(Mathematical Operations with Arrays) Applied Linear Algebra in Geoscience Using MATLAB
|
|
|
- Laurence Perry
- 5 years ago
- Views:
Transcription
1 Applied Linear Algebra in Geoscience Using MATLAB (Mathematical Operations with Arrays)
2 Contents Getting Started Matrices Creating Arrays Linear equations Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Determinants Eigenvalues and eigenvectors Orthogonal vectors and matrices Vector and matrix norms Programming in MATLAB Gaussian elimination and the LU User-Defined Functions and Function Files dec. Polynomials, Curve Fitting, and Interpolation Linear system applications Gram-Schmidt decomposition Applications in Numerical Analysis The singular value decomposition Three-Dimensional Plots Least-squares problems Symbolic Math Implementing the QR factorization TheLinear algebraic problem Applied Algebraeigenvalue in Geoscience Using MATLAB
3 Addition and Subtraction The operations + and can be used to add (subtract) arrays of identical size and to add (subtract) a scalar to an array.
4 Array Multiplication The multiplication operation * is executed by MATLAB according to the rules of linear algebra. This means that if A and B are two matrices, the operation A*B can be carried out only if the number of columns in matrix A is equal to the number of rows in matrix B The power operation can be executed only with a square matrix The multiplication of a row vector by a column vector gives a scalar MATLAB also has a built-in function, dot(a,b)
5 Array Multiplication Linear algebra rules of array multiplication provide a convenient way for writing a system of linear equations. EX.1 the system of three equations with three unknowns
6 Array Division Inverse of a matrix: The matrix B is the inverse of the matrix A if, when the two matrices are multiplied, the product is the identity matrix. Both matrices must be square and the multiplication order can be The inverse of a matrix A is typically written as A-1. In MATLAB the inverse of a matrix can be obtained either by raising A to the power of 1, or with the inv(a) function. Not every matrix has an inverse. A matrix has an inverse only if it is square and its determinant is not equal to zero. det(a)
7 Array Division MATLAB has two types of array division, right division and left division. Left division, \ : Left division is used to solve the matrix equation AX = B. In this equation X and B are column vectors. So the solution of AX =B is: In MATLAB Right division, / : The right division is used to solve the matrix equation XC = D. In this equation X and D are row vectors. The solution In MATLAB
8 Array Division EX.1 Use matrix operations to solve the following system of linear equations.
9 Element-By-Element Operation The regular symbols for multiplication and division (*and/) The mathematical operations follow the rules of linear algebra Many situations that require element-byelement operations. Element-by-element calculations are very useful for calculating the value of a function at many values of its argument.
10 Built in Fun. For Analyzing Array
11 Random Numbers MATLAB has three commands rand, randn, and randi that can be used to assign random numbers to variables.
12 Example EX.2 The coefficient of friction,, can be determined in an experiment by measuring the force F required to move a mass m. When F is measured and m is known, the coefficient of friction can be calculated by: Results from measuring F in six tests are given in the table below. Determine the coefficient of friction in each test, and the average from all tests.
Applied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
Applied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
(Linear equations) Applied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB (Linear equations) Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots
Applied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
CITS2401 Computer Analysis & Visualisation
FACULTY OF ENGINEERING, COMPUTING AND MATHEMATICS CITS2401 Computer Analysis & Visualisation SCHOOL OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING Topic 7 Matrix Algebra Material from MATLAB for Engineers,
CITS2401 Computer Analysis & Visualisation
FACULTY OF ENGINEERING, COMPUTING AND MATHEMATICS CITS2401 Computer Analysis & Visualisation SCHOOL OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING Topic 7 Matrix Algebra Material from MATLAB for Engineers,
1 Number Systems and Errors 1
Contents 1 Number Systems and Errors 1 1.1 Introduction................................ 1 1.2 Number Representation and Base of Numbers............. 1 1.2.1 Normalized Floating-point Representation...........
