nd prtil differentil equtions we look t next qurter lwys generte such mtrices. Therefore, the cost of one itertion of CG will be O(n, where n is the dimension of the problem, i.e., A is n n n mtrix. Wht is the overll complexity (in terms of n nd κ A (BA to solve the problem to given tolernce ɛ? If κ A (BA cn be bounded by constnt, independent of the problem size n, wht is the complexity? Is this then n optiml method? 8. As noted erlier, mny types of mtrices hve O(1 non-zeros per row (the finite difference nd finite element discretiztions of ordinry nd prtil differentil equtions we looked t briefly in 70A, nd will look t more in 70C, lwys generte such mtrices. Assume tht the cost to store nd pply B is O(n. Prove tht the cost of one itertion of CG will be O(n, where n is the dimension of the problem, i.e., A is n n n mtrix. 9. Wht is the overll complexity (in terms of n nd κ A (BA to solve the problem with CG to given tolernce ɛ? If κ A (BA cn be bounded by constnt, independent of the problem size n, wht is the complexity? Is this then n optiml method? Exercise 3.4. (Properties of the Chebyshev Polynomils. The Chebyshev polynomils re defined s: t n (x = cos(n cos 1 x, n = 0, 1,,.... Tking t 0 (x = 1, t 1 (x = x, it cn be show tht the Chebyshev polynomils re n orthogonl fmily tht cn be generted by the stndrd recursion (which holds for ny orthogonl polynomil fmily: t n+1 (x = t 1 (xt n (x t n 1 (x, n = 1,, 3,.... Prove the following extremely useful reltionships: t k (x = 1 [ ( x + k ( x 1 + x ] k x 1, x, (3.1 ( α + 1 t k > 1 ( k α + 1, α 1 α 1 α > 1. (3. (These two results re fundmentl in the convergence nlysis of the conjugte grdient method in the erlier prts of the homework. Hint: For the first result, use the fce tht cos kθ = (e ikθ + e ikθ /. The second result will follow from the first fter lot of lgebr. 6

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1

MATH34032: 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 information

Best Approximation. Chapter The General Case

Best 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

Discrete Least-squares Approximations

Discrete Least-squares Approximations Discrete Lest-squres Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve

More information

Lecture 19: Continuous Least Squares Approximation

Lecture 19: Continuous Least Squares Approximation Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for

More information

Best Approximation in the 2-norm

Best Approximation in the 2-norm Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion

More information

Orthogonal Polynomials and Least-Squares Approximations to Functions

Orthogonal Polynomials and Least-Squares Approximations to Functions Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny

More information

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS Throughout, we let [, b] be bounded intervl in R. C 2 ([, b]) denotes the spce of functions with derivtives of second order continuous up to the endpoints. Cc 2

More information

New Expansion and Infinite Series

New Expansion and Infinite Series Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

More information

MATH 174A: PROBLEM SET 5. Suggested Solution

MATH 174A: PROBLEM SET 5. Suggested Solution MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion

More information

Lecture 14: Quadrature

Lecture 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 rel-vlues nd smooth The pproximtion of n integrl by numericl

More information

Ordinary differential equations

Ordinary differential equations Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview Introduction-Modelling Bsic concepts to understnd n ODE. Description nd properties

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

Math 360: A primitive integral and elementary functions

Math 360: A primitive integral and elementary functions Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

8 Laplace s Method and Local Limit Theorems

8 Laplace s Method and Local Limit Theorems 8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved

More information

STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH

STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH STURM-LIOUVILLE THEORY, VARIATIONAL APPROACH XIAO-BIAO LIN. Qudrtic functionl nd the Euler-Jcobi Eqution The purpose of this note is to study the Sturm-Liouville problem. We use the vritionl problem s

More information

Lecture 1. Functional series. Pointwise and uniform convergence.

Lecture 1. Functional series. Pointwise and uniform convergence. 1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

More information

2 Sturm Liouville Theory

2 Sturm Liouville Theory 2 Sturm Liouville Theory So fr, we ve exmined the Fourier decomposition of functions defined on some intervl (often scled to be from π to π). We viewed this expnsion s n infinite dimensionl nlogue of expnding

More information

Variational Techniques for Sturm-Liouville Eigenvalue Problems

Variational Techniques for Sturm-Liouville Eigenvalue Problems Vritionl Techniques for Sturm-Liouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment

More information

Lecture 3. Limits of Functions and Continuity

Lecture 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 information

20 MATHEMATICS POLYNOMIALS

20 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 information

ECON 331 Lecture Notes: Ch 4 and Ch 5

ECON 331 Lecture Notes: Ch 4 and Ch 5 Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve

