Row Reduction
|
|
- Bathsheba Barker
- 6 years ago
- Views:
Transcription
1 Row Reduction Row reduction (or Gaussian elimination) is the process of using row operations to reduce a matrix to row reduced echelon form This procedure is used to solve systems of linear equations, invert matrices, compute determinants, and do many other things There are three kinds of row operations (Actually, there is some redundancy here you can get away with two of them) (a) You may swap two rows Here is a swap of rows 2 and 3 I ll denote it by r 2 r (b) You may multiply (or divide) a row by a nonzero number Here is row 2 multiplied by π I ll denote this operation by r 2 πr π 5π 6π (c) You may add a multiple of a row to another row I ll subtract 4 times row 1 from row 2 Notation: r 2 r 2 4r Notice that row 1 was not affected by this operation Likewise, if you do r 17 r 17 56r 31, row 17 changes and row 31 does not Example In each case, tell whether the operation is a valid row operation If it is, say what it does (in words) (a) r 5 r 3 This operation swaps row 5 and row 3 (b) r 6 r 6 7 This isn t a valid row operation You can t add or subtract a number from the elements in a row (c) r 3 r 3 +πr 17 This adds pi times row 17 to row 3 (and replaces row 3 with the result) Row 17 is not changed (d) r 6 5r 6 +11r 2 This isn t a valid row operation, though you could accomplish it using two row operations: First, multiply row 6 by 5; next, add 11 times row 2 to the new row 6 1
2 (e) r 3 r 3 +r 4 and r 4 r 4 +r 3 This is two row operations, not one The only row operation that changes two rows at once is swapping two rows Matrices can be used to represent systems of linear equations Row operations are intended to mimic the algebraic operations you use to solve a system Row-reduced echelon form corresponds to the solved form of a system A matrix is in row reduced echelon form if the following conditions are satisfied: (a) The first nonzero element in each row (if any) is a 1 (a leading coefficient) (b) Each leading coefficient is the only nonzero element in its column (c) All the all-zero rows (if any) are at the bottom of the matrix (d) The leading coefficients form a stairstep pattern from northwest to southeast: In this matrix, the leading coefficients are in positions (1,2), (2,4), (3,5), Here some more matrices in row-reduced echelon form Notice the stairstep pattern made by the leading coefficients The * s indicate that the numbers in those positions can be anything Example (Row-reduced echelon form) This matrix is not in row reduced echelon form: The first nonzero element in row 2 is a 7, rather than a 1 This matrix is not in row reduced echelon form: The leading coefficient in row 3 is not the only nonzero element in its column This matrix is not in row reduced echelon form:
3 There is an all-zero row above a nonzero row This matrix is not in row reduced echelon form: The leading coefficient in row 2 is not the only nonzero element in its column This matrix is not in row reduced echelon form: The leading coefficients do not form a stairstep pattern from northwest to southeast The following matrices are in row-reduced echelon form You should go through the definition and check that all the properties are satisfied , , The last one is called the 3 3 identity matrix, because it acts like an identity element for matrix multiplication Row reduction is the process of using row operations to transform a matrix into a row reduced echelon matrix As the algorithm proceeds, you move in stairstep fashion through different positions in the matrix In the description below, when I say that the current position is (i,j), I mean that your current location is in row i and column j The current position refers to a location, not the element at that location (which I ll sometimes call the current element) The current row means the row of the matrix containing the current position and the current column means the column of the matrix containing the current position Some notes: 1 There are many ways to arrange the algorithm For instance, another approach gives the LUdecomposition of a matrix 2 Trying to learn to row reduce by following the steps below is pretty tedious, and most people will want to learn by doing examples The steps are there so that, as you re learning to do this, you have some idea of what to do if you get stuck 3 There are shortcuts you can take which don t follow the steps below But you can get very confused if you focus on shortcuts before you ve really absorbed the sense of the algorithm The test is whether you can reliably and accurately row reduce a matrix! 4 There s no point in doing row reductions by hand forever, and for larger matrices (as would occur in real world applications) it s impractical At some point, you ll use a computer However, I think it s importnat to do enough examples by hand that you understand the algorithm Algorithm: Row Reducing a Matrix Step 1 Start with the current position at (1,1) Step 2 Test the element at the current position If it s nonzero, go to Step 2(a); if it s 0, go to Step 2(b) Step 2(a) If the element at the current position is nonzero, then: 3
