MA 3280 Lecture 05 - Generalized Echelon Form and Free Variables. Friday, January 31, 2014.
|
|
- Anastasia Knight
- 5 years ago
- Views:
Transcription
1 MA 3280 Lecture 05 - Generalized Echelon Form and Free Variables Friday, January 31, Objectives: Generalize echelon form, and introduce free variables. Material from Section 3.5 starting on page 64 of the Schaum s Outline book. Consider this example from the book. (1) 2x 1 + 6x 2 x 3 + 4x 4 2x 5 = 15 x 3 + 2x 4 + 2x 5 = 5 3x 4 9x 5 = 6 We will call the first unknown/variable in each of these equations (i.e. the first unknown/variable with a non-zero coefficient) a leading unknown/variable. The book uses the term unknown, and we ll use the terms unknown and variable interchangeably. I ll probably start saying variable most of the time. In particular, x 1 is the leading variable for the first equation, x 3 is the leading variable for the second equation, and x 4 is the leading variable for the third. We can also say that x 1, x 3, and x 4 are pivot variables. It won t be a big deal to me, but there is a slight distinction between these terms. Leading refers to a particular equation, and pivot refers to the system as a whole. The book describes this system of equations as being in echelon form, and we will too. A system of equations is in echelon form if (1) No equation is degenerate (i.e., of the form 0 = c with c 0). (2) Each leading variable lies to the right of the leading variables above it. (3) (optional) Each leading variable has coefficient 1. For a system in echelon form, the variables that are not pivot/leading variables are called free variables. This definitely looks different from the examples we ve looked at so far. There are more variables than equations. We can see what s going on by trying to do some back-substitution, and solving for our pivot variables. In the last equation, we can solve for x 4 to get (2) x 4 = 2 + 3x 5. Substituting this into the second equation, we get (3) x 3 + 2(2 + 3x 5 ) + 2x 5 = 5, and so (4) x 3 = 1 8x 5. And finally, we can substitute these values into the first equation to get (5) 2x 1 + 6x 2 (1 8x 5 ) + 4(2 + 3x 5 ) 2x 5 = 15, and then solving for x 1 to get (6) x 1 = 4 3x 2 9x 5. We have all of our pivot variables solved in terms of our free variables. It s common to think of the free variables as parameters, and give them different names, like x 2 = a and x 5 = b. Then (7) x 1 x 2 x 3 x 4 x 5 = 4 3a 9b = a = 1 8b = 2 + 3b = b In this formulation, x 2 and x 5 are free in that they are free to take any value, and the pivot variables depend on them. So no matter what values a and b take, the 5-tuple (8) (4 3a 9b, a, 1 8b, 2 + 3b, b) 1
2 MA 3280 Lecture 05 - Generalized Echelon Form and Free Variables 2 is a solution to the system of equations. For example, pick any two numbers for a and b, like a = 2 and b = 3, and you ll have a solution to the system of equations, (9) (4 3(2) 9( 3), (2), 1 8( 3), 2 + 3( 3), ( 3)) = (25, 2, 25, 7, 3). All of the solutions to this system will be a plane in 5-dimensional space. Basic Principle 1. Given a system of linear equations, if you can get it into echelon form, then that system will have solutions. The dimension of the solution space will correspond to the number of free variables. A 0-dimensional solution space is a single point, a 1-dimensional solution space is a line, a 2-dimensional solution space is a plane, and in general, an n-dimensional solution space is an n-dimensional hyper-plane (we ll talk about this more later). Example 1. Let s look at a few really simple examples. Consider the system (10) 4x 1 + 6x 2 = 12. You can see that the second equation is just a multiple of the first, so there s really only one equation here. Let s get this to echelon form. We can avoid fractions by not worrying about getting 1 s on the leading variables, so let s do that. In that case, our first operation is 2 E 1 + E 2, which gives us (11) 0x 1 + 0x 2 = 0. The second equation is 0 = 0, which is OK, and we re in echelon form. We have that x 1 is a pivot variable, and x 2 is free. Solving the first equation for x 1 gives us (12) x 1 = x 2. We have one free variable, so our solution set will have dimension 1, a line. The points on this line will take the form ( (13) 3 32 ) a, a We can graph the line by finding two points. Two easy values for a are a = 0 and a = 2, which correspond to (3, 0) and (0, 2). Of course, all we re doing is graphing the equation, which has x 1 -intercept 3 and x 2 -intercept 2.
