Sections 1.5, 1.7. Ma 322 Spring Ma 322. Jan 24-28
|
|
- Lenard Burke
- 5 years ago
- Views:
Transcription
1 Sections 1.5, 1.7 Ma 322 Spring 217 Ma 322 Jan Summary ˆ Text: Solution Sets of Linear Systems (1.5),Linear Independence (1.7) ˆ Solutions of homogeneous equations AX =. ˆ Using the rank. ˆ Parametric solution of AX = B. ˆ Linear dependence and independence of vectors in R n. ˆ Using REF and RREF as convenient.
2 The Homogeneous Equation AX =. ˆ Def. 1: Homogeneous System of Equations. A linear system (A B) is said to be homogeneous when B =, i.e. the RHS entries in B are all. In this case, the REF of M = (A ) can be seen to be M = (A ), i.e. the column can be omitted through the reduction process, since it will never change. ˆ Clearly, rank(m) = rank(a), so a homogeneous system is always consistent. Indeed, it is also clear that X = is a solution to AX = and hence consistency is directly obvious! ˆ Let the common rank be r. Then there are exactly r pivot variables and n r free variables. ˆ The final solution will consist of solving the pivot variables in terms of the free variables and reporting the conclusion. ˆ Here is a worked example. An example. ˆ Consider a system in REF: x y z w t RHS ˆ Identify pc list as (1, 3, 5, ), pivot variables as x, z, t and the free variables y, w. ˆ The fourth equation is ignored. The third gives t =, the second gives z =3w + 2t = 3w and the first gives x = 3y + 2w 3t = 3y + 2w.
3 Reporting the Solution. ˆ The above solution is best reported as a vector: x y z w t = 3y + 2w y 3w w = yv 1 + wv 2 where v 1 = 3 1, v 2 = Conclusion. ˆ Thus the solution is seen to be a member of Span{v 1, v 2 }. ˆ Denoting a general member of the span as v h, we write X = v h. It is important to remember that v h stands for any one of an infinite collection of vectors and should not be confused with a specific single vector or with the whole span! ˆ In general, the solution of AX = is always a member of the span of n r vectors, where n is the number of variables and r = rank(a).
4 The Non homogeneous Case. ˆ More generally, if we put the augmented matrix M = (A B) of a non homogeneous system into REF, say M = (A B ). then we need to verify the consistency condition first. ˆ If the condition fails, then there is no solution. ˆ If the condition holds, then the solution process and reporting is just as above, except the final answer is a fixed vector plus a span of (n r) vectors as before. Here is an example. A Non Homogeneous Example. ˆ Consider a system in REF: x y z w t RHS ˆ Identify pc list as (1, 3, 5, ), pivot variables as x, z, t and the free variables y, w. ˆ The fourth equation is ignored. The third gives t = 5, the second gives z =3w + 2t + 5 = 3w = 3w + 15 and the first gives x = 3y + 2w 3t + 9 = 3y + 2w = 3y + 2w 6.
5 Reporting the Full Solution. ˆ The above solution is best reported as a vector: x y z w t = 3y + 2w 6 y 3w + 15 w 5 = v p + yv 1 + wv 2 where v p = , v 1 = 3 1, v 2 = Conclusion. ˆ Thus the solution is seen to be v p plus a member of Span{v 1, v 2 }. ˆ Denoting a general member of the span as v h, we write X = v p + v h. It is important to remember that v h stands for any one of an infinite collection of vectors and should not be confused with a specific single vector or with the whole span! ˆ In general, the solution of AX = B is always X = v p + v h, where v h is a general member of a span of n r vectors, with n is the number of variables and r = rank(a). ˆ Def. 2: Homogeneous and Particular Solutions. We call v p as a particular solution and v h as a homogeneous solution. Note that neither of these are unique, but with proper identifications, they exhibit all the solutions of the system in a parametric form.