4 1 (i) Divide all the elements in the current row by the current element This makes the current element (ii) Add or subtract multiples of the current row from the other rows in the matrix so that all the elements in the current column (except for the current element) are 0 (iii) Move the current position to the next row (down) and the next column (right) If doing either of these things would take you out of the matrix, then stop: The matrix is in row-reduced echelon form Otherwise, return to the beginning of Step 2 Step 2(b) If the element at the current position is 0, then look at the elements in the current column below the current element There are two possibilities (i) If all the elements below the current element are 0, then move the current position to the next column (in the same row) If doing this would take you out of the matrix, then stop: The matrix is in row-reduced echelon form Otherwise, return to the beginning of Step 2 (ii) If some element below the current element is nonzero, then swap the current row and the row containing the nonzero element Then return to the beginning of Step 2 Example In each of the following cases, assume that the current position is (1,1) (a) The element in the current position is nonzero So I divide the first row by 2: r r Next, I subtract 3 times row 1 from row 2, and I add row 1 to row 3 This makes the other elements in the first column equal to r r 2 3r 1 r r 3 +r Finally, I move the current position to the next row and the next column and return to the start of Step 2: (b) The element in the current position is 0 I look below it and see a nonzero element in the same column in row 3 So I swap row 1 and row 3; the current position remains the same, and I return to the start of Step r 1 r
5 (c) The element in the current position is 0 There are no nonzero elements below it in the same column I don t perform any row operations; I just move the current position to the next column (in the same row) and return to the start of Step 2: There are two questions which arise with this algorithm: Why does the algorithm terminate? (Could you ever get stuck and never finish?) When the algorithm does terminate, why is the final matrix in row-reduced echelon form? The first question is easy to answer As you execute the algorithm, the current position moves through the matrix in stairstep fashion: The cases in Step 2 cover all the possibilities, and in each case, you perform a finite number of row operations (no larger than the number of rows in the matrix, plus one) before you move the current position Since you re always moving the current position to the right or to the right and down, and since the matrix has only finitely many rows and columns, you must eventually reach the edge of the matrix and the algorithm will terminate As for the second question, I ll give an informal argument using the matrix with the stairstep path pictured above First, if you moved the current position down and to the right, the previous current element was a 1, and every other element in its column must be 0 In the matrix with the stairstep path I gave above, this means that each spot where a curved arrow starts must be a 1, and all the other elements in the column with a 1 must be 0 Hence, the matrix must look like this: (The s stand for elements which I don t know) Next, notice that if you moved the current position to the right (but not down), then the previous current element and everything below it must have been 0 In terms of the picture, every spot where a right 5