3 MA 3280 Lecture 05 - Generalized Echelon Form and Free Variables 3 Example 2. We can change this last system slightly by making the two lines parallel. (14) Trying the same operation, 2 E 1 + E 2, gives us (15) 4x 1 + 6x 2 = 10. 0x 1 + 0x 2 = 2. The second equation is 0 = 2, which is degenerate. This system has no solutions. The solution set is empty, which is less than 0-dimensional in our view. Example 3. Here s one more example. Consider the system (16) Let s do this one using matrix notation. (17) 2x 1 + x 2 + 5x 3 = 8 4x 1 + 2x x 3 = 22 2x 1 + x 2 + 6x 3 = Doing the operations 2 R 1 + R 2 and 1 R 1 + R 3 gives us (18) We can then do 1 3 R 2 + R 3 to get (19) Now, a row of 0 s is OK. We just don t want 0 s and something non-zero in the last column, which would mean, no solutions. We can see from the last matrix that x 1 and x 3 are pivot variables, and x 2 is free. The second equation is (20) 3x 3 = 6 which simplifies to (21) x 3 = 2. Substituting this into the first equation gives us (22) 2x 1 + x 2 + 5(2) = 8, and solving for x 1 gives us (23) x 1 = 1 x 2 2. Our solutions, therefore, take the form ( (24) 1 a ) 2, a, 2. These points form a line in x 1 x 2 x 3 -space.. Find three solutions to this last system of equations. Quiz 05
4 MA 3280 Lecture 05 - Generalized Echelon Form and Free Variables 4 Homework Consider the system of equations that corresponds to the following matrix. (25) Consider the system of equations that corresponds to the following matrix. [ ] (26) Consider the system of equations that corresponds to the following matrix. [ ] (27)
5 MA 3280 Lecture 05 - Generalized Echelon Form and Free Variables 5 4. Consider the system of equations that corresponds to the following matrix. (28) Quiz 05: There are infinitely many answers. For a = 0, 1, 2, you d get ( 1, 0, 2), ( 3, 1, 2), and ( 2, 2, 2). 2 HW: 1) a) or b) x 1, x 3, and x 4. c) x 2. d) (2 2a, a, 4, 3) e) A line. 2) a) Already in echelon form. b) x 1 and x 3. c) x 2 and x 4. d) ( 7 a + 3b, a, 4 2b, b). e) A plane. 3) a) Already in echelon form. b) x 1 and x 2. c) x 3 and x 4. d) ( 4 a b, 4, a, b). e) A plane. 4) a) b) x 1. c) x 2, x 3, x 4. d) (1 a b c, a, b, c). e) A 3-D hyper-plane.
Exercise 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 informationMA 1125 Lecture 15 - The Standard Normal Distribution. Friday, October 6, Objectives: Introduce the standard normal distribution and table.
MA 1125 Lecture 15 - The Standard Normal Distribution Friday, October 6, 2017. Objectives: Introduce the standard normal distribution and table. 1. The Standard Normal Distribution We ve been looking at
More informationPart III: A Simplex pivot
MA 3280 Lecture 31 - More on The Simplex Method Friday, April 25, 2014. Objectives: Analyze Simplex examples. We were working on the Simplex tableau The matrix form of this system of equations is called
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 informationObjectives: Review open, closed, and mixed intervals, and begin discussion of graphing points in the xyplane. Interval notation
MA 0090 Section 18 - Interval Notation and Graphing Points Objectives: Review open, closed, and mixed intervals, and begin discussion of graphing points in the xyplane. Interval notation Last time, we
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 informationHomework Set #1 Solutions
Homework Set #1 Solutions Exercises 1.1 (p. 10) Assignment: Do #33, 34, 1, 3,, 29-31, 17, 19, 21, 23, 2, 27 33. (a) True. (p. 7) (b) False. It has five rows and six columns. (c) False. The definition given
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 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 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 informationOne-to-one functions and onto functions
MA 3362 Lecture 7 - One-to-one and Onto Wednesday, October 22, 2008. Objectives: Formalize definitions of one-to-one and onto One-to-one functions and onto functions At the level of set theory, there are
More informationMath 3361-Modern Algebra Lecture 08 9/26/ Cardinality
Math 336-Modern Algebra Lecture 08 9/26/4. Cardinality I started talking about cardinality last time, and you did some stuff with it in the Homework, so let s continue. I said that two sets have the same
More informationSection 20: Arrow Diagrams on the Integers
Section 0: Arrow Diagrams on the Integers Most of the material we have discussed so far concerns the idea and representations of functions. A function is a relationship between a set of inputs (the leave
More informationWhat if the characteristic equation has a double root?
