MATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.


 Ethelbert White
 1 years ago
 Views:
Transcription
1 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
2 Rowcolumn rule: ijth entry of AB: row i column j 2
3 Properties of Matrix Multiplication: A : m n matrix. If the following are defined A( BC) = ( AB) C A( B + C) = AB + AC ( B + C) A = BA + CA r( AB) = ( ra) B = A( rb) I A = A = AI m n Question: Is AB = BA? 3
4 Warnings: In general, AB BA If AB = AC, then you can t cancel. If AB = 0, it is possible that A and B are both nonzero. 4
5 Definition: A : m n matrix. The transpose of A is the n m matrix whose columns are formed from the corresponding rows of A. Notation: T A 5
6
7 Theorem: Let A and B be matrices so that the following operations are defined. T ( A ) T = A ( A + B) T T = A T + B ( ra) T T = ra ( AB) T T T = B A 6
8
9
10 MATH 2331 Linear Algebra Section 2.2 The Inverse of a Matrix Let A be an n n matrix. Definition: A is said to be invertible if there exists an n n matrix B such that AB = BA = I n. If A is not invertible, then it is said to be singular. 1 If A has an inverse, then the inverse is unique  notation: A. 1
11 2 Let a b A c d =. Theorem: Let a b A c d =. If 0 ad bc, then A is invertible and 1 1 d b A ad bc c a = If 0 ad bc = = = =, then A is not invertible. Example:
12 3 Let A be n n ; Example: If exists, find the inverse of A =
13 Theorem 5: If A is an invertible matrix, then for any b R n, the equation Ax = b has the unique solution 1 =. x A b Proof: FACTS: 1) If A is invertible, then so is 1 A 1. Moreover, ( A ) 1 = A. 4
14 2) If A, B are invertible n n matrices, then AB is invertible. Moreover, ( ) AB = B A 3) If A is invertible, then so is 1 T T 1 A. Moreover, ( A ) ( A ) T =. 5
15 Elementary Matrices An elementary matrix is one that is obtained by performing a single elementary operation on an identity matrix. For example: 6
16 7 Example: a b c A d e f g h i =
17 Fact: Each elementary matrix is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I. 8
18 Theorem: An n n matrix is invertible if and only if A is row equivalent to I n. Any sequence of row operations that reduces A to I n also transforms I n into 1 A. 9
19 Algorithm: Rowreduce the augmented matrix [ A I ]. If A is row equivalent to I, then [ A I ] is row equivalent to I 1 A. Otherwise, A is not invertible. 10
20 Section 2.3 Characterizations of Invertible Matrices Theorem 8: The Invertible Matrix Theorem Let A be an n n matrix. The following are equivalent: a. A is an invertible matrix. b. A is rowequivalent to the n n identity matrix. c. A has n pivot positions d. The equation 0 Ax = has only the trivial solution. e. The columns of A are linearly independent. f. The linear transformation x g. The equation h. The columns of A span R n. Ax is onetoone. Ax = b has at least one solution for each b R n. i. The linear transformation x Ax maps R n onto R n. j. There is an nxn matrix C such that CA=I. k. There is an nxn matrix D such that AD=I l. T A is invertible. 11
21 12 Two tests to check whether a matrix is invertible or not: Test 1: The only solution to the system Ax=[0] is [0]. Test 2: Reduced form has n pivots ( full pivots). Example: Is the following matrix invertible? A =
22 13 Example: Invertible? A = B = C = D = E =
23 F = G = Example: Let A be an n n upper triangular matrix. Is it invertible? FACT:
24
25
26
27
28
29 Definition: A linear transformation :R n n T R is invertible if there exists a linear transformation : R n n S R such that S( T( x)) = T( S( x)) = x x. = =, for all R n 1 S is the inverse of T, notation: T. Theorem: Let induced by T. :R n n T R be a linear transformation and A be the standard matrix T is invertible if and only if A is an invertible matrix. 1 1 In this case, ( ) T x = A x. 15