Computational Methods CMSC/AMSC/MAPL 460. Eigenvalues and Eigenvectors. Ramani Duraiswami, Dept. of Computer Science
Computational Methods CMSC/AMSC/MAPL 460 Eigenvalues and Eigenvectors Ramani Duraiswami, Dept. of Computer Science Eigen Values of a Matrix Recap: A N N matrix A has an eigenvector x (non-zero) with corresponding
MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra
MATH.2720 Introduction to Programming with MATLAB Vector and Matrix Algebra A. Vectors A vector is a quantity that has both magnitude and direction, like velocity. The location of a vector is irrelevant;
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
Introduction to Applied Linear Algebra with MATLAB
Sigam Series in Applied Mathematics Volume 7 Rizwan Butt Introduction to Applied Linear Algebra with MATLAB Heldermann Verlag Contents Number Systems and Errors 1 1.1 Introduction 1 1.2 Number Representation
Cheat Sheet for MATH461
Cheat Sheet for MATH46 Here is the stuff you really need to remember for the exams Linear systems Ax = b Problem: We consider a linear system of m equations for n unknowns x,,x n : For a given matrix A
Total 170. Name. Final Examination M340L-CS
1 10 2 10 3 15 4 5 5 10 6 10 7 20 8 10 9 20 10 25 11 10 12 10 13 15 Total 170 Final Examination Name M340L-CS 1. Use extra paper to determine your solutions then neatly transcribe them onto these sheets.
Linear Algebra and Matrices
Linear Algebra and Matrices 4 Overview In this chapter we studying true matrix operations, not element operations as was done in earlier chapters. Working with MAT- LAB functions should now be fairly routine.
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
Linear Algebra Practice Problems
Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a
Chap 3. Linear Algebra
Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions
EXAM. Exam 1. Math 5316, Fall December 2, 2012
EXAM Exam Math 536, Fall 22 December 2, 22 Write all of your answers on separate sheets of paper. You can keep the exam questions. This is a takehome exam, to be worked individually. You can use your notes.
Conceptual Questions for Review
Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.
Math 307 Learning Goals. March 23, 2010
Math 307 Learning Goals March 23, 2010 Course Description The course presents core concepts of linear algebra by focusing on applications in Science and Engineering. Examples of applications from recent
There are six more problems on the next two pages
Math 435 bg & bu: Topics in linear algebra Summer 25 Final exam Wed., 8/3/5. Justify all your work to receive full credit. Name:. Let A 3 2 5 Find a permutation matrix P, a lower triangular matrix L with
Chapter 3 Transformations
Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases
LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 12,
LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 12, 2000 74 6 Summary Here we summarize the most important information about theoretical and numerical linear algebra. MORALS OF THE STORY: I. Theoretically
Econ Slides from Lecture 7
Econ 205 Sobel Econ 205 - Slides from Lecture 7 Joel Sobel August 31, 2010 Linear Algebra: Main Theory A linear combination of a collection of vectors {x 1,..., x k } is a vector of the form k λ ix i for
ANSWERS. E k E 2 E 1 A = B
MATH 7- Final Exam Spring ANSWERS Essay Questions points Define an Elementary Matrix Display the fundamental matrix multiply equation which summarizes a sequence of swap, combination and multiply operations,
Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
LINEAR ALGEBRA KNOWLEDGE SURVEY
LINEAR ALGEBRA KNOWLEDGE SURVEY Instructions: This is a Knowledge Survey. For this assignment, I am only interested in your level of confidence about your ability to do the tasks on the following pages.
Quiz ) Locate your 1 st order neighbors. 1) Simplify. Name Hometown. Name Hometown. Name Hometown.
Quiz 1) Simplify 9999 999 9999 998 9999 998 2) Locate your 1 st order neighbors Name Hometown Me Name Hometown Name Hometown Name Hometown Solving Linear Algebraic Equa3ons Basic Concepts Here only real
Note: The command name is upper case in the description given by help, but must be lower case in actual use. And the backslash Anb is dierent when A i
MATLAB Tutorial You need a small number of basic commands to start using MATLAB. This short tutorial describes those fundamental commands. You need to create vectors and matrices, to change them, and to
Numerical 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
MATH 304 Linear Algebra Lecture 34: Review for Test 2.
MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1
Linear Algebra- Final Exam Review
Linear Algebra- Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.
Review of Some Concepts from Linear Algebra: Part 2
Review of Some Concepts from Linear Algebra: Part 2 Department of Mathematics Boise State University January 16, 2019 Math 566 Linear Algebra Review: Part 2 January 16, 2019 1 / 22 Vector spaces A set
1 Last time: least-squares problems
MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that
MAT Linear Algebra Collection of sample exams
MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system
NUMERICAL METHODS USING MATLAB
NUMERICAL METHODS USING MATLAB Dr John Penny George Lindfield Department of Mechanical Engineering, Aston University ELLIS HORWOOD NEW YORK LONDON TORONTO SYDNEY TOKYO SINGAPORE Preface 1 An introduction
Review problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them.