MA 360 Lecture 17 - Summary of Recurrence Relations Friday, November 30, 018. Objectives: Prove basic facts about basic recurrence relations. Last time, we looked at the relational formula for a sequence
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 informationMA 0090 Section 21 - Slope-Intercept Wednesday, October 31, Objectives: Review the slope of the graph of an equation in slope-intercept form.
MA 0090 Section 21 - Slope-Intercept Wednesday, October 31, 2018 Objectives: Review the slope of the graph of an equation in slope-intercept form. Last time, we looked at the equation Slope (1) y = 2x
More information2, or x 5, 3 x 0, x 2
Pre-AP Algebra 2 Lesson 2 End Behavior and Polynomial Inequalities Objectives: Students will be able to: use a number line model to sketch polynomials that have repeated roots. use a number line model
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 informationSolving and Graphing Inequalities
Solving and Graphing Inequalities Graphing Simple Inequalities: x > 3 When finding the solution for an equation we get one answer for x. (There is only one number that satisfies the equation.) For 3x 5
More informationWhat if the characteristic equation has complex roots?
MA 360 Lecture 18 - Summary of Recurrence Relations (cont. and Binomial Stuff Thursday, November 13, 01. Objectives: Examples of Recurrence relation solutions, Pascal s triangle. A quadratic equation What
More informationMatrix-Vector Products and the Matrix Equation Ax = b
Matrix-Vector Products and the Matrix Equation Ax = b A. Havens Department of Mathematics University of Massachusetts, Amherst January 31, 2018 Outline 1 Matrices Acting on Vectors Linear Combinations
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 informationDefinition: A "system" of equations is a set or collection of equations that you deal with all together at once.
System of Equations Definition: A "system" of equations is a set or collection of equations that you deal with all together at once. There is both an x and y value that needs to be solved for Systems
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 informationMath 31 Lesson Plan. Day 2: Sets; Binary Operations. Elizabeth Gillaspy. September 23, 2011
Math 31 Lesson Plan Day 2: Sets; Binary Operations Elizabeth Gillaspy September 23, 2011 Supplies needed: 30 worksheets. Scratch paper? Sign in sheet Goals for myself: Tell them what you re going to tell
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 informationCS 301. Lecture 18 Decidable languages. Stephen Checkoway. April 2, 2018
CS 301 Lecture 18 Decidable languages Stephen Checkoway April 2, 2018 1 / 26 Decidable language Recall, a language A is decidable if there is some TM M that 1 recognizes A (i.e., L(M) = A), and 2 halts
More informationHomework 1.1 and 1.2 WITH SOLUTIONS
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
More informationLesson 3-2: Solving Linear Systems Algebraically
Yesterday we took our first look at solving a linear system. We learned that a linear system is two or more linear equations taken at the same time. Their solution is the point that all the lines have
More informationMath 138: Introduction to solving systems of equations with matrices. The Concept of Balance for Systems of Equations
Math 138: Introduction to solving systems of equations with matrices. Pedagogy focus: Concept of equation balance, integer arithmetic, quadratic equations. The Concept of Balance for Systems of Equations
More informationLecture 4: Applications of Orthogonality: QR Decompositions
Math 08B Professor: Padraic Bartlett Lecture 4: Applications of Orthogonality: QR Decompositions Week 4 UCSB 204 In our last class, we described the following method for creating orthonormal bases, known
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 informationMath Lecture 4 Limit Laws
Math 1060 Lecture 4 Limit Laws Outline Summary of last lecture Limit laws Motivation Limits of constants and the identity function Limits of sums and differences Limits of products Limits of polynomials
More informationSec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes
Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Infinite
More information( v 1 + v 2 ) + (3 v 1 ) = 4 v 1 + v 2. and ( 2 v 2 ) + ( v 1 + v 3 ) = v 1 2 v 2 + v 3, for instance.