6 arrow starts must be a 0, and all the elements below it must be 0 Now I know that the matrix looks like this: Notice that this matrix is in row-reduced echelon form Row reduction is a key algorithm in linear algebra, and you should work through enough examples so that you understand how it works Once you understand the algorithm, for more complex problems you may use a computer to carry out the row reduction Example Row reduce the following matrix with real entries: r 1 r 2 r 1 r 1 +3r r 3 r 3 r 2 r 2 r 2 3r r 2 r 2 11r r 3 r 3 r 3 r 3 2r Noticethat thefirst operation swapping thefirst two rows isn t really in accord with thealgorithm So I cheated a little, but not too much: It accomplished the aim of creating a 1 in the (1,1) position, and following the algorithm by dividing the given first row by 3 would have created a lot of ugly fractions Example (Row reduction over Z n ) Row reduce the following matrix over Z 3 : In the computation that follows, remember that in Z r 3 r 3 r 1 r 1 r 1 r = 0, 1 = 2, and 2 2 = r 2 r 2 +r 1 r 3 2r r 3 r 3 r 2 r 1 r 1 +r 3
7 Example Row reduction is used to solve systems of linear equations Consider the following system of linear equations over R: 2x + 3y = 5 x + 2y = 4 Form the auxiliary matrix of coefficients and row reduce: The final matrix yields the equations Thus, x = 22 and y = x = 22 y = Example It s possible for a system of linear equations to have no solutions Such a system is said to be inconsistent Consider the following system of equations over R: 3x + y + 3z = 2 x + 2z = 3 2x + y + z = 4 Form the coefficient matrix and row reduce: The corresponding equations are x + 2z = 0 y 3z = 0 0 = 1 The last equation says 0 = 1 Hence, the system has no solutions Example Consider the case of a system over the real numbers with matrix Suppose the variables are a, b, c, d, and e The corresponding equations are a + 2b + d = 3 c + d 2e = 1 a and c correspond to leading coefficients b, d, and e are called free variables In this case, you get a parametrized solution: a = 2b d+3 c = d+2e+1 7
8 Each assignment of numbers to the parameters b, d, and e produces a solution For example, if b = 1, d = 0, and e = 2, a = ( 2) = 1, c = = 5 The solution (in this case) is (a,b,c,d,e) = (1,1,5,0,2) Since you can assign any real number to each of b, d, and e, there are infinitely many solutions Example (Solving a system of equations over Z n ) Solve the following system over Z 5 : r 3 r 3 2r 1 r 3 r 3 3r 2 The corresponding equations are w + x + y + 2z = 1 2x + 2y + z = 0 2w + 2y + z = 1 r 2 3r r 3 3r 3 w+4z = 1, x+4z = 3, y +4z = 2 Set z = t and solve for w, x, and y (remembering that 4 = 1): w = t+1, x = t+3, y = t+2, z = t r 1 r 1 r 2 r 2 r 2 r 3 Remark A system of linear equations over the real numbers can have no solutions, a unique solution, or infinitely many solutions (That is, such a system can t have exactly 3 solutions) On the other hand, if n is prime, a system of linear equations over Z n will have n k solutions, for k 0, or no solutions For example, a system over Z 5 can have no solutions, one solution, 5 solutions, 25 solutions, Example (Inverting a matrix) I ll discuss matrix inversion in more detail later However, it s easy to describe how row reduction provides a systematic way to find the inverse of a matrix To invert a matrix, adjoin a copy of the identity matrix and row reduce the augmented matrix When the block corresponding the original matrix becomes the identity, the block corresponding to the identity will have become the inverse For example, suppose you want to invert Adjoin a copy of the identity matrix, then do the following row reduction:
9 The inverse is Of course, there is a formula for inverting a 2 2 matrix But this procedure works with (square) matrices of any size To explain why this algorithm works, I ll need to examine the relationship between row operations and inverses more closely Example Find the line of intersection of the planes x+2y z = 4 and x y z = 3 Think of the equations as a system Write down the coefficient matrix and row reduce: [ The solution is x = z , y = 1 The parametric solution is 3 x = t+ 10 3, y = 1 3, z = t These are the parametric equations of the line of intersection ] The solution of simultaneous linear equations dates back 2000 years to the Jiuzhang Suanshu, a collection of problems compiled in China The systematic algebraic process of eliminating variables seems to be due to Isaac Newton Matrix notation developed in the second half of the 1800 s; what we call Gaussian elimination, applied to matrices, was developed in the first half of the 1900 s by various mathematicians Grcar[1] contains a nice historical account [1] Joseph Grcar, Mathematics of Gaussian elimination, Notices of the American Mathematical Society, 58(6)(2011), c 2015 by Bruce Ikenaga 9
Chapter 4. Solving Systems of Equations. Chapter 4
Solving Systems of Equations 3 Scenarios for Solutions There are three general situations we may find ourselves in when attempting to solve systems of equations: 1 The system could have one unique solution.
More informationLECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS
LECTURES 4/5: SYSTEMS OF LINEAR EQUATIONS MA1111: LINEAR ALGEBRA I, MICHAELMAS 2016 1 Linear equations We now switch gears to discuss the topic of solving linear equations, and more interestingly, systems
More informationLinear Algebra Basics
Linear Algebra Basics For the next chapter, understanding matrices and how to do computations with them will be crucial. So, a good first place to start is perhaps What is a matrix? A matrix A is an array
More information4 Elementary matrices, continued
4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. If you check the previous examples, you ll find that these matrices are constructed
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 informationSection 1.2. Row Reduction and Echelon Forms
Section 1.2 Row Reduction and Echelon Forms Row Echelon Form Let s come up with an algorithm for turning an arbitrary matrix into a solved matrix. What do we mean by solved? A matrix is in row echelon
More informationInverses and Elementary Matrices
Inverses and Elementary Matrices 1-12-2013 Matrix inversion gives a method for solving some systems of equations Suppose 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 a n1 x
More informationRow Reduced Echelon Form
Math 40 Row Reduced Echelon Form Solving systems of linear equations lies at the heart of linear algebra. In high school we learn to solve systems in or variables using elimination and substitution of
More informationMatrices and RRE Form
Matrices and RRE Form Notation R is the real numbers, C is the complex numbers (we will only consider complex numbers towards the end of the course) is read as an element of For instance, x R means that
More informationExample: 2x y + 3z = 1 5y 6z = 0 x + 4z = 7. Definition: Elementary Row Operations. Example: Type I swap rows 1 and 3
Math 0 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 and substitution
More informationSolving Linear Systems Using Gaussian Elimination
Solving Linear Systems Using Gaussian Elimination DEFINITION: A linear equation in the variables x 1,..., x n is an equation that can be written in the form a 1 x 1 +...+a n x n = b, where a 1,...,a n
More informationSection 6.2 Larger Systems of Linear Equations
Section 6.2 Larger Systems of Linear Equations Gaussian Elimination In general, to solve a system of linear equations using its augmented matrix, we use elementary row operations to arrive at a matrix
More informationGauss-Jordan Row Reduction and Reduced Row Echelon Form
Gauss-Jordan Row Reduction and Reduced Row Echelon Form If we put the augmented matrix of a linear system in reduced row-echelon form, then we don t need to back-substitute to solve the system. To put
More information4 Elementary matrices, continued
4 Elementary matrices, continued We have identified 3 types of row operations and their corresponding elementary matrices. To repeat the recipe: These matrices are constructed by performing the given row
More informationMath 308 Midterm Answers and Comments July 18, Part A. Short answer questions
Math 308 Midterm Answers and Comments July 18, 2011 Part A. Short answer questions (1) Compute the determinant of the matrix a 3 3 1 1 2. 1 a 3 The determinant is 2a 2 12. Comments: Everyone seemed to
More informationFinal Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2
Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch
More informationMATH 54 - WORKSHEET 1 MONDAY 6/22
MATH 54 - WORKSHEET 1 MONDAY 6/22 Row Operations: (1 (Replacement Add a multiple of one row to another row. (2 (Interchange Swap two rows. (3 (Scaling Multiply an entire row by a nonzero constant. A matrix
More information4.3 Row operations. As we have seen in Section 4.1 we can simplify a system of equations by either:
4.3 Row operations As we have seen in Section 4.1 we can simplify a system of equations by either: 1. Swapping the order of the equations around. For example: can become 3x 1 + 7x 2 = 9 x 1 2x 1 = 2 x
More informationMarch 19 - Solving Linear Systems
March 19 - Solving Linear Systems Welcome to linear algebra! Linear algebra is the study of vectors, vector spaces, and maps between vector spaces. It has applications across data analysis, computer graphics,
More informationLinear Algebra Handout
Linear Algebra Handout References Some material and suggested problems are taken from Fundamentals of Matrix Algebra by Gregory Hartman, which can be found here: http://www.vmi.edu/content.aspx?id=779979.
More informationRecall, we solved the system below in a previous section. Here, we learn another method. x + 4y = 14 5x + 3y = 2
We will learn how to use a matrix to solve a system of equations. College algebra Class notes Matrices and Systems of Equations (section 6.) Recall, we solved the system below in a previous section. Here,
More informationGroups. s t or s t or even st rather than f(s,t).
Groups Definition. A binary operation on a set S is a function which takes a pair of elements s,t S and produces another element f(s,t) S. That is, a binary operation is a function f : S S S. Binary operations
More informationRow Reduction and Echelon Forms
Row Reduction and Echelon Forms 1 / 29 Key Concepts row echelon form, reduced row echelon form pivot position, pivot, pivot column basic variable, free variable general solution, parametric solution existence
More information22A-2 SUMMER 2014 LECTURE 5
A- SUMMER 0 LECTURE 5 NATHANIEL GALLUP Agenda Elimination to the identity matrix Inverse matrices LU factorization Elimination to the identity matrix Previously, we have used elimination to get a system
More informationMatrices. 1 a a2 1 b b 2 1 c c π
Matrices 2-3-207 A matrix is a rectangular array of numbers: 2 π 4 37 42 0 3 a a2 b b 2 c c 2 Actually, the entries can be more general than numbers, but you can think of the entries as numbers to start
More information[Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty.]
