Homework 1.1 and 1.2 WITH SOLUTIONS
|
|
- Carmella Baldwin
- 5 years ago
- Views:
Transcription
1 Math 220 Linear Algebra (Spring 2018) Homework 1.1 and 1.2 WITH SOLUTIONS Due Thursday January 25 These will be graded in detail and will count as two (TA graded) homeworks. Be sure to start each of these problems on a new sheet of paper, summarize the problem, and explain what you are doing so that a classmate who is struggling can follow what you are doing by just reading your work (do not refer to any external references, including the text). 1. (9 points) Determine whether each of the following equations are linear or nonlinear. If it is linear, put it into the canonical form, a 1 x 1 + a 2 x a n x n = b and if it is nonlinear, explain why it is nonlinear. (a) e cos(ln 1) x 1 + x 2 = 0.9x 3 +.1x 2 Solution: The equation is linear note that e cos(ln(1)) = e cos(0) = e 1 = e is just a constant and the canonical form is ex x x 3 = 0. (b) x 1 + 2x 2 3x 2 x 3 = 1 Solution: This equation is not linear because of the x 2 x 3 term in the left-hand side. (c) (x 1 x 2 ) 2 = 15x 3 Solution: This equation is not linear either, though it might not be readily visible before expanding the square in the left-hand side. Doing so yields x 2 1 x 1 x 2 +x 2 2 = 15x 3, which is evidently nonlinear because of the x 1 x 2 term. 2. (15 points) Solve the system of equations, unless the system of equations is inconsistent, and then write inconsistent. Use elementary row operations and be sure to get the augmented matrix in at least row echelon form. (No points if the augmented matrix is not, at some point, in row echelon form).
2 (a) x 1 + 2x 2 + 2x 3 = x 1 + 3x 2 + 3x 3 = 5 2x 1 + 6x 2 + 5x 3 = 6 Solution: We set up the augmented matrix We add 1 times the first row to the second row, and times the first row to the second row, yielding Next add times the second row to the third row, giving us row echelon form: From here on we can either do backwards substitution to solve for x 3, x 2, and x 1, or we can keep on performing elementary row operations to reach reduced row echelon form. We will do the latter in order to practice matrix manipulation. We multiply the last row by 1, making the pivot element 1 there, and then add 1 times the last row to the second row, and times the last row to the first row: We are almost done, it only remains to get right of the 2 in the first row, which we accomplish by adding times the second row to the first row, leaving us with Since this is now in reduced row echelon form we can read off the unique solution x 1 = 2, x 2 = 3, and x 3 =, and we see also that this system is consistent.
3 (b) x 1 + 2x 2 + x = 7 x 1 + x 2 + x 3 x = 3 3x 1 + x 2 + 5x 3 7x = 1 Solution: We set up the augmented matrix Adding 1 times the first row and 3 times the first row to the second and third row respectively we get , and then add 5 times row two to row three, whence Multiplying the second row by 1 and subtracting 2 times this new row from row 1 we end up with Reading off this matrix we now see that the system is consistent with infinitely many solutions, namely x 1 = 1 2x 3 + 3x, x 2 = + x 3 2x, x 3 is free, x is free. (c) x 1 + 2x 2 + 2x 3 = 3x 1 2x 2 + 3x 3 = x 1 + 5x 3 = 0
4 Solution: As per usual we set up an augmented coefficient matrix: In an effort to get a pivot in the first column we add 3 times row 1 to row 2, and times row 1 to row 3: To get a pivot in the second column we now add 1 times row two to row three: The augmented matrix is now in row echelon form, but note that the last column has a pivot. This means that the system is not consistent. Another way of seeing the same is to spell out what the last row is really saying: 0x 1 + 0x 2 + 0x 3 = 6, or in other words 0 = 6, which is impossible. All this to say, the system is inconsistent. 3. (9 points) (a) Explain the difference between a coefficient matrix and an augmented matrix. Solution: A coefficient matrix is a matrix made up of the coefficients from a system of linear equations. An augmented matrix is similar in that it, too, is a coefficient matrix, but in addition it is augmented with a column consisting of the values on the right-hand side of the equations of the linear system. (b) If you know that a system of linear equations has at least two solutions, how many solutions must it actually have? Explain. Solution: It must have infinitely many solutions. There are three options for a system of linear equations: it has no solutions at all (called inconsistent), exactly one solution, or infinitely many solutions. Hence if it has more than one solution, it has infinitely many. (c) How many solutions can a consistent system of linear equations have? Discuss the different possibilities. Solution: A system being consistent means it has solutions at all. There are now two possibilities: either there is exactly one unique solution, or there are infinitely many solutions.. (12 points) (a) Give an example of two row equivalent matrices in echelon form. Note that although the reduced echelon form is unique, there will be several different row equivalent matrices in echelon form.
