CHAPTER 6. Direct Methods for Solving Linear Systems
|
|
- Berniece Harper
- 6 years ago
- Views:
Transcription
1 CHAPTER 6 Direct Methods for Solving Linear Systems. Introduction A direct method for approximating the solution of a system of n linear equations in n unknowns is one that gives the exact solution to the system, if it is assumed that all calculations can be performed without round-o error e ects. We consider the role of finite-digit arithmetic error in the approiximation to the solution to the system and how to arrange the calculations to minimize its e ect. Note that we assume that we are working on a system with a unique solution.. Gaussian Elimination Two systems of equations are equivalent systems if they have the same solution set. The following three operations on a system of equations result in an equivalent system. Operations on Systems of Equations Yielding an Equivalent System () Equation E i can be multiplied by any nonzero constant, with the resulting equation used in place of E i. This operation is denoted ( E i )! (E i ). () Equation E j can be multiplied by any constant and added to equation E i, with the resulting equation used in place of E i. This operation is denoted (E i + E j )! (E i ). () Equations E i and E j can be transposed in order. This operation is denoted (E i ) $ (E j ).
2 . GAUSSIAN ELIMINATION 9 Example (Gaussian Elimination with Backward Substitution). Notice how the row operations on the matrices mirror the corresponding operations on the equations. Notice also that the coding can apply to either the equations or the matrix. >< >: () x + x + x = 9 () x + x x = 5 () x + x x = >< >: >< >: () x + x x = 9 (E ) $ (E ) () x + x x = 5 () x + x + x = 9 (E ) $ (E ) () x + x 6x = (E )! (E ) () x + x x = 5 () x + x + x = >< >: () x + x 6x = () x + 5x = (E E )! (E ) () x + x = 5 (E E )! (E ) 0 5 >< >: () x + x 6x = () x + 5x = () x = 9 (E + E )! (E ) This system of equations is (upper) triangular or reduced, and can be solved by backward substitution. The solution found thus applies to all five equivalent systems.
3 0 6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS x = 9 x = x + 5 = x = x = x = x = x = Definition. A = 0 55 is a matrix, an array of numbers in rows and columns. The plural of matrix is matrices. The numbers are the entries of A. A is a matrix since it has rows and columns. An n n matrix (same number of rows as columns) is a square matrix. B = 6 05 is a square matrix. 0 An n m matrix A may be represented as a a a m A = [a ij ] = 6a a a m a n a n a nm where a kl means the entry in row k, column l.
4 a a n. GAUSSIAN ELIMINATION The n matrix A = 6a. 5is a n-dimensional column vector. The m matrix A = a a a m is an m-dimensional row vector. Thus, ignoring the unneeded subscript, x = x 6x. 5 y = y y y m is a row vector. a a a m b If A = 6a a a m and b = 6b. 5, then a n a n a nm b n x n is a column vector and A = [A, b] = a a a m b 6a a a m b a n a n a nm b n is an augmented matrix, used to represent a system of n equations in m variables. In the future, we will often dismiss the practice of using the separators to denote the right-hand side column matrix. Note. () This method of Gaussian elimination with backward substitution can fail to give a unique solution, assuming as many equations as variables, if any a ii = 0 in the augmented matrix after it has been reduced. () The number of computations a ects computation time and round-o error in a machine with finite-digit arithmetic. The number of addition/subtractions
5 6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS in our method for n equations in n variables is n + n 5n 6 while the number of multiplications/divisions (which take more time) is n + n n. Both are O(n ). See the table on Page of the text. Maple. See linsys.mw and/or lynsys.pdf.
6 . PIVOTING STRATEGIES. Pivoting Strategies In systems with unique solutions, the a ii are the pivot elements. In , 0 5 the boxed 0 is a pivot element. But the reduced matrix can have no a ii = 0, so we swap rows and. It is often necessary to swap rows to reduce round-o error even when the pivot elements are not 0. Consider. a ii a il 6. a ki 5.. To get a ki = 0, we multiply row i by m i = a ki a ii and then add the product to row k. If a ii is much smaller than a ki, then m i is large and multiplying row i by m i magnifies round-o error in a il {z} approximate value and the other values in row i. = b il {z} exact value + il {z} error Also, for back substitution, x i = a P n i,n+ j=i+ a ij, a ii so if a ii is small, any round-o error in the numerator is magnified.