30
31 2 2 Example: T :R R ; x1 x1 + 2x2 T = x 2 4x1 + 6x2 Is T invertible? If yes, find the inverse. 16
32 Example: Let T be a onetoone and onto linear transformation. Is it invertible? Example: Let T be an invertible linear transformation. Is it onetoone? Is it onto? 17
33 Exercise: Let A,B be n n matrices. Show that if AB is invertible, then A is invertible. 18
34 Extra Ex: Let A be an nxn matrix. True or False? If the equation Ax = 0 has only the trivial solution, then A is row equivalent to I. If the columns of A span R n, then the columns are linearly independent. Ax=b is consistent for any b in R n. If Ax=0 has nontrivial solutions, then A has less than n pivots. If the columns of A are linearly independent, then the columns span R n. If the equation Ax=b is consistent for each b in R n, then the transformation is onetoone. x Ax If Ax=b has infinitely many solutions for some b in R n, then A is not invertible. 19
35
36 MATH 2331 Linear Algebra Section 2.4 Partitioned matrices 1
37 Fact: T T T A B A C = T T C D B D Fact: A block diagonal matrix is invertible if each block on the diagonal is invertible. A 0 0 B is invertible if A and B are invertible. 2
38 Section 2.8, 2.9: Basis, Dimension, Rank, Column Space, Nullspace. Definition: The column space of a matrix A is the set of all linear combinations of the columns of A. Notation: Col(A) The null space of a matrix A is the set of all solutions to Ax = 0. Notation: Null(A) Rank of a matrix A is the number of pivot columns when the matrix is reduced. Notation: rank(a) 3
39
40 Definition: A basis for a subspace H is a linearly independent set that spans H. The dimension of a subspace is the number of elements in the basis. Fact: The pivot columns of A form a basis for Col(A). Fact: To find a basis for Null(A), describe the solution of Ax=0 in parametric form. 4
41 Invertible Matrix Theorem (continued) A: nxn matrix. A is invertible if and only if n m. The columns of A form a basis for R n. Col( A ) =R o. dim( Col( A)) p. rank( A) n = n q. Null( A ) = {0} = n r. dim( Null( A )) = 0 5
42
43
2. 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 informationChapter 2 Notes, Linear Algebra 5e Lay
Contents.1 Operations with Matrices..................................1.1 Addition and Subtraction.............................1. Multiplication by a scalar............................ 3.1.3 Multiplication
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 4A Notes. Written by Victoria Kala Last updated June 11, 2017
Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...
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 informationMath 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam
Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system
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 informationMTH501 Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~
MTH501 Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~ Question No: 1 (Marks: 1) If for a linear transformation the equation T(x) =0 has only the trivial solution then T is Onetoone Onto Question
More informationReview Let A, B, and C be matrices of the same size, and let r and s be scalars. Then
1 Sec 21 Matrix Operations Review Let A, B, and C be matrices of the same size, and let r and s be scalars Then (i) A + B = B + A (iv) r(a + B) = ra + rb (ii) (A + B) + C = A + (B + C) (v) (r + s)a = ra
More informationMath Final December 2006 C. Robinson
Math 2851 Final December 2006 C. Robinson 2 5 8 5 1 2 01 0 1. (21 Points) The matrix A = 1 2 2 3 1 8 3 2 6 has the reduced echelon form U = 0 0 1 2 0 0 0 0 0 1. 2 6 1 0 0 0 0 0 a. Find a basis for the
More informationChapter 3. Directions: For questions 111 mark each statement True or False. Justify each answer.