5 Solution: For example and There are infinitely many solutions to this problem; the particular one shown above was obtained by starting with a matrix (in this case the identity matrix, which happens to be in reduced row echelon form) and adding its third row to its second row. Since this is an elementary row operation they remain row equivalent, and since we haven t introduced any nonzero elements under the pivot elements it remains in row echelon form. (b) Suppose a 6 coefficient matrix has pivot columns. Is the corresponding system of equations consistent? Justify your answer. Solution: The system is consistent with infinitely many solutions since there s a pivot in every row but not every column. The columns without pivot elements correspond to free variables. (c) if a 7 5 augmented matrix has a pivot in every column, what can you say about the solutions to the corresponding system of equations? Justify your answer. Solution: Because the matrix at hand is augmented and has a pivot in every column, the rightmost column in particular must be a pivot column. This means that the system is not consistent, since it means that the row corresponding to the last pivot element looks like b where b 0, which means that 0 = b, which is impossible. Hence the system is inconsistent. (d) If a consistent system of equations has more unknowns than equations, what can be said about the number of solutions? Solution: A system with more unknowns than equations has either no solutions or infinitely many solutions. Since our system is specified to be consistent (meaning it has solutions), it must be the second option.
Row 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 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 informationM 340L CS Homework Set 1
M 340L CS Homework Set 1 Solve each system in Problems 1 6 by using elementary row operations on the equations or on the augmented matri. Follow the systematic elimination procedure described in Lay, Section
More informationMath 2331 Linear Algebra
1.2 Echelon Forms Math 2331 Linear Algebra 1.2 Row Reduction and Echelon Forms Shang-Huan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/ schiu/ January 22, 2018 Shang-Huan
More informationElementary Matrices. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
Elementary Matrices MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Outline Today s discussion will focus on: elementary matrices and their properties, using elementary
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 informationMath "Matrix Approach to Solving Systems" Bibiana Lopez. November Crafton Hills College. (CHC) 6.3 November / 25
Math 102 6.3 "Matrix Approach to Solving Systems" Bibiana Lopez Crafton Hills College November 2010 (CHC) 6.3 November 2010 1 / 25 Objectives: * Define a matrix and determine its order. * Write the augmented
More informationMATH10212 Linear Algebra B Homework Week 3. Be prepared to answer the following oral questions if asked in the supervision class
MATH10212 Linear Algebra B Homework Week Students are strongly advised to acquire a copy of the Textbook: D. C. Lay Linear Algebra its Applications. Pearson, 2006. ISBN 0-521-2871-4. Normally, homework
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 information9.1 - Systems of Linear Equations: Two Variables
9.1 - Systems of Linear Equations: Two Variables Recall that a system of equations consists of two or more equations each with two or more variables. A solution to a system in two variables is an ordered
More information0.0.1 Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros.