7 6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS Partial Pivoting or Maximal Column Pivoting When a ii is the pivot, swap rows so that the element of largest magnitude from a ii thru a ni is moved to a ii. But this is still sometimes inadequate. Scaled Partial Pivoting This method places in the pivot position a ii the element from a ii thru a ni that is largest relative to the entries in its row. The e ect of scaling is to ensure that the largest element in each row has a relative magnitude of before the comparison for row interchange is performed. () For each row k, define the scale factor s k = max applejapplen a kj. If some s k = 0, then there is no unique solution. () For a, find the least integer p such that a p s p a k = max applekapplen s k and then swap (E ) $ (E p ) (and also swap s $ s p ) if p 6=. () For a ii, i >, find the least integer p i such that a pi s p a ki = max iapplekapplen s k and then swap (E i ) $ (E p ) (and also swap s i $ s p ) if p 6= i. Note. The scale factors s, s,..., s n are computed only once at the start of the procedure and must be swapped along with the row swaps.
8 . LINEAR ALGEBRA AND MATRIX INVERSION 5 Maximal (or Total or Full) Pivoting Do row and column swapping to bring the element of largest magnitude below and to the right of a ii to a ii, provided its magnitude is larger than that of a ii. n n Gaussian elimination requires O multiplications/divisions and O additions/subtractions. Partial Pivoting is about the same. Scaled partial pivoting adds n(n + ) n(n ) comparisons and so does not significantly add to computation time. divisions, Complete pivoting adds n(n )(n+5) 6 comparisons, which approximately doubles the amount of addition/subtraction time over Gausian elimination. Maple. See pivoting.mw and/or pivoting.pdf.. Linear Algebra and Matrix Inversion Matrices A and B are equal if both are n m and a ij = b ij for i =,..., n and j =,..., m. The sum A + B of n m matrices A = [a ij ] and B = [b ij ] is the n m matrix obtained by adding corresponding entries. A + B = [a ij + b ij ], for i =,..., n and j =,..., m. The scalar multiplication of the real (or complex) scalar A = [a ij ] is the n m matrix A = [ a ij ]. 0 n m is the n m zero matrix where all entrie are 0. and the n m matrix If A = [a ij ] is an n m matrix, the A = [ a ij ] is also an n m matrix.
9 6 6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS Theorem. The set of all n m matrices with real entries is a vector space over the field of real numbers. If A, B, C are n m matrices and, µ R: (a) A + B = B + A (b) (A + B) + C = A + (B + C) (c) A + 0 = 0 + A = A (d) A + ( A) = A + A = 0 (e) (A + B) = A + B (f) ( + µ)a = A + µa (g) (µa) = ( µ)a (h) A = A The matrix product AB where A is an n m matrix and B is an m p matrix is the n p matrix C where a a m b a p c c p = a n a nm b m b mp c n c np with c ij = a i a i a im 6 b j b j. b mj m 5 = X k= a ik b kj, i.e., to get the entry in row i, column j of the product, just multiply each of the numbers in row i of A with the corresponding number in column j of B, and then add the m products. Note. {z} A {z} B = {z} C. The m s must match, and the outside numbers n m m p n p give the new dimensions.
10 r r n. LINEAR ALGEBRA AND MATRIX INVERSION If A = 6r. 5 with the r i m row vectors and B = c c c p with the c j m column vectors, then c c c p r c r c r c p C = 6c c c p = 6r c r c r c p c n c n c np r n c r n c r n c p Example. apple apple apple 0 = and apple apple 0 apple 0 =, so matrix multiplication is not commutative. 5 5 = {z } 6 {z } 6 {z } These matrices cannot even be multiplied in the other direction due to dimension mismatch. A square n n matrix has the same number of rows as columns. A diagonal matrix is a square matrix D = [d ij ] with d ij = 0 for i 6= j. Example. D =
11 6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS An n n identity matrix is a square matrix I n = [ ij ] where ( for i = j ij = 0 for i 6= j. Example. I = A square matrix is upper-triangular if all entries below the diagonal are 0 and lower-triangular if all entries above the diagonal are 0. Theorem. Let A n m, B m k, C k p, D m k be matrices an (a) A(BC) = (AB)C (b) A(B + D) = AB + AD (c) I m B = B = BI k (d) (AB) = ( A)B = A( B) R. Then An n n matrix A is invertible or nonsingular if there exists an n n matrix A such that AA = A A = I n. We say A is the inverse of A. A matrix without an inverse is singular or noninvertible. Theorem. For any n n nonsingular matrix A: (a) A is unique (b) A is nonsingular and (A ) = A (c) If B is also an n n nonsingular matrix, (AB) = B A