More information

Continuous Random Variables

Continuous 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 rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

More information

Inner-product spaces

Inner-product spaces Inner-product spces Definition: Let V be rel or complex liner spce over F (here R or C). An inner product is n opertion between two elements of V which results in sclr. It is denoted by u, v nd stisfies:

More information

Numerical integration

Numerical 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 information

CS667 Lecture 6: Monte Carlo Integration 02/10/05

CS667 Lecture 6: Monte Carlo Integration 02/10/05 CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of

More information

MA Handout 2: Notation and Background Concepts from Analysis

MA Handout 2: Notation and Background Concepts from Analysis MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,

More information

Linear Systems with Constant Coefficients

Linear Systems with Constant Coefficients Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

More information

Chapter 2. Determinants

Chapter 2. Determinants Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if d-bc0. The expression d-bc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is

More information

MATH 423 Linear Algebra II Lecture 28: Inner product spaces.

MATH 423 Linear Algebra II Lecture 28: Inner product spaces. MATH 423 Liner Algebr II Lecture 28: Inner product spces. Norm The notion of norm generlizes the notion of length of vector in R 3. Definition. Let V be vector spce over F, where F = R or C. A function

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

a a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.

a a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants. Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting

More information

Numerical Integration

Numerical Integration Chpter 1 Numericl Integrtion Numericl differentition methods compute pproximtions to the derivtive of function from known vlues of the function. Numericl integrtion uses the sme informtion to compute numericl

More information

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0. STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

More information

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,

More information

Lecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature

Lecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics - A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the

More information

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by. NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

More information

NWI: Mathematics. Various books in the library with the title Linear Algebra I, or Analysis I. (And also Linear Algebra II, or Analysis II.

NWI: Mathematics. Various books in the library with the title Linear Algebra I, or Analysis I. (And also Linear Algebra II, or Analysis II. NWI: Mthemtics Literture These lecture notes! Vrious books in the librry with the title Liner Algebr I, or Anlysis I (And lso Liner Algebr II, or Anlysis II) The lecture notes of some of the people who

More information

Sturm-Liouville Theory

Sturm-Liouville Theory LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory

More information

M597K: Solution to Homework Assignment 7

M597K: Solution to Homework Assignment 7 M597K: Solution to Homework Assignment 7 The following problems re on the specified pges of the text book by Keener (2nd Edition, i.e., revised nd updted version) Problems 3 nd 4 of Section 2.1 on p.94;

More information

Quantum Physics II (8.05) Fall 2013 Assignment 2

Quantum Physics II (8.05) Fall 2013 Assignment 2 Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More 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

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

Integrals along Curves.

Integrals along Curves. Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the

More information

Lecture Notes: Orthogonal Polynomials, Gaussian Quadrature, and Integral Equations

Lecture Notes: Orthogonal Polynomials, Gaussian Quadrature, and Integral Equations 18330 Lecture Notes: Orthogonl Polynomils, Gussin Qudrture, nd Integrl Equtions Homer Reid My 1, 2014 In the previous set of notes we rrived t the definition of Chebyshev polynomils T n (x) vi the following

More information

Semigroup of generalized inverses of matrices

Semigroup of generalized inverses of matrices Semigroup of generlized inverses of mtrices Hnif Zekroui nd Sid Guedjib Abstrct. The pper is divided into two principl prts. In the first one, we give the set of generlized inverses of mtrix A structure

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

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

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

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230 Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

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

LECTURE 19. Numerical Integration. Z b. is generally thought of as representing the area under the graph of fèxè between the points x = a and

LECTURE 19. Numerical Integration. Z b. is generally thought of as representing the area under the graph of fèxè between the points x = a and LECTURE 9 Numericl Integrtion Recll from Clculus I tht denite integrl is generlly thought of s representing the re under the grph of fèxè between the points x = nd x = b, even though this is ctully only

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

Mapping the delta function and other Radon measures

Mapping the delta function and other Radon measures Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support

More information

Math 554 Integration

Math 554 Integration Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we

More information

MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL

MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL DR. RITU AGARWAL MALVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR, INDIA-302017 Tble of Contents Contents Tble of Contents 1 1. Introduction 1 2. Prtition

More information

3.4 Numerical integration

3.4 Numerical integration 3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,

More information

Math 211A Homework. Edward Burkard. = tan (2x + z)

Math 211A Homework. Edward Burkard. = tan (2x + z) Mth A Homework Ewr Burkr Eercises 5-C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z =

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

Numerical quadrature based on interpolating functions: A MATLAB implementation

Numerical quadrature based on interpolating functions: A MATLAB implementation SEMINAR REPORT Numericl qudrture bsed on interpolting functions: A MATLAB implementtion by Venkt Ayylsomyjul A seminr report submitted in prtil fulfillment for the degree of Mster of Science (M.Sc) in

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

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a). The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

More information

MATH 573 FINAL EXAM. May 30, 2007

MATH 573 FINAL EXAM. May 30, 2007 MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.