Prof A Suciu MTH U37 LINEAR ALGEBRA Spring 2005 PRACTICE FINAL EXAM Are the following vectors independent or dependent? If they are independent, say why If they are dependent, exhibit a linear dependence
Linear Algebra Practice Problems
Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,
Mathematical Operations with Arrays
Mathematical Operations with Arrays. Introduction. Addition and Subtraction. Array Multiplication. Array Division. Element by Element Operations. Using Arrays in Built in Math Function.6 Built in Functions
Math 307 Learning Goals
Math 307 Learning Goals May 14, 2018 Chapter 1 Linear Equations 1.1 Solving Linear Equations Write a system of linear equations using matrix notation. Use Gaussian elimination to bring a system of linear
Linear Algebra, part 3 QR and SVD
Linear Algebra, part 3 QR and SVD Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2012 Going back to least squares (Section 1.4 from Strang, now also see section 5.2). We
Midterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015
Midterm for Introduction to Numerical Analysis I, AMSC/CMSC 466, on 10/29/2015 The test lasts 1 hour and 15 minutes. No documents are allowed. The use of a calculator, cell phone or other equivalent electronic
Notes on Eigenvalues, Singular Values and QR
Notes on Eigenvalues, Singular Values and QR Michael Overton, Numerical Computing, Spring 2017 March 30, 2017 1 Eigenvalues Everyone who has studied linear algebra knows the definition: given a square
Chapter 8: Linear Algebraic Equations
Chapter 8: Linear Algebraic Equations Matrix Methods for Linear Equations Uniqueness and Existence of Solutions Under-Determined Systems Over-Determined Systems Linear Algebraic Equations For 2 Equations
MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.
MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v
Linear 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
Numerical Methods for Engineers. and Scientists. Applications using MATLAB. An Introduction with. Vish- Subramaniam. Third Edition. Amos Gilat.
Numerical Methods for Engineers An Introduction with and Scientists Applications using MATLAB Third Edition Amos Gilat Vish- Subramaniam Department of Mechanical Engineering The Ohio State University Wiley
Review of Linear Algebra
Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=
Matrix 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
Chapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1
Linear Algebra Primer
Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................
AM205: Assignment 2. i=1
AM05: Assignment Question 1 [10 points] (a) [4 points] For p 1, the p-norm for a vector x R n is defined as: ( n ) 1/p x p x i p ( ) i=1 This definition is in fact meaningful for p < 1 as well, although
Reduction to the associated homogeneous system via a particular solution
June PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 5) Linear Algebra This study guide describes briefly the course materials to be covered in MA 5. In order to be qualified for the credit, one
Example Linear Algebra Competency Test
Example Linear Algebra Competency Test The 4 questions below are a combination of True or False, multiple choice, fill in the blank, and computations involving matrices and vectors. In the latter case,
SOLVING 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
Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Exam 2 will be held on Tuesday, April 8, 7-8pm in 117 MacMillan What will be covered The exam will cover material from the lectures
Solution of Linear Equations
Solution of Linear Equations (Com S 477/577 Notes) Yan-Bin Jia Sep 7, 07 We have discussed general methods for solving arbitrary equations, and looked at the special class of polynomial equations A subclass
Inverses. Stephen Boyd. EE103 Stanford University. October 28, 2017
Inverses Stephen Boyd EE103 Stanford University October 28, 2017 Outline Left and right inverses Inverse Solving linear equations Examples Pseudo-inverse Left and right inverses 2 Left inverses a number
MATH 2050 Assignment 8 Fall [10] 1. Find the determinant by reducing to triangular form for the following matrices.
![MATH 2050 Assignment 8 Fall [10] 1. Find the determinant by reducing to triangular form for the following matrices.](/thumbs/95/123052681.jpg "MATH 2050 Assignment 8 Fall [10] 1. Find the determinant by reducing to triangular form for the following matrices.")