Chapter 3 Directions: For questions 111 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 informationMatrix operations Linear Algebra with Computer Science Application
Linear Algebra with Computer Science Application February 14, 2018 1 Matrix operations 11 Matrix operations If A is an m n matrix that is, a matrix with m rows and n columns then the scalar entry in the
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.7 LINEAR INDEPENDENCE LINEAR INDEPENDENCE Definition: An indexed set of vectors {v 1,, v p } in n is said to be linearly independent if the vector equation x x x
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 informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW SPENCER BECKERKAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for
More informationMath 1553 Introduction to Linear Algebra
Math 1553 Introduction to Linear Algebra Lecture Notes Chapter 2 Matrix Algebra School of Mathematics The Georgia Institute of Technology Math 1553 Lecture Notes for Chapter 2 Introduction, Slide 1 Section
More informationStudy Guide for Linear Algebra Exam 2
Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real
More information2018 Fall 2210Q Section 013 Midterm Exam II Solution
08 Fall 0Q Section 0 Midterm Exam II Solution True or False questions points 0 0 points) ) Let A be an n n matrix. If the equation Ax b has at least one solution for each b R n, then the solution is unique
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)
More informationGlossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the
More informationChapter 2: Matrix Algebra
Chapter 2: Matrix Algebra (Last Updated: October 12, 2016) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). Write A = 1. Matrix operations [a 1 a n. Then entry
More informationMATH10212 Linear Algebra B Homework 6. Be prepared to answer the following oral questions if asked in the supervision class:
MATH0 Linear Algebra B Homework 6 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra its Applications Pearson, 006 (or other editions) Normally, homework assignments
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 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 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 informationftuiowamath2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST
me me ftuiowamath255 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 2/3/2 at :3pm CST. ( pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following
More informationFind the solution set of 2x 3y = 5. Answer: We solve for x = (5 + 3y)/2. Hence the solution space consists of all vectors of the form
Math 2 Homework #7 March 4, 2 7.3.3. Find the solution set of 2x 3y = 5. Answer: We solve for x = (5 + 3y/2. Hence the solution space consists of all vectors of the form ( ( ( ( x (5 + 3y/2 5/2 3/2 x =
More informationICS 6N Computational Linear Algebra Vector Space
ICS 6N Computational Linear Algebra Vector Space Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 24 Vector Space Definition: A vector space is a non empty set V of
More informationAssignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work.
Assignment 1 Math 5341 Linear Algebra Review Give complete answers to each of the following questions Show all of your work Note: You might struggle with some of these questions, either because it has
More informationAnnouncements Monday, October 29
Announcements Monday, October 29 WeBWorK on determinents due on Wednesday at :59pm. The quiz on Friday covers 5., 5.2, 5.3. My office is Skiles 244 and Rabinoffice hours are: Mondays, 2 pm; Wednesdays,
More informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More informationYORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions
YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 222 3. M Test # July, 23 Solutions. For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For
More informationMath 54 First Midterm Exam, Prof. Srivastava September 23, 2016, 4:10pm 5:00pm, 155 Dwinelle Hall.
Math 54 First Midterm Exam, Prof Srivastava September 23, 26, 4:pm 5:pm, 55 Dwinelle Hall Name: SID: Instructions: Write all answers in the provided space This exam includes two pages of scratch paper,
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 informationMATH 152 Exam 1Solutions 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 information(c)
1. Find the reduced echelon form of the matrix 1 1 5 1 8 5. 1 1 1 (a) 3 1 3 0 1 3 1 (b) 0 0 1 (c) 3 0 0 1 0 (d) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (e) 1 0 5 0 0 1 3 0 0 0 0 Solution. 1 1 1 1 1 1 1 1
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 informationPractice Final Exam. Solutions.
MATH Applied Linear Algebra December 6, 8 Practice Final Exam Solutions Find the standard matrix f the linear transfmation T : R R such that T, T, T Solution: Easy to see that the transfmation T can be
More informationMATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix.
MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis.
More informationMAT Linear Algebra Collection of sample exams
MAT 342  Linear Algebra Collection of sample exams Ax. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system
More informationChapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015
Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal
More informationChapter 4  MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in
Chapter 4  MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column
More informationSolutions to Final Practice Problems Written by Victoria Kala Last updated 12/5/2015
Solutions to Final Practice Problems Written by Victoria Kala vtkala@math.ucsb.edu Last updated /5/05 Answers This page contains answers only. See the following pages for detailed solutions. (. (a x. See
More informationMATH 300, Second Exam REVIEW SOLUTIONS. NOTE: You may use a calculator for this exam You only need something that will perform basic arithmetic.
MATH 300, Second Exam REVIEW SOLUTIONS NOTE: You may use a calculator for this exam You only need something that will perform basic arithmetic. [ ] [ ] 2 2. Let u = and v =, Let S be the parallelegram
More informationMath 313 Chapter 1 Review
Math 313 Chapter 1 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 1.1 Introduction to Systems of Linear Equations 2 2 1.2 Gaussian Elimination 3 3 1.3 Matrices and Matrix Operations
More informationSection 4.5. Matrix Inverses
Section 4.5 Matrix Inverses The Definition of Inverse Recall: The multiplicative inverse (or reciprocal) of a nonzero number a is the number b such that ab = 1. We define the inverse of a matrix in almost
More informationFinal Examination 201NYC05 December and b =
. (5 points) Given A [ 6 5 8 [ and b (a) Express the general solution of Ax b in parametric vector form. (b) Given that is a particular solution to Ax d, express the general solution to Ax d in parametric
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationMath Linear Algebra Final Exam Review Sheet
Math 151 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a componentwise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationMath Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT
Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear
More informationElementary Linear Algebra Review for Exam 2 Exam is Monday, November 16th.