6 Trivial Solutions. ˆ If we consider a homogeneous equation AX = and C 1, C 2,, C n are n columns of A, then we are solving the equation: x 1 C 1 + x 2 C x n C n =. ˆ As already observed, this system is consistent and always has a solution x 1 = x 2 = = x n = or briefly X =. Def. 3: Trivial Solution. The solution X = is said to be a trivial solution of the system AX =. ˆ As already observed, the solution of AX = is X = v h, where v h is a general member of a span of n r vectors where r = rank(a). Thus, for a homogeneous system, a particular solution is the trivial solution! Linear Dependence. ˆ Def. 4: Linearly Dependent vectors The columns C 1, C 2,, C n of A are said to be linearly dependent if the system AX = has a non trivial solution, or equivalently n r >, i.e. rank of A is less than its number of columns. ˆ For future use, we restate this definition more generally thus: Def. 5(general): Linearly Dependent vectors. Any set S of vectors is said to be linearly dependent if there is a positive integer n such that n distinct vectors v 1, v 2,, v n of S satisfy c 1 v 1 +c 2 v 2 + +c n v n = where at least one of c i is non zero. ˆ Note that this makes sense even for an infinite set S.
7 Linear Independence. ˆ Def. 6: Linearly Independent Vectors. A set S of vectors is said to be linearly independent if it is not linearly dependent! ˆ Def. 7: Notations rownum, colnum. For any matrix A, we shall define colnum(a) to be the number of columns in A and rownum(a) to be the number of rows in A. ˆ Though logically clear, linear independence can be difficult to verify without better tools. We describe such a tool next. ˆ Convention: We shall often drop the word linearly from the terms linearly dependent and linearly independent. Test of Linear dependence/independence. ˆ For a finite set of vectors in R m, there is a simple criterion for dependence/independence. ˆ Given vectors v 1, v 2,, v n in R m, make a matrix A by taking these as columns and find its rank (by using REF, for example.). ˆ Suppose rank(a) = r. It is obvious that r n = colnum(a). Then we have: v 1, v 2,, v n are linearly dependent iff r < n and thus, they are linearly independent iff r = n.
8 Using REF, RREF. ˆ We have illustrated how to make and use REF - the row echelon form. ˆ We defined RREF, but did not work much with it. Roughly, it needs twice as much work as REF and hence we avoided using it, if it was not really needed. ˆ It is, however, needed for some of the later work and we include an illustration of the method to get that form. ˆ Unlike, REF, the form RREF is well defined and thus has theoretical merit. ˆ Typically, the act of solving equations, becomes, writing down the answers if we have achieved RREF. An example of RREF ˆ Consider the following augmented matrix (already in REF) and convert to RREF. Use it to write out the solution of the associated linear system. ˆ x y z RHS ˆ Notice that the pc list is (1, 2, 3, ) and the respective pivot entries are 1, 1, 9. ˆ The process is to start with the last pivot, make all entries above it equal to zero and make it 1. Then repeat with earlier pivots.
9 RREF Example concluded. ˆ Thus, the operations R R 3, R R 3, 1 9 R 3 give: ˆ x y z RHS x y z RHS ˆ Then the operation R 1 R 2 finishes off the RREF. x y z RHS ˆ The answer to the system can now be simply read off!
Sections 1.5, 1.7. Ma 322 Fall Ma 322. Sept
Sections 1.5, 1.7 Ma 322 Fall 213 Ma 322 Sept. 9-13 Summary ˆ Solutions of homogeneous equations AX =. ˆ Using the rank. ˆ Parametric solution of AX = B. ˆ Linear dependence and independence of vectors
More informationWeek 3 September 5-7.