0.0.1 Section 1.2: Row Reduction and Echelon Forms Echelon form (or row echelon form): 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry (i.e. left most nonzero entry) of a row
More informationMATH 2331 Linear Algebra. Section 1.1 Systems of Linear Equations. Finding the solution to a set of two equations in two variables: Example 1: Solve:
MATH 2331 Linear Algebra Section 1.1 Systems of Linear Equations Finding the solution to a set of two equations in two variables: Example 1: Solve: x x = 3 1 2 2x + 4x = 12 1 2 Geometric meaning: Do these
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 information5x 2 = 10. x 1 + 7(2) = 4. x 1 3x 2 = 4. 3x 1 + 9x 2 = 8
1 To solve the system x 1 + x 2 = 4 2x 1 9x 2 = 2 we find an (easier to solve) equivalent system as follows: Replace equation 2 with (2 times equation 1 + equation 2): x 1 + x 2 = 4 Solve equation 2 for
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 informationMA 1B PRACTICAL - HOMEWORK SET 3 SOLUTIONS. Solution. (d) We have matrix form Ax = b and vector equation 4
MA B PRACTICAL - HOMEWORK SET SOLUTIONS (Reading) ( pts)[ch, Problem (d), (e)] Solution (d) We have matrix form Ax = b and vector equation 4 i= x iv i = b, where v i is the ith column of A, and 4 A = 8
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 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 informationChapter 1. Vectors, Matrices, and Linear Spaces
1.4 Solving Systems of Linear Equations 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.4. Solving Systems of Linear Equations Note. We give an algorithm for solving a system of linear equations (called
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 informationMath 3C Lecture 20. John Douglas Moore
Math 3C Lecture 20 John Douglas Moore May 18, 2009 TENTATIVE FORMULA I Midterm I: 20% Midterm II: 20% Homework: 10% Quizzes: 10% Final: 40% TENTATIVE FORMULA II Higher of two midterms: 30% Homework: 10%
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 informationHandout 1 EXAMPLES OF SOLVING SYSTEMS OF LINEAR EQUATIONS
22M:33 J. Simon page 1 of 7 22M:33 Summer 06 J. Simon Example 1. Handout 1 EXAMPLES OF SOLVING SYSTEMS OF LINEAR EQUATIONS 2x 1 + 3x 2 5x 3 = 10 3x 1 + 5x 2 + 6x 3 = 16 x 1 + 5x 2 x 3 = 10 Step 1. Write
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 informationMath 314H EXAM I. 1. (28 points) The row reduced echelon form of the augmented matrix for the system. is the matrix
Math 34H EXAM I Do all of the problems below. Point values for each of the problems are adjacent to the problem number. Calculators may be used to check your answer but not to arrive at your answer. That
More informationSolutions to Exam I MATH 304, section 6
Solutions to Exam I MATH 304, section 6 YOU MUST SHOW ALL WORK TO GET CREDIT. Problem 1. Let A = 1 2 5 6 1 2 5 6 3 2 0 0 1 3 1 1 2 0 1 3, B =, C =, I = I 0 0 0 1 1 3 4 = 4 4 identity matrix. 3 1 2 6 0
More informationMath 51, Homework-2. Section numbers are from the course textbook.
SSEA Summer 2017 Math 51, Homework-2 Section numbers are from the course textbook. 1. Write the parametric equation of the plane that contains the following point and line: 1 1 1 3 2, 4 2 + t 3 0 t R.
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 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 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 2. Systems
More informationChapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra Sections 5.1 5.3 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
More informationChapter 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 information1300 Linear Algebra and Vector Geometry Week 2: Jan , Gauss-Jordan, homogeneous matrices, intro matrix arithmetic
1300 Linear Algebra and Vector Geometry Week 2: Jan 14 18 1.2, 1.3... Gauss-Jordan, homogeneous matrices, intro matrix arithmetic R. Craigen Office: MH 523 Email: craigenr@umanitoba.ca Winter 2019 What
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 informationSystem of Linear Equations
Math 20F Linear Algebra Lecture 2 1 System of Linear Equations Slide 1 Definition 1 Fix a set of numbers a ij, b i, where i = 1,, m and j = 1,, n A system of m linear equations in n variables x j, is given
More informationChapter 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
More informationLinear Algebra Exam 1 Spring 2007
Linear Algebra Exam 1 Spring 2007 March 15, 2007 Name: SOLUTION KEY (Total 55 points, plus 5 more for Pledged Assignment.) Honor Code Statement: Directions: Complete all problems. Justify all answers/solutions.