4.2. Linear Combinations and Linear Independence If we know that the vectors v 1, v 2,..., v k are are in a subspace W, then the Subspace Test gives us more vectors which must also be in W ; for instance,
More informationSec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes
Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Sec 2.2: Infinite Limits / Vertical Asymptotes Sec 2.6: Limits At Infinity / Horizontal Asymptotes Infinite
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More informationI am trying to keep these lessons as close to actual class room settings as possible.
Greetings: I am trying to keep these lessons as close to actual class room settings as possible. They do not intend to replace the text book actually they will involve the text book. An advantage of a
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 informationMA 1128: Lecture 08 03/02/2018. Linear Equations from Graphs And Linear Inequalities
MA 1128: Lecture 08 03/02/2018 Linear Equations from Graphs And Linear Inequalities Linear Equations from Graphs Given a line, we would like to be able to come up with an equation for it. I ll go over
More informationMatrices and Systems of Equations
M CHAPTER 3 3 4 3 F 2 2 4 C 4 4 Matrices and Systems of Equations Probably the most important problem in mathematics is that of solving a system of linear equations. Well over 75 percent of all mathematical
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 informationBusiness Statistics. Lecture 9: Simple Regression
Business Statistics Lecture 9: Simple Regression 1 On to Model Building! Up to now, class was about descriptive and inferential statistics Numerical and graphical summaries of data Confidence intervals
More informationLecture 10: Powers of Matrices, Difference Equations
Lecture 10: Powers of Matrices, Difference Equations Difference Equations A difference equation, also sometimes called a recurrence equation is an equation that defines a sequence recursively, i.e. each
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 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 informationWe ll start today by learning how to change a decimal to a fraction on our calculator! Then we will pick up our Unit 1-5 Review where we left off!
Welcome to math! We ll start today by learning how to change a decimal to a fraction on our calculator! Then we will pick up our Unit 1-5 Review where we left off! So go back to your normal seat and get
More informationPhysics 6303 Lecture 22 November 7, There are numerous methods of calculating these residues, and I list them below. lim
Physics 6303 Lecture 22 November 7, 208 LAST TIME:, 2 2 2, There are numerous methods of calculating these residues, I list them below.. We may calculate the Laurent series pick out the coefficient. 2.
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 informationTheory of Computation Lecture 1. Dr. Nahla Belal
Theory of Computation Lecture 1 Dr. Nahla Belal Book The primary textbook is: Introduction to the Theory of Computation by Michael Sipser. Grading 10%: Weekly Homework. 30%: Two quizzes and one exam. 20%:
More informationMatrix Factorization Reading: Lay 2.5
Matrix Factorization Reading: Lay 2.5 October, 20 You have seen that if we know the inverse A of a matrix A, we can easily solve the equation Ax = b. Solving a large number of equations Ax = b, Ax 2 =
More informationMath 220 F11 Lecture Notes
Math 22 F Lecture Notes William Chen November 4, 2. Lecture. Firstly, lets just get some notation out of the way. Notation. R, Q, C, Z, N,,,, {},, A B. Everyone in high school should have studied equations
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 informationMath 220 Some Exam 1 Practice Problems Fall 2017
Math Some Exam Practice Problems Fall 7 Note that this is not a sample exam. This is much longer than your exam will be. However, the ideas and question types represented here (along with your homework)
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 informationSystems of Equations Homework Solutions
Systems of Equations Homework Solutions Olena Bormashenko October 5, 2011 Find all solutions to the following systems of equations by writing the system as an augmented matrix and row-reducing it until
More informationPolynomial and Synthetic Division
Polynomial and Synthetic Division Polynomial Division Polynomial Division is very similar to long division. Example: 3x 3 5x 3x 10x 1 3 Polynomial Division 3x 1 x 3x 3 3 x 5x 3x x 6x 4 10x 10x 7 3 x 1
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 informationCSC321 Lecture 4 The Perceptron Algorithm
CSC321 Lecture 4 The Perceptron Algorithm Roger Grosse and Nitish Srivastava January 17, 2017 Roger Grosse and Nitish Srivastava CSC321 Lecture 4 The Perceptron Algorithm January 17, 2017 1 / 1 Recap:
More informationVector calculus background