MA322 Weekl topics and quiz preparations Week 3 September 5-7. Topics These are alread partl covered in lectures. We collect the details for convenience.. Solutions of homogeneous equations AX =. 2. Using
More informationMa 322 Spring Ma 322. Jan 18, 20
Ma 322 Spring 2017 Ma 322 Jan 18, 20 Summary ˆ Review of the Standard Gauss Elimination Algorithm: REF+ Backsub ˆ The rank of a matrix. ˆ Vectors and Linear combinations. ˆ Span of a set of vectors. ˆ
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 informationRank, Homogenous Systems, Linear Independence
Sept 11,1 Rank, Homogenous Systems, Linear Independence If (A B) is a linear system represented as an augmented matrix then A is called the COEFFICIENT MATRIX and B is (usually) called the RIGHT HAND SIDE
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 informationSolutions of Linear system, vector and matrix equation
Goals: Solutions of Linear system, vector and matrix equation Solutions of linear system. Vectors, vector equation. Matrix equation. Math 112, Week 2 Suggested Textbook Readings: Sections 1.3, 1.4, 1.5
More informationLinear Independence Reading: Lay 1.7
Linear Independence Reading: Lay 17 September 11, 213 In this section, we discuss the concept of linear dependence and independence I am going to introduce the definitions and then work some examples and
More information1. Determine by inspection which of the following sets of vectors is linearly independent. 3 3.
1. Determine by inspection which of the following sets of vectors is linearly independent. (a) (d) 1, 3 4, 1 { [ [,, 1 1] 3]} (b) 1, 4 5, (c) 3 6 (e) 1, 3, 4 4 3 1 4 Solution. The answer is (a): v 1 is
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 informationLinear Equation: a 1 x 1 + a 2 x a n x n = b. x 1, x 2,..., x n : variables or unknowns
Linear Equation: a x + a 2 x 2 +... + a n x n = b. x, x 2,..., x n : variables or unknowns a, a 2,..., a n : coefficients b: constant term Examples: x + 4 2 y + (2 5)z = is linear. x 2 + y + yz = 2 is
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 informationReview for Chapter 1. Selected Topics
Review for Chapter 1 Selected Topics Linear Equations We have four equivalent ways of writing linear systems: 1 As a system of equations: 2x 1 + 3x 2 = 7 x 1 x 2 = 5 2 As an augmented matrix: ( 2 3 ) 7
More informationDEPARTMENT OF MATHEMATICS
DEPARTMENT OF MATHEMATICS. Points: 4+7+4 Ma 322 Solved First Exam February 7, 207 With supplements You are given an augmented matrix of a linear system of equations. Here t is a parameter: 0 4 4 t 0 3
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 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 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 informationMTH 35, SPRING 2017 NIKOS APOSTOLAKIS
MTH 35, SPRING 2017 NIKOS APOSTOLAKIS 1. Linear independence Example 1. Recall the set S = {a i : i = 1,...,5} R 4 of the last two lectures, where a 1 = (1,1,3,1) a 2 = (2,1,2, 1) a 3 = (7,3,5, 5) a 4
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 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 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 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 informationSystem of Linear Equations
Chapter 7 - S&B Gaussian and Gauss-Jordan Elimination We will study systems of linear equations by describing techniques for solving such systems. The preferred solution technique- Gaussian elimination-
More informationReview Notes for Linear Algebra True or False Last Updated: February 22, 2010
Review Notes for Linear Algebra True or False Last Updated: February 22, 2010 Chapter 4 [ Vector Spaces 4.1 If {v 1,v 2,,v n } and {w 1,w 2,,w n } are linearly independent, then {v 1 +w 1,v 2 +w 2,,v n
More informationLinear Combination. v = a 1 v 1 + a 2 v a k v k
Linear Combination Definition 1 Given a set of vectors {v 1, v 2,..., v k } in a vector space V, any vector of the form v = a 1 v 1 + a 2 v 2 +... + a k v k for some scalars a 1, a 2,..., a k, is called
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 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 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 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 informationLinear Independence x
Linear Independence A consistent system of linear equations with matrix equation Ax = b, where A is an m n matrix, has a solution set whose graph in R n is a linear object, that is, has one of only n +
More informationW2 ) = dim(w 1 )+ dim(w 2 ) for any two finite dimensional subspaces W 1, W 2 of V.