More informationLecture 6 & 7. Shuanglin Shao. September 16th and 18th, 2013
Lecture 6 & 7 Shuanglin Shao September 16th and 18th, 2013 1 Elementary matrices 2 Equivalence Theorem 3 A method of inverting matrices Def An n n matrice is called an elementary matrix if it can be obtained
More informationLinear Algebra I Lecture 10
Linear Algebra I Lecture 10 Xi Chen 1 1 University of Alberta January 30, 2019 Outline 1 Gauss-Jordan Algorithm ] Let A = [a ij m n be an m n matrix. To reduce A to a reduced row echelon form using elementary
More informationMATH 2360 REVIEW PROBLEMS
MATH 2360 REVIEW PROBLEMS Problem 1: In (a) (d) below, either compute the matrix product or indicate why it does not exist: ( )( ) 1 2 2 1 (a) 0 1 1 2 ( ) 0 1 2 (b) 0 3 1 4 3 4 5 2 5 (c) 0 3 ) 1 4 ( 1
More informationHomework 1 Due: Wednesday, August 27. x + y + z = 1. x y = 3 x + y + z = c 2 2x + cz = 4
Homework 1 Due: Wednesday, August 27 1. Find all values of c for which the linear system: (a) has no solutions. (b) has exactly one solution. (c) has infinitely many solutions. (d) is consistent. x + y
More informationMath 1314 Week #14 Notes
Math 3 Week # Notes Section 5.: A system of equations consists of two or more equations. A solution to a system of equations is a point that satisfies all the equations in the system. In this chapter,
More informationMATH 2050 Assignment 6 Fall 2018 Due: Thursday, November 1. x + y + 2z = 2 x + y + z = c 4x + 2z = 2
MATH 5 Assignment 6 Fall 8 Due: Thursday, November [5]. For what value of c does have a solution? Is it unique? x + y + z = x + y + z = c 4x + z = Writing the system as an augmented matrix, we have c R
More informationMA 242 LINEAR ALGEBRA C1, Solutions to First Midterm Exam
MA 242 LINEAR ALGEBRA C Solutions to First Midterm Exam Prof Nikola Popovic October 2 9:am - :am Problem ( points) Determine h and k such that the solution set of x + = k 4x + h = 8 (a) is empty (b) contains
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 informationR2 R2 2R R2 R1 3R This is the reduced echelon form of our matrix. It corresponds to the linear system of equations
We label the rows R1 and R2. This is echelon form. [ 4 12 ] 8 2 10 0 R1 R1/4 2 10 0 R2 R2 2R1 0 4 4 R2 R2/4 [ ] 1 0 5 R2 R1 3R2 [ 1 0 ] 5 This is the reduced echelon form of our matrix. It corresponds
More informationChapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra Sections 5.1 5.3 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
More informationMATH 1553, C.J. JANKOWSKI MIDTERM 1
MATH 155, C.J. JANKOWSKI MIDTERM 1 Name Section Please read all instructions carefully before beginning. You have 5 minutes to complete this exam. There are no aids of any kind (calculators, notes, text,
More informationMath 2331 Linear Algebra
1.1 Linear System Math 2331 Linear Algebra 1.1 Systems of Linear Equations Shang-Huan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/ schiu/ Shang-Huan Chiu, University
More informationChapter 2: Approximating Solutions of Linear Systems
Linear of Chapter 2: Solutions of Linear Peter W. White white@tarleton.edu Department of Mathematics Tarleton State University Summer 2015 / Numerical Analysis Overview Linear of Linear of Linear of Linear
More informationRank and Nullity. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
Rank and Nullity MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives We have defined and studied the important vector spaces associated with matrices (row space,
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 informationLecture 22: Section 4.7
Lecture 22: Section 47 Shuanglin Shao December 2, 213 Row Space, Column Space, and Null Space Definition For an m n, a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn, the vectors r 1 = [ a 11 a 12 a 1n
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 I Lecture 8
Linear Algebra I Lecture 8 Xi Chen 1 1 University of Alberta January 25, 2019 Outline 1 2 Gauss-Jordan Elimination Given a system of linear equations f 1 (x 1, x 2,..., x n ) = 0 f 2 (x 1, x 2,..., x n
More informationReview Solutions for Exam 1
Definitions Basic Theorems. Finish the definition: Review Solutions for Exam (a) A linear combination of vectors {v,..., v n } is: any vector of the form c v + c v + + c n v n (b) A set of vectors {v,...,
More information3.4 Elementary Matrices and Matrix Inverse
Math 220: Summer 2015 3.4 Elementary Matrices and Matrix Inverse A n n elementary matrix is a matrix which is obtained from the n n identity matrix I n n by a single elementary row operation. Elementary
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 informationSolving Linear Systems
Math 240 TA: Shuyi Weng Winter 2017 January 12, 2017 Solving Linear Systems Linear Systems You have dealt with linear equations and systems of linear equations since you first learn mathematics in elementary
More information1. (7pts) Find the points of intersection, if any, of the following planes. 3x + 9y + 6z = 3 2x 6y 4z = 2 x + 3y + 2z = 1
Math 125 Exam 1 Version 1 February 20, 2006 1. (a) (7pts) Find the points of intersection, if any, of the following planes. Solution: augmented R 1 R 3 3x + 9y + 6z = 3 2x 6y 4z = 2 x + 3y + 2z = 1 3 9
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 information7.5 Operations with Matrices. Copyright Cengage Learning. All rights reserved.