12 . LINEAR ALGEBRA AND MATRIX INVERSION 9 The system of linear equations corresponds to a x + + a n x n = b a n x + + a nn x n = b n a a n x b =. 5 or Ax = b. a n a nn x n b n Ax = b has a unique solution () A is invertible. In that case, A (Ax) = A b =) (A A)x = A b =) I n x = A b =) x = A b Problem (Page 6 #b(ii)). Find the inverse of A = find B such that 0 5 b b b b b b 5 = b b b = b + b b + b b + b b + b b b + b b b + b b b + b + b b + b + b b + b + b 5 = 0 5, i.e., Matching up columns, we have systems of linear equations, all with the same coe cient matrix. As a result, we can use Gaussian elimination on a larger augmented matrix to solve all systems at once:
13 50 6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS E E! E E E! E E E! E Now, using back substitution. b = b b = 5 b = b = b = 5 b = = b + 5 = b = 0 b = 5 b = b = b = 5 b = b = b + 0 = b = 0 b = 0 b = b = b = Thus B = Checking, 0 5 and =
14 5. LINEAR ALGEBRA AND MATRIX INVERSION = The transpose of A = [a ij ], an n m matrix, is the m n matrix A t = [a ji ] obtained from A by interchanging the rows an columns of A. A square matrix is symmetric if A = A t. Theorem. The following hold whenever the operation is possible: (a) (A t ) t = A (b) (A + B) t = A t + B t (c) (AB) t = B t A t (d) If A exists, (A ) t = (A t ) Definition (Determinant of a matrix A, denoted det A or A ). (a) If A = [a], det A = a. (b) If A is n n, the minor M ij is the determinant of the (n ) (n ) submatrix of A obtained by deleting row i and column j from A. (c) The cofactor A ij associated with M ij is A ij = ( ) i+j M ij. (d) For an n n matrix A, n >, nx nx det A = a ij A ij = ( ) i+j a ij M ij for any i =,..., n or det A = j= nx a ij A ij = i= j= nx ( ) i+j a ij M ij for any j =,..., n. i=
15 5 6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS Now Example. det () det 0 5 = det 0 n o () det[] () det[] 5 + () det 5 + () det 0 5. apple 5 = ( ) det apple + ( ) det = n o () det[] () det[] ( ) ( ) = ( 5) ( 5) = 0. We will use a special shortcut for the other two determinants. We rewrite the first two columns to the right of the matrix as an aid. det 5 = & z } { ( ) + & z } { ( ) + & z } { ( ). z } { ( ). z } { ( ) =. z } { ( ) = = 5. Once the pattern is known, this can be simplified further by thinking of rolling up the matrix into a vertical cylinder. We do this for the third determinant. det 0 5 = + 0 ( 6) 0 ( ) =. Putting everything together,
16 . LINEAR ALGEBRA AND MATRIX INVERSION 5 0 det = ( ) = apple a b Note. In general, det = ad bc. c d Theorem. Suppose A is an n n matrix: (a) If any row or column of A has only 0 entries, then det A = 0. (b) If e A is obtained from A by the operation (E i ) $ (E k ), then det e A = det A. (c) If A has two rows or two columns the same, then det A = 0. (d) If e A is obtained from A by the operation ( E i )! (E i ), then det e A = det A. (e) If e A is obtained from A by the operation (E i + det e A = det A. E k )! (E i ), then (f) If B is also an n n matrix, then (g) det A t = det A. det(ab) = det A det B. (h) If A exists, then det A = det A. (i) If A is an upper triangular, lower triangular, or diagonal matrix, then ny det A = a ii = a a a nn. i=
17 5 6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS Theorem. The following statements are equivalent for an n n matrix A: (a) The equation Ax = 0 has the unique solution x = 0. (b) The system Ax = b has a unique solution for any n-dimensional column vector b. (c) The matrix A is nonsingular; that is A exists. (d) det A 6= 0. (e) Gaussian elimination with row interchanges can be performed on the system Ax = b for any n-dimensional column vector b. Maple. See inverses.mw and/or inverses.pdf. 5. Matrix Factorization We want to be able to factor a matrix A as A = LU where L is lower triangular and U is upper triangular. This can be done if Ax = b can be solved by Gaussian elimination without row interchanges. We will use Maple to do the factoring. Then Ax = b becomes LUx = b. Let y = Ux =) Ly = b. Solve for y by forward substitution. Then solve Ux = y for x by backward substitution.