MA322 Sathaye Final Preparations Spring 2017 The final MA 322 exams will be given as described in the course web site (following the Registrar s listing. You should check and verify that you do not have
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 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 informationLinear Algebra MATH20F Midterm 1
University of California San Diego NAME TA: Linear Algebra Wednesday, October st, 9 :am - :5am No aids are allowed Be sure to write all row operations used Remember that you can often check your answers
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 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 informationWhat is on this week. 1 Vector spaces (continued) 1.1 Null space and Column Space of a matrix
Professor Joana Amorim, jamorim@bu.edu What is on this week Vector spaces (continued). Null space and Column Space of a matrix............................. Null Space...........................................2
More informationMATH 1553, SPRING 2018 SAMPLE MIDTERM 2 (VERSION B), 1.7 THROUGH 2.9
MATH 155, SPRING 218 SAMPLE MIDTERM 2 (VERSION B), 1.7 THROUGH 2.9 Name Section 1 2 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 1 points. The maximum score
More informationMath 3A Winter 2016 Midterm
Math 3A Winter 016 Midterm Name Signature UCI ID # E-mail address There are 7 problems for a total of 115 points. Present your work as clearly as possible. Partial credit will be awarded, and you must
More informationColumn 3 is fine, so it remains to add Row 2 multiplied by 2 to Row 1. We obtain
Section Exercise : We are given the following augumented matrix 3 7 6 3 We have to bring it to the diagonal form The entries below the diagonal are already zero, so we work from bottom to top Adding 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 informationMATH240: Linear Algebra Exam #1 solutions 6/12/2015 Page 1
MATH4: Linear Algebra Exam # solutions 6//5 Page Write legibly and show all work. No partial credit can be given for an unjustified, incorrect answer. Put your name in the top right corner and sign the
More informationMath 369 Exam #2 Practice Problem Solutions
Math 369 Exam #2 Practice Problem Solutions 2 5. Is { 2, 3, 8 } a basis for R 3? Answer: No, it is not. To show that it is not a basis, it suffices to show that this is not a linearly independent set.
More informationMath 308 Spring Midterm Answers May 6, 2013
Math 38 Spring Midterm Answers May 6, 23 Instructions. Part A consists of questions that require a short answer. There is no partial credit and no need to show your work. In Part A you get 2 points per
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 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 informationMATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.
MATH 2331 Linear Algebra Section 2.1 Matrix Operations Definition: A : m n, B : n p ( 1 2 p ) ( 1 2 p ) AB = A b b b = Ab Ab Ab Example: Compute AB, if possible. 1 Row-column rule: i-j-th entry of AB:
More informationT ((x 1, x 2,..., x n )) = + x x 3. , x 1. x 3. Each of the four coordinates in the range is a linear combination of the three variables x 1
MATH 37 Linear Transformations from Rn to Rm Dr. Neal, WKU Let T : R n R m be a function which maps vectors from R n to R m. Then T is called a linear transformation if the following two properties are
More informationEXAM. Exam #2. Math 2360 Summer II, 2000 Morning Class. Nov. 15, 2000 ANSWERS
EXAM Exam # Math 6 Summer II Morning Class Nov 5 ANSWERS i Problem Consider the matrix 6 pts A = 6 4 9 5 7 6 5 5 5 4 The RREF of A is the matrix R = A Find a basis for the nullspace of A Solve the homogeneous
More informationThe definition of a vector space (V, +, )
The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element
More informationMATH10212 Linear Algebra B Homework Week 4
MATH22 Linear Algebra B Homework Week 4 Students are strongly advised to acquire a copy of the Textbook: D. C. Lay Linear Algebra and its Applications. Pearson, 26. ISBN -52-2873-4. Normally, homework
More information1 Systems of equations
Highlights from linear algebra David Milovich, Math 2 TA for sections -6 November, 28 Systems of equations A leading entry in a matrix is the first (leftmost) nonzero entry of a row. For example, the leading
More informationVector Spaces and Dimension. Subspaces of. R n. addition and scalar mutiplication. That is, if u, v in V and alpha in R then ( u + v) Exercise: x
Vector Spaces and Dimension Subspaces of Definition: A non-empty subset V is a subspace of if V is closed under addition and scalar mutiplication. That is, if u, v in V and alpha in R then ( u + v) V and
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 informationQuizzes for Math 304
Quizzes for Math 304 QUIZ. A system of linear equations has augmented matrix 2 4 4 A = 2 0 2 4 3 5 2 a) Write down this system of equations; b) Find the reduced row-echelon form of A; c) What are the pivot
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 information2 Eigenvectors and Eigenvalues in abstract spaces.