MATH 2050 Assignment 8 Fall 2016 [10] 1. Find the determinant by reducing to triangular form for the following matrices. 0 1 2 (a) A = 2 1 4. ANS: We perform the Gaussian Elimination on A by the following
MATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION
MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether
MATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
Properties of Linear Transformations from R n to R m
Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation
Problem 1: Solving a linear equation
Math 38 Practice Final Exam ANSWERS Page Problem : Solving a linear equation Given matrix A = 2 2 3 7 4 and vector y = 5 8 9. (a) Solve Ax = y (if the equation is consistent) and write the general solution
OR MSc Maths Revision Course
OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:30-12:30 Mathematics revision
Applied Linear Algebra
Applied Linear Algebra Peter J. Olver School of Mathematics University of Minnesota Minneapolis, MN 55455 olver@math.umn.edu http://www.math.umn.edu/ olver Chehrzad Shakiban Department of Mathematics University
Orthonormal Bases; Gram-Schmidt Process; QR-Decomposition
Orthonormal Bases; Gram-Schmidt Process; QR-Decomposition MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 205 Motivation When working with an inner product space, the most
Matrix decompositions
Matrix decompositions Zdeněk Dvořák May 19, 2015 Lemma 1 (Schur decomposition). If A is a symmetric real matrix, then there exists an orthogonal matrix Q and a diagonal matrix D such that A = QDQ T. The
FF505 Computational Science. Matrix Calculus. Marco Chiarandini
FF505 Computational Science Matrix Calculus Marco Chiarandini (marco@imada.sdu.dk) Department of Mathematics and Computer Science (IMADA) University of Southern Denmark Resume MATLAB, numerical computing
JUST THE MATHS SLIDES NUMBER 9.6. MATRICES 6 (Eigenvalues and eigenvectors) A.J.Hobson
JUST THE MATHS SLIDES NUMBER 96 MATRICES 6 (Eigenvalues and eigenvectors) by AJHobson 96 The statement of the problem 962 The solution of the problem UNIT 96 - MATRICES 6 EIGENVALUES AND EIGENVECTORS 96
Lecture 6. Numerical methods. Approximation of functions
Lecture 6 Numerical methods Approximation of functions Lecture 6 OUTLINE 1. Approximation and interpolation 2. Least-square method basis functions design matrix residual weighted least squares normal equation
Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the
Using MATLAB. Linear Algebra
Using MATLAB in Linear Algebra Edward Neuman Department of Mathematics Southern Illinois University at Carbondale One of the nice features of MATLAB is its ease of computations with vectors and matrices.
UNIT 6: The singular value decomposition.
UNIT 6: The singular value decomposition. María Barbero Liñán Universidad Carlos III de Madrid Bachelor in Statistics and Business Mathematical methods II 2011-2012 A square matrix is symmetric if A T
Math 224, Fall 2007 Exam 3 Thursday, December 6, 2007
Math 224, Fall 2007 Exam 3 Thursday, December 6, 2007 You have 1 hour and 20 minutes. No notes, books, or other references. You are permitted to use Maple during this exam, but you must start with a blank
Linear least squares problem: Example
Linear least squares problem: Example We want to determine n unknown parameters c,,c n using m measurements where m n Here always denotes the -norm v = (v + + v n) / Example problem Fit the experimental
Introduction to Matrices
POLS 704 Introduction to Matrices Introduction to Matrices. The Cast of Characters A matrix is a rectangular array (i.e., a table) of numbers. For example, 2 3 X 4 5 6 (4 3) 7 8 9 0 0 0 Thismatrix,with4rowsand3columns,isoforder
1. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal
. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal 3 9 matrix D such that A = P DP, for A =. 3 4 3 (a) P = 4, D =. 3 (b) P = 4, D =. (c) P = 4 8 4, D =. 3 (d) P
Matrix 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
COMP 558 lecture 18 Nov. 15, 2010
Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to
Linear Algebra. and
Instructions Please answer the six problems on your own paper. These are essay questions: you should write in complete sentences. 1. Are the two matrices 1 2 2 1 3 5 2 7 and 1 1 1 4 4 2 5 5 2 row equivalent?
+ MATRIX VARIABLES AND TWO DIMENSIONAL ARRAYS
+ MATRIX VARIABLES AND TWO DIMENSIONAL ARRAYS Matrices are organized rows and columns of numbers that mathematical operations can be performed on. MATLAB is organized around the rules of matrix operations.