More information

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C. A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c

More information

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE

UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence

More information

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

More information

Introduction to Numerical Analysis

Introduction to Numerical Analysis Introduction to Numericl Anlysis Doron Levy Deprtment of Mthemtics nd Center for Scientific Computtion nd Mthemticl Modeling (CSCAMM) University of Mrylnd June 14, 2012 D. Levy CONTENTS Contents 1 Introduction

More information

If deg(num) deg(denom), then we should use long-division of polynomials to rewrite: p(x) = s(x) + r(x) q(x), q(x)

If deg(num) deg(denom), then we should use long-division of polynomials to rewrite: p(x) = s(x) + r(x) q(x), q(x) Mth 50 The method of prtil frction decomposition (PFD is used to integrte some rtionl functions of the form p(x, where p/q is in lowest terms nd deg(num < deg(denom. q(x If deg(num deg(denom, then we should

More information

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014 SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.

More information

International Jour. of Diff. Eq. and Appl., 3, N1, (2001),

International Jour. of Diff. Eq. and Appl., 3, N1, (2001), Interntionl Jour. of Diff. Eq. nd Appl., 3, N1, (2001), 31-37. 1 New proof of Weyl s theorem A.G. Rmm Mthemtics Deprtment, Knss Stte University, Mnhttn, KS 66506-2602, USA rmm@mth.ksu.edu http://www.mth.ksu.edu/

More information

A Convergence Theorem for the Improper Riemann Integral of Banach Space-valued Functions

A Convergence Theorem for the Improper Riemann Integral of Banach Space-valued Functions Interntionl Journl of Mthemticl Anlysis Vol. 8, 2014, no. 50, 2451-2460 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2014.49294 A Convergence Theorem for the Improper Riemnn Integrl of Bnch

More information

4.1. Probability Density Functions

4.1. Probability Density Functions STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of

More information

Math 100 Review Sheet

Math 100 Review Sheet Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s

More information

Math 3B Final Review

Math 3B Final Review Mth 3B Finl Review Written by Victori Kl vtkl@mth.ucsb.edu SH 6432u Office Hours: R 9:45-10:45m SH 1607 Mth Lb Hours: TR 1-2pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems

More information

ON BERNOULLI BOUNDARY VALUE PROBLEM

ON BERNOULLI BOUNDARY VALUE PROBLEM LE MATEMATICHE Vol. LXII (2007) Fsc. II, pp. 163 173 ON BERNOULLI BOUNDARY VALUE PROBLEM FRANCESCO A. COSTABILE - ANNAROSA SERPE We consider the boundry vlue problem: x (m) (t) = f (t,x(t)), t b, m > 1

More information

MAC-solutions of the nonexistent solutions of mathematical physics

MAC-solutions of the nonexistent solutions of mathematical physics Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE

More information

Ordinary Differential Equations- Boundary Value Problem

Ordinary Differential Equations- Boundary Value Problem Ordinry Differentil Equtions- Boundry Vlue Problem Shooting method Runge Kutt method Computer-bsed solutions o BVPFD subroutine (Fortrn IMSL subroutine tht Solves (prmeterized) system of differentil equtions

More information

Designing Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon.

Designing Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon. EECS 16A Designing Informtion Devices nd Systems I Fll 2016 Bk Ayzifr, Vldimir Stojnovic Homework 6 This homework is due Octoer 11, 2016, t Noon. 1. Homework process nd study group Who else did you work

More information

PARTIAL FRACTION DECOMPOSITION

PARTIAL FRACTION DECOMPOSITION PARTIAL FRACTION DECOMPOSITION LARRY SUSANKA 1. Fcts bout Polynomils nd Nottion We must ssemble some tools nd nottion to prove the existence of the stndrd prtil frction decomposition, used s n integrtion

More information

MTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008

MTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008 MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul

More information

Riemann Integrals and the Fundamental Theorem of Calculus

Riemann Integrals and the Fundamental Theorem of Calculus Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums

More information

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1 Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