MA322 Sathaye Notes on Eigenvalues Spring 27 Introduction In these notes, we start with the definition of eigenvectors in abstract vector spaces and follow with the more common definition of eigenvectors
More information10. Rank-nullity Definition Let A M m,n (F ). The row space of A is the span of the rows. The column space of A is the span of the columns.
10. Rank-nullity Definition 10.1. Let A M m,n (F ). The row space of A is the span of the rows. The column space of A is the span of the columns. The nullity ν(a) of A is the dimension of the kernel. The
More informationFebruary 20 Math 3260 sec. 56 Spring 2018
February 20 Math 3260 sec. 56 Spring 2018 Section 2.2: Inverse of a Matrix Consider the scalar equation ax = b. Provided a 0, we can solve this explicity x = a 1 b where a 1 is the unique number such that
More information2. Every linear system with the same number of equations as unknowns has a unique solution.
1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations
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 informationSolutions to Math 51 First Exam April 21, 2011
Solutions to Math 5 First Exam April,. ( points) (a) Give the precise definition of a (linear) subspace V of R n. (4 points) A linear subspace V of R n is a subset V R n which satisfies V. If x, y V then
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 informationMath 240, 4.3 Linear Independence; Bases A. DeCelles. 1. definitions of linear independence, linear dependence, dependence relation, basis
Math 24 4.3 Linear Independence; Bases A. DeCelles Overview Main ideas:. definitions of linear independence linear dependence dependence relation basis 2. characterization of linearly dependent set using
More informationINVERSE OF A MATRIX [2.2]
INVERSE OF A MATRIX [2.2] The inverse of a matrix: Introduction We have a mapping from R n to R n represented by a matrix A. Can we invert this mapping? i.e. can we find a matrix (call it B for now) such
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 informationMTH Linear Algebra. Study Guide. Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education
MTH 3 Linear Algebra Study Guide Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education June 3, ii Contents Table of Contents iii Matrix Algebra. Real Life
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 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 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 informationMAT 242 CHAPTER 4: SUBSPACES OF R n
MAT 242 CHAPTER 4: SUBSPACES OF R n JOHN QUIGG 1. Subspaces Recall that R n is the set of n 1 matrices, also called vectors, and satisfies the following properties: x + y = y + x x + (y + z) = (x + y)
More informationSolutions to Math 51 Midterm 1 July 6, 2016
Solutions to Math 5 Midterm July 6, 26. (a) (6 points) Find an equation (of the form ax + by + cz = d) for the plane P in R 3 passing through the points (, 2, ), (2,, ), and (,, ). We first compute two
More informationSpan & Linear Independence (Pop Quiz)
Span & Linear Independence (Pop Quiz). Consider the following vectors: v = 2, v 2 = 4 5, v 3 = 3 2, v 4 = Is the set of vectors S = {v, v 2, v 3, v 4 } linearly independent? Solution: Notice that the number
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 information1. Solve each linear system using Gaussian elimination or Gauss-Jordan reduction. The augmented matrix of this linear system is
Solutions to Homework Additional Problems. Solve each linear system using Gaussian elimination or Gauss-Jordan reduction. (a) x + y = 8 3x + 4y = 7 x + y = 3 The augmented matrix of this linear system
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 informationVector Spaces 4.4 Spanning and Independence
Vector Spaces 4.4 and Independence Summer 2017 Goals Discuss two important basic concepts: Define linear combination of vectors. Define Span(S) of a set S of vectors. Define linear Independence of a set
More informationKevin James. MTHSC 3110 Section 4.3 Linear Independence in Vector Sp
MTHSC 3 Section 4.3 Linear Independence in Vector Spaces; Bases Definition Let V be a vector space and let { v. v 2,..., v p } V. If the only solution to the equation x v + x 2 v 2 + + x p v p = is the
More informationSection 1.5. Solution Sets of Linear Systems