3. Array and Matrix Operations
3. Array and Matrix Operations Almost anything you learned about in your linear algebra classmatlab has a command to do. Here is a brief summary of the most useful ones for physics. In MATLAB matrices
Linear Algebra, part 3. Going back to least squares. Mathematical Models, Analysis and Simulation = 0. a T 1 e. a T n e. Anna-Karin Tornberg
Linear Algebra, part 3 Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2010 Going back to least squares (Sections 1.7 and 2.3 from Strang). We know from before: The vector
Class notes: Approximation
Class notes: Approximation Introduction Vector spaces, linear independence, subspace The goal of Numerical Analysis is to compute approximations We want to approximate eg numbers in R or C vectors in R
A Review of Linear Algebra
A Review of Linear Algebra Mohammad Emtiyaz Khan CS,UBC A Review of Linear Algebra p.1/13 Basics Column vector x R n, Row vector x T, Matrix A R m n. Matrix Multiplication, (m n)(n k) m k, AB BA. Transpose
AMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 1: Course Overview & Matrix-Vector Multiplication Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 20 Outline 1 Course
CS 143 Linear Algebra Review
CS 143 Linear Algebra Review Stefan Roth September 29, 2003 Introductory Remarks This review does not aim at mathematical rigor very much, but instead at ease of understanding and conciseness. Please see
lecture 2 and 3: algorithms for linear algebra
lecture 2 and 3: algorithms for linear algebra STAT 545: Introduction to computational statistics Vinayak Rao Department of Statistics, Purdue University August 27, 2018 Solving a system of linear equations
Math 310 Final Exam Solutions
Math 3 Final Exam Solutions. ( pts) Consider the system of equations Ax = b where: A, b (a) Compute deta. Is A singular or nonsingular? (b) Compute A, if possible. (c) Write the row reduced echelon form
Linear Algebra Review
Linear Algebra Review CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Doug James (and Justin Solomon) CS 205A: Mathematical Methods Linear Algebra Review 1 / 16 Midterm Exam Tuesday Feb
Chapter 9: Gaussian Elimination
Uchechukwu Ofoegbu Temple University Chapter 9: Gaussian Elimination Graphical Method The solution of a small set of simultaneous equations, can be obtained by graphing them and determining the location
SECTION 2: VECTORS AND MATRICES. ENGR 112 Introduction to Engineering Computing
SECTION 2: VECTORS AND MATRICES ENGR 112 Introduction to Engineering Computing 2 Vectors and Matrices The MAT in MATLAB 3 MATLAB The MATrix (not MAThematics) LABoratory MATLAB assumes all numeric variables
LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS
LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,
Math 265 Linear Algebra Sample Spring 2002., rref (A) =
Math 265 Linear Algebra Sample Spring 22. It is given that A = rref (A T )= 2 3 5 3 2 6, rref (A) = 2 3 and (a) Find the rank of A. (b) Find the nullityof A. (c) Find a basis for the column space of A.
5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.
Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the
SOLUTIONS: ASSIGNMENT Use Gaussian elimination to find the determinant of the matrix. = det. = det = 1 ( 2) 3 6 = 36. v 4.
SOLUTIONS: ASSIGNMENT 9 66 Use Gaussian elimination to find the determinant of the matrix det 1 1 4 4 1 1 1 1 8 8 = det = det 0 7 9 0 0 0 6 = 1 ( ) 3 6 = 36 = det = det 0 0 6 1 0 0 0 6 61 Consider a 4
lecture 3 and 4: algorithms for linear algebra
lecture 3 and 4: algorithms for linear algebra STAT 545: Introduction to computational statistics Vinayak Rao Department of Statistics, Purdue University August 30, 2016 Solving a system of linear equations
No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question.
Math 304 Final Exam (May 8) Spring 206 No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question. Name: Section: Question Points
EE5120 Linear Algebra: Tutorial 1, July-Dec Solve the following sets of linear equations using Gaussian elimination (a)
EE5120 Linear Algebra: Tutorial 1, July-Dec 2017-18 1. Solve the following sets of linear equations using Gaussian elimination (a) 2x 1 2x 2 3x 3 = 2 3x 1 3x 2 2x 3 + 5x 4 = 7 x 1 x 2 2x 3 x 4 = 3 (b)