Math 43 Review Notes [Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty Dot Product If v (v, v, v 3 and w (w, w, w 3, then the
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 informationChapter 1. Linear Equations
Chapter 1. Linear Equations We ll start our study of linear algebra with linear equations. Lost of parts of mathematics rose out of trying to understand the solutions of different types of equations. Linear
More informationDesigning Information Devices and Systems I Fall 2018 Lecture Notes Note Introduction to Linear Algebra the EECS Way
EECS 16A Designing Information Devices and Systems I Fall 018 Lecture Notes Note 1 1.1 Introduction to Linear Algebra the EECS Way In this note, we will teach the basics of linear algebra and relate it
More informationLecture 1 Systems of Linear Equations and Matrices
Lecture 1 Systems of Linear Equations and Matrices Math 19620 Outline of Course Linear Equations and Matrices Linear Transformations, Inverses Bases, Linear Independence, Subspaces Abstract Vector Spaces
More informationPre-Calculus I. For example, the system. x y 2 z. may be represented by the augmented matrix
Pre-Calculus 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 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 information3 Fields, Elementary Matrices and Calculating Inverses
3 Fields, Elementary Matrices and Calculating Inverses 3. Fields So far we have worked with matrices whose entries are real numbers (and systems of equations whose coefficients and solutions are real numbers).
More information2 Systems of Linear Equations
2 Systems of Linear Equations A system of equations of the form or is called a system of linear equations. x + 2y = 7 2x y = 4 5p 6q + r = 4 2p + 3q 5r = 7 6p q + 4r = 2 Definition. An equation involving
More informationis a 3 4 matrix. It has 3 rows and 4 columns. The first row is the horizontal row [ ]
Matrices: Definition: An m n matrix, A m n is a rectangular array of numbers with m rows and n columns: a, a, a,n a, a, a,n A m,n =...... a m, a m, a m,n Each a i,j is the entry at the i th row, j th column.
More informationLinear Independence. e 1 = (1,0,0,...0) e 2 = 0,1,0,...0). e n = (0,0,0,...1) a 1 e 1 = (a 1,0,0,...0) a 2 e 2 = 0,a 2,0,...0)
Linear Independence --17 Definition Let V be a vector space over a field F, and let S V The set S is linearly independent if v 1,,v n S,,, F, and v 1 + + v n = implies = = = An equation like the one above
More informationRings If R is a commutative ring, a zero divisor is a nonzero element x such that xy = 0 for some nonzero element y R.
Rings 10-26-2008 A ring is an abelian group R with binary operation + ( addition ), together with a second binary operation ( multiplication ). Multiplication must be associative, and must distribute over
More informationLecture 2 Systems of Linear Equations and Matrices, Continued
Lecture 2 Systems of Linear Equations and Matrices, Continued Math 19620 Outline of Lecture Algorithm for putting a matrix in row reduced echelon form - i.e. Gauss-Jordan Elimination Number of Solutions
More informationLast Time. x + 3y = 6 x + 2y = 1. x + 3y = 6 y = 1. 2x + 4y = 8 x 2y = 1. x + 3y = 6 2x y = 7. Lecture 2
January 9 Last Time 1. Last time we ended with saying that the following four systems are equivalent in the sense that we can move from one system to the other by a special move we discussed. (a) (b) (c)
More informationRelationships Between Planes
Relationships Between Planes Definition: consistent (system of equations) A system of equations is consistent if there exists one (or more than one) solution that satisfies the system. System 1: {, System
More informationLinear Algebra for Beginners Open Doors to Great Careers. Richard Han
Linear Algebra for Beginners Open Doors to Great Careers Richard Han Copyright 2018 Richard Han All rights reserved. CONTENTS PREFACE... 7 1 - INTRODUCTION... 8 2 SOLVING SYSTEMS OF LINEAR EQUATIONS...
More informationDesigning Information Devices and Systems I Spring 2018 Lecture Notes Note Introduction to Linear Algebra the EECS Way
EECS 16A Designing Information Devices and Systems I Spring 018 Lecture Notes Note 1 1.1 Introduction to Linear Algebra the EECS Way In this note, we will teach the basics of linear algebra and relate
More informationMatrix Inverses. November 19, 2014
Matrix Inverses November 9, 204 22 The Inverse of a Matrix Now that we have discussed how to multiply two matrices, we can finally have a proper discussion of what we mean by the expression A for a matrix
More information3. Replace any row by the sum of that row and a constant multiple of any other row.