18 5. MATRIX FACTORIZATION 55 Example x x 5 = {z } 0 {z x } {z } {z } L U x b y y 5 = y 5 =) y + y 5 = 5 0 y {z } {z } {z } y + y {z } {z } L y b Ly b Forward substitution: y = + y = =) y = 5 + y = =) y = 0 5 x x 5 = 55 =) x + x x x + x 5 = 55 =) 0 0 {z x } {z } {z } x {z } {z } U x y Ux y Backward substitution: Thus x = =) x = x + = 5 =) x = x + 9 = =) x = x = 5.
19 56 6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS Why do this factorization? n Ax = b requires O arithmetic operations. Ly = b and Ux = y each require O(n ) arithmetic operations for a total of O(n ). In systems greater than 00 00, this reduces the amount of calculation by more than 9%. n But: LU factorization requires O operations. Still, once factored, LU can be used with any b. What if row interchanges are necessary? We use an n n permutation matrix P formed by rearranging the rows of I n. Suppose A = 5 65 and P = P A = = P has rows of I n in order,,, so P A has rows of A in order,,. AP = = P has columns of I n in order,,, so AP has columns of A in order,,. Method Find a permutation matrix P such that P Ax = b can be solved without row interchanges. Then, noting P = P t, P A = LU =) A = P t LU =) P t LUx = b =) LUx = P b = c. Now solve as above. Maple. See lu.mw and/or lu.pdf.
1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i )
Direct Methods for Linear Systems Chapter Direct Methods for Solving Linear Systems Per-Olof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical
More informationDirect Methods for Solving Linear Systems. Matrix Factorization
Direct Methods for Solving Linear Systems Matrix Factorization Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011
More informationDirect Methods for Solving Linear Systems. Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le
Direct Methods for Solving Linear Systems Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le 1 Overview General Linear Systems Gaussian Elimination Triangular Systems The LU Factorization
More informationLinear Algebra and Matrix Inversion
Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much
More informationMATH2210 Notebook 2 Spring 2018
MATH2210 Notebook 2 Spring 2018 prepared by Professor Jenny Baglivo c Copyright 2009 2018 by Jenny A. Baglivo. All Rights Reserved. 2 MATH2210 Notebook 2 3 2.1 Matrices and Their Operations................................
More informationLinear Algebra Section 2.6 : LU Decomposition Section 2.7 : Permutations and transposes Wednesday, February 13th Math 301 Week #4
Linear Algebra Section. : LU Decomposition Section. : Permutations and transposes Wednesday, February 1th Math 01 Week # 1 The LU Decomposition We learned last time that we can factor a invertible matrix
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationNumerical Linear Algebra
Numerical Linear Algebra Direct Methods Philippe B. Laval KSU Fall 2017 Philippe B. Laval (KSU) Linear Systems: Direct Solution Methods Fall 2017 1 / 14 Introduction The solution of linear systems is one
More informationFundamentals of Engineering Analysis (650163)
Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is
More informationChapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in
Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column
More informationGaussian Elimination and Back Substitution
Jim Lambers MAT 610 Summer Session 2009-10 Lecture 4 Notes These notes correspond to Sections 31 and 32 in the text Gaussian Elimination and Back Substitution The basic idea behind methods for solving
More informationLinear Systems and Matrices
Department of Mathematics The Chinese University of Hong Kong 1 System of m linear equations in n unknowns (linear system) a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.......
More informationA Review of Matrix Analysis
Matrix Notation Part Matrix Operations Matrices are simply rectangular arrays of quantities Each quantity in the array is called an element of the matrix and an element can be either a numerical value
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
More informationPOLI270 - Linear Algebra
POLI7 - Linear Algebra Septemer 8th Basics a x + a x +... + a n x n b () is the linear form where a, b are parameters and x n are variables. For a given equation such as x +x you only need a variable and
More information1 Determinants. 1.1 Determinant
1 Determinants [SB], Chapter 9, p.188-196. [SB], Chapter 26, p.719-739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to
More informationDirect Methods for Solving Linear Systems
Chapter 6 Direct Methods for Solving Linear Systems 6. Introduction Systems of equations are used to represent physical problems that involve the interaction of various properties. The variables in the
More information2.1 Gaussian Elimination
2. Gaussian Elimination A common problem encountered in numerical models is the one in which there are n equations and n unknowns. The following is a description of the Gaussian elimination method for
More informationLinear Equations and Matrix
1/60 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Gaussian Elimination 2/60 Alpha Go Linear algebra begins with a system of linear
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More informationMatrix Algebra Determinant, Inverse matrix. Matrices. A. Fabretti. Mathematics 2 A.Y. 2015/2016. A. Fabretti Matrices
Matrices A. Fabretti Mathematics 2 A.Y. 2015/2016 Table of contents Matrix Algebra Determinant Inverse Matrix Introduction A matrix is a rectangular array of numbers. The size of a matrix is indicated
More informationSection 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns.