Vector calculus background Jiří Lebl January 18, 2017 This class is really the vector calculus that you haven t really gotten to in Calc III. Let us start with a very quick review of the concepts from
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 informationQuiz 07a. Integers Modulo 12
MA 3260 Lecture 07 - Binary Operations Friday, September 28, 2018. Objectives: Continue with binary operations. Quiz 07a We have a machine that is set to run for x hours, turn itself off for 3 hours, and
More informationFactored State Spaces 3/2/178
Factored State Spaces 3/2/178 Converting POMDPs to MDPs In a POMDP: Action + observation updates beliefs Value is a function of beliefs. Instead we can view this as an MDP where: There is a state for every
More informationSystems of Linear Equations
Systems of Linear Equations Linear Equation Definition Any equation that is equivalent to the following format a a ann b (.) where,,, n are unknown variables and a, a,, an, b are known numbers (the so
More informationMath 31 Lesson Plan. Day 16: Review; Start Section 8. Elizabeth Gillaspy. October 18, Supplies needed: homework. Colored chalk. Quizzes!
Math 31 Lesson Plan Day 16: Review; Start Section 8 Elizabeth Gillaspy October 18, 2011 Supplies needed: homework Colored chalk Quizzes! Goals for students: Students will: improve their understanding of
More information22A-2 SUMMER 2014 LECTURE Agenda
22A-2 SUMMER 204 LECTURE 2 NATHANIEL GALLUP The Dot Product Continued Matrices Group Work Vectors and Linear Equations Agenda 2 Dot Product Continued Angles between vectors Given two 2-dimensional vectors
More informationAnswers in blue. If you have questions or spot an error, let me know. 1. Find all matrices that commute with A =. 4 3
Answers in blue. If you have questions or spot an error, let me know. 3 4. Find all matrices that commute with A =. 4 3 a b If we set B = and set AB = BA, we see that 3a + 4b = 3a 4c, 4a + 3b = 3b 4d,
More informationO.K. But what if the chicken didn t have access to a teleporter.
The intermediate value theorem, and performing algebra on its. This is a dual topic lecture. : The Intermediate value theorem First we should remember what it means to be a continuous function: A function
More informationI started to think that maybe I could just distribute the log so that I get:
2.3 Chopping Logs A Solidify Understanding Task Abe and Mary were working on their math homework together when Abe has a brilliant idea Abe: I was just looking at this log function that we graphed in Falling
More informationMATH 341 MIDTERM 2. (a) [5 pts] Demonstrate that A and B are row equivalent by providing a sequence of row operations leading from A to B.
11/01/2011 Bormashenko MATH 341 MIDTERM 2 Show your work for all the problems. Good luck! (1) Let A and B be defined as follows: 1 1 2 A =, B = 1 2 3 0 2 ] 2 1 3 4 Name: (a) 5 pts] Demonstrate that A and
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 informationMATH 115, SUMMER 2012 LECTURE 12
MATH 115, SUMMER 2012 LECTURE 12 JAMES MCIVOR - last time - we used hensel s lemma to go from roots of polynomial equations mod p to roots mod p 2, mod p 3, etc. - from there we can use CRT to construct
More informationSection 6.3. Matrices and Systems of Equations
Section 6.3 Matrices and Systems of Equations Introduction Definitions A matrix is a rectangular array of numbers. Definitions A matrix is a rectangular array of numbers. For example: [ 4 7 π 3 2 5 Definitions
More informationMath 31 Lesson Plan. Day 5: Intro to Groups. Elizabeth Gillaspy. September 28, 2011
Math 31 Lesson Plan Day 5: Intro to Groups Elizabeth Gillaspy September 28, 2011 Supplies needed: Sign in sheet Goals for students: Students will: Improve the clarity of their proof-writing. Gain confidence
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 informationContents. Contents. Matrices. Contents. Objectives. Matrices
9/8/7 Physics for Majors Class 8 Matrices and Lorentz s Space-time Four- Last Class Test Review Scalars and vectors Three-vectors and four-vectors The energy-momentum four-vector Rotations about the z
More informationSymbolic Logic Outline
Symbolic Logic Outline 1. Symbolic Logic Outline 2. What is Logic? 3. How Do We Use Logic? 4. Logical Inferences #1 5. Logical Inferences #2 6. Symbolic Logic #1 7. Symbolic Logic #2 8. What If a Premise
More informationMTHSC 3110 Section 1.1
MTHSC 3110 Section 1.1 Kevin James A system of linear equations is a collection of equations in the same set of variables. For example, { x 1 + 3x 2 = 5 2x 1 x 2 = 4 Of course, since this is a pair of
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 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 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 informationNext, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations.