Section. Solution of Linear Systems by Gauss-Jordan Method A matrix is an ordered rectangular array of numbers, letters, symbols or algebraic expressions. A matrix with m rows and n columns has size or
More informationGaussian elimination
Gaussian elimination October 14, 2013 Contents 1 Introduction 1 2 Some definitions and examples 2 3 Elementary row operations 7 4 Gaussian elimination 11 5 Rank and row reduction 16 6 Some computational
More informationLecture 3: Gaussian Elimination, continued. Lecture 3: Gaussian Elimination, continued
Definition The process of solving a system of linear equations by converting the system to an augmented matrix is called Gaussian Elimination. The general strategy is as follows: Convert the system of
More informationElementary matrices, continued. To summarize, we have identified 3 types of row operations and their corresponding
Elementary matrices, continued To summarize, we have identified 3 types of row operations and their corresponding elementary matrices. If you check the previous examples, you ll find that these matrices
More informationLinear Algebra, Summer 2011, pt. 2
Linear Algebra, Summer 2, pt. 2 June 8, 2 Contents Inverses. 2 Vector Spaces. 3 2. Examples of vector spaces..................... 3 2.2 The column space......................... 6 2.3 The null space...........................
More informationLecture 7: Introduction to linear systems
Lecture 7: Introduction to linear systems Two pictures of linear systems Consider the following system of linear algebraic equations { x 2y =, 2x+y = 7. (.) Note that it is a linear system with two unknowns
More informationNumerical Methods Lecture 2 Simultaneous Equations
Numerical Methods Lecture 2 Simultaneous Equations Topics: matrix operations solving systems of equations pages 58-62 are a repeat of matrix notes. New material begins on page 63. Matrix operations: Mathcad
More informationTopic 14 Notes Jeremy Orloff
Topic 4 Notes Jeremy Orloff 4 Row reduction and subspaces 4. Goals. Be able to put a matrix into row reduced echelon form (RREF) using elementary row operations.. Know the definitions of null and column
More informationNumber of solutions of a system
Roberto s Notes on Linear Algebra Chapter 3: Linear systems and matrices Section 7 Number of solutions of a system What you need to know already: How to solve a linear system by using Gauss- Jordan elimination.
More informationMAC Module 1 Systems of Linear Equations and Matrices I
MAC 2103 Module 1 Systems of Linear Equations and Matrices I 1 Learning Objectives Upon completing this module, you should be able to: 1. Represent a system of linear equations as an augmented matrix.
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 informationMidterm 1 Review. Written by Victoria Kala SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015
Midterm 1 Review Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015 Summary This Midterm Review contains notes on sections 1.1 1.5 and 1.7 in your
More informationEigenvalues by row operations
Eigenvalues by row operations Barton L. Willis Department of Mathematics University of Nebraska at Kearney Kearney, Nebraska 68849 May, 5 Introduction There is a wonderful article, Down with Determinants!,
More informationSection Gaussian Elimination
Section. - Gaussian Elimination A matrix is said to be in row echelon form (REF) if it has the following properties:. The first nonzero entry in any row is a. We call this a leading one or pivot one..