Elementary Linear Algebra Review for Exam Exam is Monday, November 6th. The exam will cover sections:.4,..4, 5. 5., 7., the class notes on Markov Models. You must be able to do each of the following. Section.4
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 informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Here are a slew of practice problems for the final culled from old exams:. Let P be the vector space of polynomials of degree at most. Let B = {, (t ), t + t }. (a) Show
More informationFamily Feud Review. Linear Algebra. October 22, 2013
Review Linear Algebra October 22, 2013 Question 1 Let A and B be matrices. If AB is a 4 7 matrix, then determine the dimensions of A and B if A has 19 columns. Answer 1 Answer A is a 4 19 matrix, while
More informationMATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION
MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether
More informationMATH 2210Q MIDTERM EXAM I PRACTICE PROBLEMS
MATH Q MIDTERM EXAM I PRACTICE PROBLEMS Date and place: Thursday, November, 8, inclass exam Section : : :5pm at MONT Section : 9: :5pm at MONT 5 Material: Sections,, 7 Lecture 9 8, Quiz, Worksheet 9 8,
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
More informationRemark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial
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 informationThis MUST hold matrix multiplication satisfies the distributive property.
The columns of AB are combinations of the columns of A. The reason is that each column of AB equals A times the corresponding column of B. But that is a linear combination of the columns of A with coefficients
More informationMA 511, Session 10. The Four Fundamental Subspaces of a Matrix
MA 5, Session The Four Fundamental Subspaces of a Matrix Let A be a m n matrix. (i) The row space C(A T )ofais the subspace of R n spanned by the rows of A. (ii) The null space N (A) ofa is the subspace
More informationMATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian.
MATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian. Spanning set Let S be a subset of a vector space V. Definition. The span of the set S is the smallest subspace W V that contains S. If
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 informationMath 2331 Linear Algebra
4.5 The Dimension of a Vector Space Math 233 Linear Algebra 4.5 The Dimension of a Vector Space ShangHuan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/ schiu/ ShangHuan
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University February 6, 2018 Linear Algebra (MTH
More informationMath 242 fall 2008 notes on problem session for week of This is a short overview of problems that we covered.
Math 242 fall 28 notes on problem session for week of 938 This is a short overview of problems that we covered.. For each of the following sets ask the following: Does it span R 3? Is it linearly independent?
More informationMath 123, Week 5: Linear Independence, Basis, and Matrix Spaces. Section 1: Linear Independence
Math 123, Week 5: Linear Independence, Basis, and Matrix Spaces Section 1: Linear Independence Recall that every row on the lefthand side of the coefficient matrix of a linear system A x = b which could
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 informationProblem # Max points possible Actual score Total 120
FINAL EXAMINATION  MATH 2121, FALL 2017. Name: ID#: Email: Lecture & Tutorial: Problem # Max points possible Actual score 1 15 2 15 3 10 4 15 5 15 6 15 7 10 8 10 9 15 Total 120 You have 180 minutes to
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 5228734. Normally, homework
More informationEK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016
EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016 Answer the questions in the spaces provided on the question sheets. You must show your work to get credit for your answers. There will
More informationORIE 6300 Mathematical Programming I August 25, Recitation 1
ORIE 6300 Mathematical Programming I August 25, 2016 Lecturer: Calvin Wylie Recitation 1 Scribe: Mateo Díaz 1 Linear Algebra Review 1 1.1 Independence, Spanning, and Dimension Definition 1 A (usually infinite)
More informationWarmup. True or false? Baby proof. 2. The system of normal equations for A x = y has solutions iff A x = y has solutions