Section 6.3 - Solving Trigonometric Equations Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations. These are equations from algebra: Linear Equation: Solve:
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 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 informationMatrices, Row Reduction of Matrices
Matrices, Row Reduction of Matrices October 9, 014 1 Row Reduction and Echelon Forms In the previous section, we saw a procedure for solving systems of equations It is simple in that it consists of only
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 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 informationFirst Derivative Test
MA 2231 Lecture 22 - Concavity and Relative Extrema Wednesday, November 1, 2017 Objectives: Introduce the Second Derivative Test and its limitations. First Derivative Test When looking for relative extrema
More informationPhysics Motion Math. (Read objectives on screen.)
Physics 302 - Motion Math (Read objectives on screen.) Welcome back. When we ended the last program, your teacher gave you some motion graphs to interpret. For each section, you were to describe the motion
More informationMath 416, Spring 2010 Coordinate systems and Change of Basis February 16, 2010 COORDINATE SYSTEMS AND CHANGE OF BASIS. 1.
Math 46 Spring Coordinate systems and Change of asis February 6 COORDINAE SYSEMS AND CHANGE OF ASIS Announcements Don t forget that we have a quiz on hursday and test coming up the following hursday Finishing
More informationAnnouncements Wednesday, October 04
Announcements Wednesday, October 04 Please fill out the mid-semester survey under Quizzes on Canvas. WeBWorK 1.8, 1.9 are due today at 11:59pm. The quiz on Friday covers 1.7, 1.8, and 1.9. My office is
More informationMATH 310, REVIEW SHEET
MATH 310, REVIEW SHEET These notes are a 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, so please
More informationProbability (Devore Chapter Two)
Probability (Devore Chapter Two) 1016-345-01: Probability and Statistics for Engineers Fall 2012 Contents 0 Administrata 2 0.1 Outline....................................... 3 1 Axiomatic Probability 3
More informationMITOCW ocw f99-lec09_300k
MITOCW ocw-18.06-f99-lec09_300k OK, this is linear algebra lecture nine. And this is a key lecture, this is where we get these ideas of linear independence, when a bunch of vectors are independent -- or
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 information. As x gets really large, the last terms drops off and f(x) ½x
Pre-AP Algebra 2 Unit 8 -Lesson 3 End behavior of rational functions Objectives: Students will be able to: Determine end behavior by dividing and seeing what terms drop out as x Know that there will be
More information36 What is Linear Algebra?
36 What is Linear Algebra? The authors of this textbook think that solving linear systems of equations is a big motivation for studying linear algebra This is certainly a very respectable opinion as systems
More informationMAT 211, Spring 2015, Introduction to Linear Algebra.
MAT 211, Spring 2015, Introduction to Linear Algebra. Lecture 04, 53103: MWF 10-10:53 AM. Location: Library W4535 Contact: mtehrani@scgp.stonybrook.edu Final Exam: Monday 5/18/15 8:00 AM-10:45 AM The aim
More information2.4 The Extreme Value Theorem and Some of its Consequences
2.4 The Extreme Value Theorem and Some of its Consequences The Extreme Value Theorem deals with the question of when we can be sure that for a given function f, (1) the values f (x) don t get too big or
More informationLinear Algebra (wi1403lr) Lecture no.3
Linear Algebra (wi1403lr) Lecture no.3 EWI / DIAM / Numerical Analysis group Matthias Möller 25/04/2014 M. Möller (EWI/NA group) LA (wi1403lr) 25/04/2014 1 / 18 Review of lecture no.2 1.3 Vector Equations
More information