Section 9.2: Matrices Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. That is, a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn A
More informationLU Factorization. Marco Chiarandini. DM559 Linear and Integer Programming. Department of Mathematics & Computer Science University of Southern Denmark
DM559 Linear and Integer Programming LU Factorization Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark [Based on slides by Lieven Vandenberghe, UCLA] Outline
More informationMatrix notation. A nm : n m : size of the matrix. m : no of columns, n: no of rows. Row matrix n=1 [b 1, b 2, b 3,. b m ] Column matrix m=1
Matrix notation A nm : n m : size of the matrix m : no of columns, n: no of rows Row matrix n=1 [b 1, b 2, b 3,. b m ] Column matrix m=1 n = m square matrix Symmetric matrix Upper triangular matrix: matrix
More informationMath 240 Calculus III
The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A
More informationMATH 3511 Lecture 1. Solving Linear Systems 1
MATH 3511 Lecture 1 Solving Linear Systems 1 Dmitriy Leykekhman Spring 2012 Goals Review of basic linear algebra Solution of simple linear systems Gaussian elimination D Leykekhman - MATH 3511 Introduction
More informationMatrix & Linear Algebra
Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A
More informationUndergraduate Mathematical Economics Lecture 1
Undergraduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 15, 2014 Outline 1 Courses Description and Requirement 2 Course Outline ematical techniques used in economics courses
More informationMath Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT
Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear
More informationMATRICES. a m,1 a m,n A =
MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of
More informationLinear Algebra. James Je Heon Kim
Linear lgebra James Je Heon Kim (jjk9columbia.edu) If you are unfamiliar with linear or matrix algebra, you will nd that it is very di erent from basic algebra or calculus. For the duration of this session,
More informationSOLVING LINEAR SYSTEMS
SOLVING LINEAR SYSTEMS We want to solve the linear system a, x + + a,n x n = b a n, x + + a n,n x n = b n This will be done by the method used in beginning algebra, by successively eliminating unknowns
More informationElementary Row Operations on Matrices
King Saud University September 17, 018 Table of contents 1 Definition A real matrix is a rectangular array whose entries are real numbers. These numbers are organized on rows and columns. An m n matrix
More informationLinear Systems of n equations for n unknowns
Linear Systems of n equations for n unknowns In many application problems we want to find n unknowns, and we have n linear equations Example: Find x,x,x such that the following three equations hold: x
More informationLinear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.
Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the
More informationNumerical Linear Algebra
Chapter 3 Numerical Linear Algebra We review some techniques used to solve Ax = b where A is an n n matrix, and x and b are n 1 vectors (column vectors). We then review eigenvalues and eigenvectors and
More informationLecture Notes in Linear Algebra
Lecture Notes in Linear Algebra Dr. Abdullah Al-Azemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................
More informationGraduate Mathematical Economics Lecture 1
Graduate Mathematical Economics Lecture 1 Yu Ren WISE, Xiamen University September 23, 2012 Outline 1 2 Course Outline ematical techniques used in graduate level economics courses Mathematics for Economists
More informationReview of matrices. Let m, n IN. A rectangle of numbers written like A =
Review of matrices Let m, n IN. A rectangle of numbers written like a 11 a 12... a 1n a 21 a 22... a 2n A =...... a m1 a m2... a mn where each a ij IR is called a matrix with m rows and n columns or an
More informationVectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes.
Vectors and matrices: matrices (Version 2) This is a very brief summary of my lecture notes Matrices and linear equations A matrix is an m-by-n array of numbers A = a 11 a 12 a 13 a 1n a 21 a 22 a 23 a
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 6
CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 6 GENE H GOLUB Issues with Floating-point Arithmetic We conclude our discussion of floating-point arithmetic by highlighting two issues that frequently
More informationIntroduction. Vectors and Matrices. Vectors [1] Vectors [2]
Introduction Vectors and Matrices Dr. TGI Fernando 1 2 Data is frequently arranged in arrays, that is, sets whose elements are indexed by one or more subscripts. Vector - one dimensional array Matrix -
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More informationCS412: Lecture #17. Mridul Aanjaneya. March 19, 2015
CS: Lecture #7 Mridul Aanjaneya March 9, 5 Solving linear systems of equations Consider a lower triangular matrix L: l l l L = l 3 l 3 l 33 l n l nn A procedure similar to that for upper triangular systems
More informationThings we can already do with matrices. Unit II - Matrix arithmetic. Defining the matrix product. Things that fail in matrix arithmetic
Unit II - Matrix arithmetic matrix multiplication matrix inverses elementary matrices finding the inverse of a matrix determinants Unit II - Matrix arithmetic 1 Things we can already do with matrices equality
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION x 1,, x n A linear equation in the variables equation that can be written in the form a 1 x 1 + a 2 x 2 + + a n x n
More informationEquality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.
Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read
More informationLinear Algebra: Lecture notes from Kolman and Hill 9th edition.
Linear Algebra: Lecture notes from Kolman and Hill 9th edition Taylan Şengül March 20, 2019 Please let me know of any mistakes in these notes Contents Week 1 1 11 Systems of Linear Equations 1 12 Matrices
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University February 6, 2018 Linear Algebra (MTH
More informationAMS 209, Fall 2015 Final Project Type A Numerical Linear Algebra: Gaussian Elimination with Pivoting for Solving Linear Systems
AMS 209, Fall 205 Final Project Type A Numerical Linear Algebra: Gaussian Elimination with Pivoting for Solving Linear Systems. Overview We are interested in solving a well-defined linear system given
More informationMatrix operations Linear Algebra with Computer Science Application
Linear Algebra with Computer Science Application February 14, 2018 1 Matrix operations 11 Matrix operations If A is an m n matrix that is, a matrix with m rows and n columns then the scalar entry in the
More informationGAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511)
GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) D. ARAPURA Gaussian elimination is the go to method for all basic linear classes including this one. We go summarize the main ideas. 1.
More informationSection 9.2: Matrices.. a m1 a m2 a mn
Section 9.2: Matrices Definition: A matrix is a rectangular array of numbers: a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn In general, a ij denotes the (i, j) entry of A. That is, the entry in
More information. =. a i1 x 1 + a i2 x 2 + a in x n = b i. a 11 a 12 a 1n a 21 a 22 a 1n. i1 a i2 a in
Vectors and Matrices Continued Remember that our goal is to write a system of algebraic equations as a matrix equation. Suppose we have the n linear algebraic equations a x + a 2 x 2 + a n x n = b a 2
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J Olver 3 Review of Matrix Algebra Vectors and matrices are essential for modern analysis of systems of equations algebrai, differential, functional, etc In this
More informationII. Determinant Functions
Supplemental Materials for EE203001 Students II Determinant Functions Chung-Chin Lu Department of Electrical Engineering National Tsing Hua University May 22, 2003 1 Three Axioms for a Determinant Function
More informationMatrix Algebra. Matrix Algebra. Chapter 8 - S&B
Chapter 8 - S&B Algebraic operations Matrix: The size of a matrix is indicated by the number of its rows and the number of its columns. A matrix with k rows and n columns is called a k n matrix. The number
More informationNext topics: Solving systems of linear equations
Next topics: Solving systems of linear equations 1 Gaussian elimination (today) 2 Gaussian elimination with partial pivoting (Week 9) 3 The method of LU-decomposition (Week 10) 4 Iterative techniques:
More informationThe Solution of Linear Systems AX = B
Chapter 2 The Solution of Linear Systems AX = B 21 Upper-triangular Linear Systems We will now develop the back-substitution algorithm, which is useful for solving a linear system of equations that has
More informationNotes on Determinants and Matrix Inverse
Notes on Determinants and Matrix Inverse University of British Columbia, Vancouver Yue-Xian Li March 17, 2015 1 1 Definition of determinant Determinant is a scalar that measures the magnitude or size of
More informationMATH 106 LINEAR ALGEBRA LECTURE NOTES
MATH 6 LINEAR ALGEBRA LECTURE NOTES FALL - These Lecture Notes are not in a final form being still subject of improvement Contents Systems of linear equations and matrices 5 Introduction to systems of
More informationPhys 201. Matrices and Determinants
Phys 201 Matrices and Determinants 1 1.1 Matrices 1.2 Operations of matrices 1.3 Types of matrices 1.4 Properties of matrices 1.5 Determinants 1.6 Inverse of a 3 3 matrix 2 1.1 Matrices A 2 3 7 =! " 1
More informationTOPIC III LINEAR ALGEBRA
[1] Linear Equations TOPIC III LINEAR ALGEBRA (1) Case of Two Endogenous Variables 1) Linear vs. Nonlinear Equations Linear equation: ax + by = c, where a, b and c are constants. 2 Nonlinear equation:
More informationReview for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions.