Section 1.5 Solution Sets of Linear Systems Plan For Today Today we will learn to describe and draw the solution set of an arbitrary system of linear equations Ax = b, using spans. Ax = b Recall: the solution
More informationSection 3.3. Matrix Equations
Section 3.3 Matrix Equations Matrix Vector the first number is the number of rows the second number is the number of columns Let A be an m n matrix A = v v v n with columns v, v,..., v n Definition The
More informationMath Linear algebra, Spring Semester Dan Abramovich
Math 52 - Linear algebra, Spring Semester 22-23 Dan Abramovich Review We saw: an algorithm - row reduction, bringing to reduced echelon form - answers the questions: - consistency (no pivot on right) -
More informationAlgorithms to Compute Bases and the Rank of a Matrix
Algorithms to Compute Bases and the Rank of a Matrix Subspaces associated to a matrix Suppose that A is an m n matrix The row space of A is the subspace of R n spanned by the rows of A The column space
More informationEigenvalues, Eigenvectors. Eigenvalues and eigenvector will be fundamentally related to the nature of the solutions of state space systems.
Chapter 3 Linear Algebra In this Chapter we provide a review of some basic concepts from Linear Algebra which will be required in order to compute solutions of LTI systems in state space form, discuss
More information1 Linear systems, existence, uniqueness
Jor-el Briones / Math 2F, 25 Summer Session, Practice Midterm Page of 9 Linear systems, existence, uniqueness For each part, construct an augmented matrix for a linear system with the given properties,
More informationLinear Algebra: Sample Questions for Exam 2
Linear Algebra: Sample Questions for Exam 2 Instructions: This is not a comprehensive review: there are concepts you need to know that are not included. Be sure you study all the sections of the book and
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 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 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 information1111: Linear Algebra I
1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 13 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 13 1 / 8 The coordinate vector space R n We already used vectors in n dimensions
More informationLinear independence, span, basis, dimension - and their connection with linear systems
Linear independence span basis dimension - and their connection with linear systems Linear independence of a set of vectors: We say the set of vectors v v..v k is linearly independent provided c v c v..c
More informationAdditional Problems for Midterm 1 Review
Additional Problems for Midterm Review About This Review Set As stated in the syllabus, a goal of this course is to prepare students for more advanced courses that have this course as a pre-requisite.
More informationMATH 1553 PRACTICE FINAL EXAMINATION
MATH 553 PRACTICE FINAL EXAMINATION Name Section 2 3 4 5 6 7 8 9 0 Total Please read all instructions carefully before beginning. The final exam is cumulative, covering all sections and topics on the master
More informationChapter 3. Vector spaces
Chapter 3. Vector spaces Lecture notes for MA1111 P. Karageorgis pete@maths.tcd.ie 1/22 Linear combinations Suppose that v 1,v 2,...,v n and v are vectors in R m. Definition 3.1 Linear combination We say
More informationMath 2331 Linear Algebra
4.3 Linearly Independent Sets; Bases Math 233 Linear Algebra 4.3 Linearly Independent Sets; Bases Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math233
More informationSolving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
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 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 informationMath 2174: Practice Midterm 1
Math 74: Practice Midterm Show your work and explain your reasoning as appropriate. No calculators. One page of handwritten notes is allowed for the exam, as well as one blank page of scratch paper.. Consider
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 informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 8 MATRICES III Rank of a matrix 2 General systems of linear equations 3 Eigenvalues and eigenvectors Rank of a matrix
More information