More information

Numerical Methods I. Olof Widlund Transcribed by Ian Tobasco

Numerical Methods I. Olof Widlund Transcribed by Ian Tobasco Numericl Methods I Olof Widlund Trnscribed by In Tobsco Abstrct. This is prt one of two semester course on numericl methods. The course ws offered in Fll 011 t the Cournt Institute for Mthemticl Sciences,

More information

POSITIVE SOLUTIONS FOR SINGULAR THREE-POINT BOUNDARY-VALUE PROBLEMS

POSITIVE SOLUTIONS FOR SINGULAR THREE-POINT BOUNDARY-VALUE PROBLEMS Electronic Journl of Differentil Equtions, Vol. 27(27), No. 156, pp. 1 8. ISSN: 172-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu (login: ftp) POSITIVE SOLUTIONS

More information

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s). Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different

More information

Data Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading

Data Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading Dt Assimiltion Aln O Neill Dt Assimiltion Reserch Centre University of Reding Contents Motivtion Univrite sclr dt ssimiltion Multivrite vector dt ssimiltion Optiml Interpoltion BLUE 3d-Vritionl Method

More information

PLANAR NORMAL SECTIONS OF FOCAL MANIFOLDS OF ISOPARAMETRIC HYPERSURFACES IN SPHERES

PLANAR NORMAL SECTIONS OF FOCAL MANIFOLDS OF ISOPARAMETRIC HYPERSURFACES IN SPHERES REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 56, No. 2, 2015, Pges 119 133 Published online: September 30, 2015 PLANAR NORMAL SECTIONS OF FOCAL MANIFOLDS OF ISOPARAMETRIC HYPERSURFACES IN SPHERES Abstrct.

More information

The Wave Equation I. MA 436 Kurt Bryan

The Wave Equation I. MA 436 Kurt Bryan 1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

More information

Eigenfunction Expansions for a Sturm Liouville Problem on Time Scales

Eigenfunction Expansions for a Sturm Liouville Problem on Time Scales Interntionl Journl of Difference Equtions (IJDE). ISSN 0973-6069 Volume 2 Number 1 (2007), pp. 93 104 Reserch Indi Publictions http://www.ripubliction.com/ijde.htm Eigenfunction Expnsions for Sturm Liouville

More information

1 Ordinary Differential Equations

1 Ordinary Differential Equations 1 Ordinry Differentil Equtions 1. Mthemticl Bckground 1..1 Smoothness Definition 1.1 A function f defined on [, b] is continuous t x [, b] if lim x x f(x) = f(x ). Remrk Note tht this implies existence

More information

MATH 222 Second Semester Calculus. Fall 2015

MATH 222 Second Semester Calculus. Fall 2015 MATH Second Semester Clculus Fll 5 Typeset:August, 5 Mth nd Semester Clculus Lecture notes version. (Fll 5) This is self contined set of lecture notes for Mth. The notes were written by Sigurd Angenent,

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

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

440-2 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam

440-2 Geometry/Topology: Differentiable Manifolds Northwestern University Solutions of Practice Problems for Final Exam 440-2 Geometry/Topology: Differentible Mnifolds Northwestern University Solutions of Prctice Problems for Finl Exm 1) Using the cnonicl covering of RP n by {U α } 0 α n, where U α = {[x 0 : : x n ] RP

More information

Determinants Chapter 3

Determinants Chapter 3 Determinnts hpter Specil se : x Mtrix Definition : the determinnt is sclr quntity defined for ny squre n x n mtrix nd denoted y or det(). x se ecll : this expression ppers in the formul for x mtrix inverse!

More information

Intuitionistic Fuzzy Lattices and Intuitionistic Fuzzy Boolean Algebras

Intuitionistic Fuzzy Lattices and Intuitionistic Fuzzy Boolean Algebras Intuitionistic Fuzzy Lttices nd Intuitionistic Fuzzy oolen Algebrs.K. Tripthy #1, M.K. Stpthy *2 nd P.K.Choudhury ##3 # School of Computing Science nd Engineering VIT University Vellore-632014, TN, Indi

More information

DETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ

DETERMINANTS. All Mathematical truths are relative and conditional. C.P. STEINMETZ All Mthemticl truths re reltive nd conditionl. C.P. STEINMETZ 4. Introduction DETERMINANTS In the previous chpter, we hve studied bout mtrices nd lgebr of mtrices. We hve lso lernt tht system of lgebric

More information

along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate

along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate L8 VECTOR EQUATIONS OF LINES HL Mth - Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne

More information