Review for Exam. Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. x + y z = 2 x + 2y + z = 3 x + y + (a 2 5)z = a 2 The augmented matrix for
More informationn n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full
n n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in n variables x 1, x 2,..., x n a 11 x 1 + a 12 x
More informationComputational Methods. Systems of Linear Equations
Computational Methods Systems of Linear Equations Manfred Huber 2010 1 Systems of Equations Often a system model contains multiple variables (parameters) and contains multiple equations Multiple equations
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More informationMaterials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat
Materials engineering Collage \\ Ceramic & construction materials department Numerical Analysis \\Third stage by \\ Dalya Hekmat Linear Algebra Lecture 2 1.3.7 Matrix Matrix multiplication using Falk s
More informationMatrix decompositions
Matrix decompositions How can we solve Ax = b? 1 Linear algebra Typical linear system of equations : x 1 x +x = x 1 +x +9x = 0 x 1 +x x = The variables x 1, x, and x only appear as linear terms (no powers
More informationDigital Workbook for GRA 6035 Mathematics
Eivind Eriksen Digital Workbook for GRA 6035 Mathematics November 10, 2014 BI Norwegian Business School Contents Part I Lectures in GRA6035 Mathematics 1 Linear Systems and Gaussian Elimination........................
More informationMatrices. Chapter Definitions and Notations
Chapter 3 Matrices 3. Definitions and Notations Matrices are yet another mathematical object. Learning about matrices means learning what they are, how they are represented, the types of operations which
More informationMatrix Arithmetic. j=1
An m n matrix is an array A = Matrix Arithmetic a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn of real numbers a ij An m n matrix has m rows and n columns a ij is the entry in the i-th row and j-th column
More informationAlgebra & Trig. I. For example, the system. x y 2 z. may be represented by the augmented matrix
Algebra & Trig. I 8.1 Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 12: Gaussian Elimination and LU Factorization Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 10 Gaussian Elimination
More informationFinite Mathematics Chapter 2. where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero.
Finite Mathematics Chapter 2 Section 2.1 Systems of Linear Equations: An Introduction Systems of Equations Recall that a system of two linear equations in two variables may be written in the general form
More informationMATH Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product.
MATH 311-504 Topics in Applied Mathematics Lecture 12: Evaluation of determinants. Cross product. Determinant is a scalar assigned to each square matrix. Notation. The determinant of a matrix A = (a ij
More informationLinear Algebra (part 1) : Matrices and Systems of Linear Equations (by Evan Dummit, 2016, v. 2.02)
Linear Algebra (part ) : Matrices and Systems of Linear Equations (by Evan Dummit, 206, v 202) Contents 2 Matrices and Systems of Linear Equations 2 Systems of Linear Equations 2 Elimination, Matrix Formulation
More information1.4 Gaussian Elimination Gaussian elimination: an algorithm for finding a (actually the ) reduced row echelon form of a matrix. A row echelon form
1. Gaussian Elimination Gaussian elimination: an algorithm for finding a (actually the ) reduced row echelon form of a matrix. Original augmented matrix A row echelon form 1 1 0 0 0 1!!!! The reduced row
More informationMODEL ANSWERS TO THE THIRD HOMEWORK
MODEL ANSWERS TO THE THIRD HOMEWORK 1 (i) We apply Gaussian elimination to A First note that the second row is a multiple of the first row So we need to swap the second and third rows 1 3 2 1 2 6 5 7 3
More informationChapter 1 Matrices and Systems of Equations
Chapter 1 Matrices and Systems of Equations System of Linear Equations 1. A linear equation in n unknowns is an equation of the form n i=1 a i x i = b where a 1,..., a n, b R and x 1,..., x n are variables.
More informationLectures on Linear Algebra for IT
Lectures on Linear Algebra for IT by Mgr. Tereza Kovářová, Ph.D. following content of lectures by Ing. Petr Beremlijski, Ph.D. Department of Applied Mathematics, VSB - TU Ostrava Czech Republic 11. Determinants
More informationGaussian Elimination -(3.1) b 1. b 2., b. b n
Gaussian Elimination -() Consider solving a given system of n linear equations in n unknowns: (*) a x a x a n x n b where a ij and b i are constants and x i are unknowns Let a n x a n x a nn x n a a a
More informationLinear System of Equations
Linear System of Equations Linear systems are perhaps the most widely applied numerical procedures when real-world situation are to be simulated. Example: computing the forces in a TRUSS. F F 5. 77F F.
More informationMATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics
MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam This study sheet will not be allowed during the test Books and notes will not be allowed during the test Calculators and cell phones
More informationMatrices and Linear Algebra
Contents Quantitative methods for Economics and Business University of Ferrara Academic year 2017-2018 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2 3 4 5 Contents 1 Basics 2
More informationAPPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF
ELEMENTARY LINEAR ALGEBRA WORKBOOK/FOR USE WITH RON LARSON S TEXTBOOK ELEMENTARY LINEAR ALGEBRA CREATED BY SHANNON MARTIN MYERS APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF When you are done
More informationMatrix Factorization and Analysis
Chapter 7 Matrix Factorization and Analysis Matrix factorizations are an important part of the practice and analysis of signal processing. They are at the heart of many signal-processing algorithms. Their
More informationMath 313 Chapter 1 Review
Math 313 Chapter 1 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 1.1 Introduction to Systems of Linear Equations 2 2 1.2 Gaussian Elimination 3 3 1.3 Matrices and Matrix Operations
More informationLinear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds
Linear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds These notes are meant to provide a brief introduction to the topics from Linear Algebra that will be useful in Math3315/CSE3365, Introduction
More informationNumerical Methods I Solving Square Linear Systems: GEM and LU factorization
Numerical Methods I Solving Square Linear Systems: GEM and LU factorization Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 September 18th,
More informationAppendix C Vector and matrix algebra
Appendix C Vector and matrix algebra Concepts Scalars Vectors, rows and columns, matrices Adding and subtracting vectors and matrices Multiplying them by scalars Products of vectors and matrices, scalar
More informationComputational Linear Algebra
Computational Linear Algebra PD Dr. rer. nat. habil. Ralf Peter Mundani Computation in Engineering / BGU Scientific Computing in Computer Science / INF Winter Term 2017/18 Part 2: Direct Methods PD Dr.
More informationECON 186 Class Notes: Linear Algebra
ECON 86 Class Notes: Linear Algebra Jijian Fan Jijian Fan ECON 86 / 27 Singularity and Rank As discussed previously, squareness is a necessary condition for a matrix to be nonsingular (have an inverse).
More informationCS227-Scientific Computing. Lecture 4: A Crash Course in Linear Algebra
CS227-Scientific Computing Lecture 4: A Crash Course in Linear Algebra Linear Transformation of Variables A common phenomenon: Two sets of quantities linearly related: y = 3x + x 2 4x 3 y 2 = 2.7x 2 x
More informationMIDTERM 1 - SOLUTIONS
MIDTERM - SOLUTIONS MATH 254 - SUMMER 2002 - KUNIYUKI CHAPTERS, 2, GRADED OUT OF 75 POINTS 2 50 POINTS TOTAL ) Use either Gaussian elimination with back-substitution or Gauss-Jordan elimination to solve
More informationMAC Module 2 Systems of Linear Equations and Matrices II. Learning Objectives. Upon completing this module, you should be able to :
MAC 0 Module Systems of Linear Equations and Matrices II Learning Objectives Upon completing this module, you should be able to :. Find the inverse of a square matrix.. Determine whether a matrix is invertible..
More informationMatrix decompositions
Matrix decompositions How can we solve Ax = b? 1 Linear algebra Typical linear system of equations : x 1 x +x = x 1 +x +9x = 0 x 1 +x x = The variables x 1, x, and x only appear as linear terms (no powers
More informationCOURSE Numerical methods for solving linear systems. Practical solving of many problems eventually leads to solving linear systems.
COURSE 9 4 Numerical methods for solving linear systems Practical solving of many problems eventually leads to solving linear systems Classification of the methods: - direct methods - with low number of
More informationLecture 12 (Tue, Mar 5) Gaussian elimination and LU factorization (II)
Math 59 Lecture 2 (Tue Mar 5) Gaussian elimination and LU factorization (II) 2 Gaussian elimination - LU factorization For a general n n matrix A the Gaussian elimination produces an LU factorization if
More informationExample: 2x y + 3z = 1 5y 6z = 0 x + 4z = 7. Definition: Elementary Row Operations. Example: Type I swap rows 1 and 3
Linear Algebra Row Reduced Echelon Form Techniques for solving systems of linear equations lie at the heart of linear algebra. In high school we learn to solve systems with or variables using elimination
More information