7.5 Operations with Matrices Copyright Cengage Learning. All rights reserved. What You Should Learn Decide whether two matrices are equal. Add and subtract matrices and multiply matrices by scalars. Multiply
More informationSection 5.3 Systems of Linear Equations: Determinants
Section 5. Systems of Linear Equations: Determinants In this section, we will explore another technique for solving systems called Cramer's Rule. Cramer's rule can only be used if the number of equations
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 informationMath 54 HW 4 solutions
Math 54 HW 4 solutions 2.2. Section 2.2 (a) False: Recall that performing a series of elementary row operations A is equivalent to multiplying A by a series of elementary matrices. Suppose that E,...,
More informationMTH 2530: Linear Algebra. Sec Systems of Linear Equations
MTH 0 Linear Algebra Professor Chao Huang Department of Mathematics and Statistics Wright State University Week # Section.,. Sec... Systems of Linear Equations... D examples Example Consider a system of
More informationLinear equations in linear algebra
Linear equations in linear algebra Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra Pearson Collections Samy T. Linear
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 informationMatrices and systems of linear equations
Matrices and systems of linear equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T.
More informationSolving Systems of Linear Equations Using Matrices
Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.
More informationRow Space, Column Space, and Nullspace
Row Space, Column Space, and Nullspace MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Every matrix has associated with it three vector spaces: row space
More informationExercise Sketch these lines and find their intersection.
These are brief notes for the lecture on Friday August 21, 2009: they are not complete, but they are a guide to what I want to say today. They are not guaranteed to be correct. 1. Solving systems of linear
More informationMATH 1553, SPRING 2018 SAMPLE MIDTERM 1: THROUGH SECTION 1.5
MATH 553, SPRING 28 SAMPLE MIDTERM : THROUGH SECTION 5 Name Section Please read all instructions carefully before beginning You have 5 minutes to complete this exam There are no aids of any kind (calculators,
More informationMatrix equation Ax = b
Fall 2017 Matrix equation Ax = b Authors: Alexander Knop Institute: UC San Diego Previously On Math 18 DEFINITION If v 1,..., v l R n, then a set of all linear combinations of them is called Span {v 1,...,
More informationProblem Sheet 1 with Solutions GRA 6035 Mathematics
Problem Sheet 1 with Solutions GRA 6035 Mathematics BI Norwegian Business School 2 Problems 1. From linear system to augmented matrix Write down the coefficient matrix and the augmented matrix of the following
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 informationMatrix multiplications that do row operations
May 6, 204 Matrix multiplications that do row operations page Matrix multiplications that do row operations Introduction We have yet to justify our method for finding inverse matrices using row operations:
More informationMath 102, Winter 2009, Homework 7
Math 2, Winter 29, Homework 7 () Find the standard matrix of the linear transformation T : R 3 R 3 obtained by reflection through the plane x + z = followed by a rotation about the positive x-axes by 6
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 informationMath 314 Lecture Notes Section 006 Fall 2006
Math 314 Lecture Notes Section 006 Fall 2006 CHAPTER 1 Linear Systems of Equations First Day: (1) Welcome (2) Pass out information sheets (3) Take roll (4) Open up home page and have students do same
More informationSection 2.2: The Inverse of a Matrix
Section 22: The Inverse of a Matrix Recall that a linear equation ax b, where a and b are scalars and a 0, has the unique solution x a 1 b, where a 1 is the reciprocal of a From this result, it is natural
More informationLinear Equations in Linear Algebra
Linear Equations in Linear Algebra.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v,, v p } in n is said to be linearly independent if the vector equation x x x 2 2 p
More informationChapter 1: Systems of Linear Equations
Chapter : Systems of Linear Equations February, 9 Systems of linear equations Linear systems Lecture A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where
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 informationMATH 152 Exam 1-Solutions 135 pts. Write your answers on separate paper. You do not need to copy the questions. Show your work!!!