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.1 SYSTEMS OF LINEAR EQUATIONS LINEAR EQUATION,, 1 n A linear equation in the variables equation that can be written in the form a a a b 1 1 2 2 n n a a is an where
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 informationMethods for Solving Linear Systems Part 2
Methods for Solving Linear Systems Part 2 We have studied the properties of matrices and found out that there are more ways that we can solve Linear Systems. In Section 7.3, we learned that we can use
More informationThe Gauss-Jordan Elimination Algorithm
The Gauss-Jordan Elimination Algorithm Solving Systems of Real Linear Equations A. Havens Department of Mathematics University of Massachusetts, Amherst January 24, 2018 Outline 1 Definitions Echelon Forms
More informationLecture 2e Row Echelon Form (pages 73-74)
Lecture 2e Row Echelon Form (pages 73-74) At the end of Lecture 2a I said that we would develop an algorithm for solving a system of linear equations, and now that we have our matrix notation, we can proceed
More informationSection 1.1 System of Linear Equations. Dr. Abdulla Eid. College of Science. MATHS 211: Linear Algebra
Section 1.1 System of Linear Equations College of Science MATHS 211: Linear Algebra (University of Bahrain) Linear System 1 / 33 Goals:. 1 Define system of linear equations and their solutions. 2 To represent
More informationMath 1021, Linear Algebra 1. Section: A at 10am, B at 2:30pm
Math 1021, Linear Algebra 1. Section: A at 10am, B at 2:30pm All course information is available on Moodle. Text: Nicholson, Linear algebra with applications, 7th edition. We shall cover Chapters 1,2,3,4,5:
More informationv = ( 2)
Chapter : Introduction to Vectors.. Vectors and linear combinations Let s begin by saying what vectors are: They are lists of numbers. If there are numbers in the list, there is a natural correspondence
More informationLecture 9: Elementary Matrices
Lecture 9: Elementary Matrices Review of Row Reduced Echelon Form Consider the matrix A and the vector b defined as follows: 1 2 1 A b 3 8 5 A common technique to solve linear equations of the form Ax
More informationMath Week 1 notes
Math 2270-004 Week notes We will not necessarily finish the material from a given day's notes on that day. Or on an amazing day we may get farther than I've predicted. We may also add or subtract some
More informationTopic 15 Notes Jeremy Orloff
Topic 5 Notes Jeremy Orloff 5 Transpose, Inverse, Determinant 5. Goals. Know the definition and be able to compute the inverse of any square matrix using row operations. 2. Know the properties of inverses.
More information1 - Systems of Linear Equations
1 - Systems of Linear Equations 1.1 Introduction to Systems of Linear Equations Almost every problem in linear algebra will involve solving a system of equations. ü LINEAR EQUATIONS IN n VARIABLES We are
More informationchapter 12 MORE MATRIX ALGEBRA 12.1 Systems of Linear Equations GOALS
chapter MORE MATRIX ALGEBRA GOALS In Chapter we studied matrix operations and the algebra of sets and logic. We also made note of the strong resemblance of matrix algebra to elementary algebra. The reader
More informationEBG # 3 Using Gaussian Elimination (Echelon Form) Gaussian Elimination: 0s below the main diagonal
EBG # 3 Using Gaussian Elimination (Echelon Form) Gaussian Elimination: 0s below the main diagonal [ x y Augmented matrix: 1 1 17 4 2 48 (Replacement) Replace a row by the sum of itself and a multiple
More informationFor updated version of this document see LINEAR EQUATION. Chickens and Rabbits
LINEAR EQUATION Chickens and Rabbits A farm has chickens and rabbits. The farmer counts 26 heads and 82 feet. How many chickens and rabbits are in the farm? Trial and error. Before learning algebra, you
More informationLecture 4: Gaussian Elimination and Homogeneous Equations
Lecture 4: Gaussian Elimination and Homogeneous Equations Reduced Row Echelon Form An augmented matrix associated to a system of linear equations is said to be in Reduced Row Echelon Form (RREF) if the
More informationChapter 1: Linear Equations
Chapter : Linear Equations (Last Updated: September, 6) The material for these notes is derived primarily from Linear Algebra and its applications by David Lay (4ed).. Systems of Linear Equations Before
More informationMATH 320, WEEK 6: Linear Systems, Gaussian Elimination, Coefficient Matrices
MATH 320, WEEK 6: Linear Systems, Gaussian Elimination, Coefficient Matrices We will now switch gears and focus on a branch of mathematics known as linear algebra. There are a few notes worth making before
More informationMAC1105-College Algebra. Chapter 5-Systems of Equations & Matrices
MAC05-College Algebra Chapter 5-Systems of Equations & Matrices 5. Systems of Equations in Two Variables Solving Systems of Two Linear Equations/ Two-Variable Linear Equations A system of equations is
More informationMATH 310, REVIEW SHEET 2
MATH 310, REVIEW SHEET 2 These notes are a very short summary of the key topics in the book (and follow the book pretty closely). You should be familiar with everything on here, but it s not comprehensive,