MATH Exam -Solutions pts Write your answers on separate paper. You do not need to copy the questions. Show your work!!!. ( pts) Find the reduced row echelon form of the matrix Solution : 4 4 6 4 4 R R
More information1 Last time: linear systems and row operations
1 Last time: linear systems and row operations Here s what we did last time: a system of linear equations or linear system is a list of equations a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22
More information22m:033 Notes: 1.2 Row Reduction and Echelon Forms
22m:033 Notes: 1.2 Row Reduction and Echelon Forms Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman January 25, 2010 1 1 Echelon form and reduced Echelon form Definition
More informationDetermine whether the following system has a trivial solution or non-trivial solution:
Practice Questions Lecture # 7 and 8 Question # Determine whether the following system has a trivial solution or non-trivial solution: x x + x x x x x The coefficient matrix is / R, R R R+ R The corresponding
More informationMath 344 Lecture # Linear Systems
Math 344 Lecture #12 2.7 Linear Systems Through a choice of bases S and T for finite dimensional vector spaces V (with dimension n) and W (with dimension m), a linear equation L(v) = w becomes the linear
More informationMath 308 Discussion Problems #2 (Sections ) SOLUTIONS
Math 8 Discussion Problems # (Sections.-.) SOLUTIONS () When Jake works from home, he typically spends 4 minutes of each hour on research, and on teaching, and drinks half a cup of coffee. (The remaining
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 informationChapter 1 Linear Equations. 1.1 Systems of Linear Equations
Chapter Linear Equations. Systems of Linear Equations A linear equation in the n variables x, x 2,..., x n is one that can be expressed in the form a x + a 2 x 2 + + a n x n = b where a, a 2,..., a n and
More informationChapter 2. Systems of Equations and Augmented Matrices. Creighton University
Chapter Section - Systems of Equations and Augmented Matrices D.S. Malik Creighton University Systems of Linear Equations Common ways to solve a system of equations: Eliminationi Substitution Elimination
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 informationSection 1.1: Systems of Linear Equations
Section 1.1: Systems of Linear Equations Two Linear Equations in Two Unknowns Recall that the equation of a line in 2D can be written in standard form: a 1 x 1 + a 2 x 2 = b. Definition. A 2 2 system of
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 informationChapter 3. Directions: For questions 1-11 mark each statement True or False. Justify each answer.
Chapter 3 Directions: For questions 1-11 mark each statement True or False. Justify each answer. 1. (True False) Asking whether the linear system corresponding to an augmented matrix [ a 1 a 2 a 3 b ]
More informationMATH10212 Linear Algebra B Homework Week 5
MATH Linear Algebra B Homework Week 5 Students are strongly advised to acquire a copy of the Textbook: D C Lay Linear Algebra its Applications Pearson 6 (or other editions) Normally homework assignments
More informationEvaluating Determinants by Row Reduction
Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives Reduce a matrix to row echelon form and evaluate its determinant.
More informationMath 4377/6308 Advanced Linear Algebra
3.1 Elementary Matrix Math 4377/6308 Advanced Linear Algebra 3.1 Elementary Matrix Operations and Elementary Matrix Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/
More information