More information1111: Linear Algebra I
1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 5 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 5 1 / 12 Systems of linear equations Geometrically, we are quite used to the fact
More informationMath101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:
Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know the
More informationMATH240: Linear Algebra Review for exam #1 6/10/2015 Page 1
MATH24: Linear Algebra Review for exam # 6//25 Page No review sheet can cover everything that is potentially fair game for an exam, but I tried to hit on all of the topics with these questions, as well
More informationChapter 1: Linear Equations
Chapter : Linear Equations (Last Updated: September, 7) The material for these notes is derived primarily from Linear Algebra and its applications by David Lay (4ed).. Systems of Linear Equations Before
More informationMI 4 Mathematical Induction Name. Mathematical Induction
Mathematical Induction It turns out that the most efficient solution to the Towers of Hanoi problem with n disks takes n 1 moves. If this isn t the formula you determined, make sure to check your data
More informationLinear Methods (Math 211) - Lecture 2
Linear Methods (Math 211) - Lecture 2 David Roe September 11, 2013 Recall Last time: Linear Systems Matrices Geometric Perspective Parametric Form Today 1 Row Echelon Form 2 Rank 3 Gaussian Elimination
More informationLinear 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
More informationCommutative Rings and Fields
Commutative Rings and Fields 1-22-2017 Different algebraic systems are used in linear algebra. The most important are commutative rings with identity and fields. Definition. A ring is a set R with two
More informationCHAPTER 8: MATRICES and DETERMINANTS
(Section 8.1: Matrices and Determinants) 8.01 CHAPTER 8: MATRICES and DETERMINANTS The material in this chapter will be covered in your Linear Algebra class (Math 254 at Mesa). SECTION 8.1: MATRICES and
More informationVolume in n Dimensions
Volume in n Dimensions MA 305 Kurt Bryan Introduction You ve seen that if we have two vectors v and w in two dimensions then the area spanned by these vectors can be computed as v w = v 1 w 2 v 2 w 1 (where
More informationLecture 12: Solving Systems of Linear Equations by Gaussian Elimination
Lecture 12: Solving Systems of Linear Equations by Gaussian Elimination Winfried Just, Ohio University September 22, 2017 Review: The coefficient matrix Consider a system of m linear equations in n variables.
More informationLinear equations The first case of a linear equation you learn is in one variable, for instance:
Math 52 0 - Linear algebra, Spring Semester 2012-2013 Dan Abramovich Linear equations The first case of a linear equation you learn is in one variable, for instance: 2x = 5. We learned in school that this
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 informationCH 54 PREPARING FOR THE QUADRATIC FORMULA
1 CH 54 PREPARING FOR THE QUADRATIC FORMULA Introduction W e re pretty good by now at solving equations like (3x 4) + 8 10(x + 1), and we ve had a whole boatload of word problems which can be solved by
More informationDeterminants - Uniqueness and Properties
Determinants - Uniqueness and Properties 2-2-2008 In order to show that there s only one determinant function on M(n, R), I m going to derive another formula for the determinant It involves permutations
More information1300 Linear Algebra and Vector Geometry
1300 Linear Algebra and Vector Geometry R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca May-June 2017 Introduction: linear equations Read 1.1 (in the text that is!) Go to course, class webpages.
More informationAnnouncements Wednesday, August 30
Announcements Wednesday, August 30 WeBWorK due on Friday at 11:59pm. The first quiz is on Friday, during recitation. It covers through Monday s material. Quizzes mostly test your understanding of the homework.
More informationLECTURES 14/15: LINEAR INDEPENDENCE AND BASES
LECTURES 14/15: LINEAR INDEPENDENCE AND BASES MA1111: LINEAR ALGEBRA I, MICHAELMAS 2016 1. Linear Independence We have seen in examples of span sets of vectors that sometimes adding additional vectors
More informationA Quick Introduction to Row Reduction
A Quick Introduction to Row Reduction Gaussian Elimination Suppose we are asked to solve the system of equations 4x + 5x 2 + 6x 3 = 7 6x + 7x 2 + 8x 3 = 9. That is, we want to find all values of x, x 2
More informationAlgebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More informationNotes on Row Reduction
Notes on Row Reduction Francis J. Narcowich Department of Mathematics Texas A&M University September The Row-Reduction Algorithm The row-reduced form of a matrix contains a great deal of information, both
More informationDifferential Equations
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More information(I.D) Solving Linear Systems via Row-Reduction
(I.D) Solving Linear Systems via Row-Reduction Turning to the promised algorithmic approach to Gaussian elimination, we say an m n matrix M is in reduced-row echelon form if: the first nonzero entry of
More information