Warmup True or false? 1. proj u proj v u = u 2. The system of normal equations for A x = y has solutions iff A x = y has solutions 3. The normal equations are always consistent Baby proof 1. Let A be
More informationMath 2940: Prelim 1 Practice Solutions
Math 294: Prelim Practice Solutions x. Find all solutions x = x 2 x 3 to the following system of equations: x 4 2x + 4x 2 + 2x 3 + 2x 4 = 6 x + 2x 2 + x 3 + x 4 = 3 3x 6x 2 + x 3 + 5x 4 = 5 Write your
More informationMath 54. Selected Solutions for Week 5
Math 54. Selected Solutions for Week 5 Section 4. (Page 94) 8. Consider the following two systems of equations: 5x + x 3x 3 = 5x + x 3x 3 = 9x + x + 5x 3 = 4x + x 6x 3 = 9 9x + x + 5x 3 = 5 4x + x 6x 3
More informationApplied Matrix Algebra Lecture Notes Section 2.2. Gerald Höhn Department of Mathematics, Kansas State University
Applied Matrix Algebra Lecture Notes Section 22 Gerald Höhn Department of Mathematics, Kansas State University September, 216 Chapter 2 Matrices 22 Inverses Let (S) a 11 x 1 + a 12 x 2 + +a 1n x n = b
More informationMath 3191 Applied Linear Algebra
Math 191 Applied Linear Algebra Lecture 9: Characterizations of Invertible Matrices Stephen Billups University of Colorado at Denver Math 191Applied Linear Algebra p.1/ Announcements Review for Exam 1
More informationChapter 6  Orthogonality
Chapter 6  Orthogonality Maggie Myers Robert A. van de Geijn The University of Texas at Austin Orthogonality Fall 2009 http://z.cs.utexas.edu/wiki/pla.wiki/ 1 Orthogonal Vectors and Subspaces http://z.cs.utexas.edu/wiki/pla.wiki/
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 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 information(a) only (ii) and (iv) (b) only (ii) and (iii) (c) only (i) and (ii) (d) only (iv) (e) only (i) and (iii)
. Which of the following are Vector Spaces? (i) V = { polynomials of the form q(t) = t 3 + at 2 + bt + c : a b c are real numbers} (ii) V = {at { 2 + b : a b are real numbers} } a (iii) V = : a 0 b is
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 informationMath 3191 Applied Linear Algebra
Math 191 Applied Linear Algebra Lecture 8: Inverse of a Matrix Stephen Billups University of Colorado at Denver Math 191Applied Linear Algebra p.1/0 Announcements We will not make it to section. tonight,
More information2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian
FE661  Statistical Methods for Financial Engineering 2. Linear algebra Jitkomut Songsiri matrices and vectors linear equations range and nullspace of matrices function of vectors, gradient and Hessian
More informationMatrix invertibility. RankNullity Theorem: For any ncolumn matrix A, nullity A +ranka = n
Matrix invertibility RankNullity Theorem: For any ncolumn matrix A, nullity A +ranka = n Corollary: Let A be an R C matrix. Then A is invertible if and only if R = C and the columns of A are linearly
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationAnnouncements Wednesday, November 01
Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section
More informationFinal Review Written by Victoria Kala SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015
Final Review Written by Victoria Kala vtkala@mathucsbedu SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015 Summary This review contains notes on sections 44 47, 51 53, 61, 62, 65 For your final,
More informationCSL361 Problem set 4: Basic linear algebra
CSL361 Problem set 4: Basic linear algebra February 21, 2017 [Note:] If the numerical matrix computations turn out to be tedious, you may use the function rref in Matlab. 1 Rowreduced echelon matrices
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 informationMath 60. Rumbos Spring Solutions to Assignment #17
Math 60. Rumbos Spring 2009 1 Solutions to Assignment #17 a b 1. Prove that if ad bc 0 then the matrix A = is invertible and c d compute A 1. a b Solution: Let A = and assume that ad bc 0. c d First consider
More informationRow Space and Column Space of a Matrix
Row Space and Column Space of a Matrix 1/18 Summary: To a m n matrix A = (a ij ), we can naturally associate subspaces of K n and of K m, called the row space of A and the column space of A, respectively.
More informationMATH 167: APPLIED LINEAR ALGEBRA Chapter 2
MATH 167: APPLIED LINEAR ALGEBRA Chapter 2 Jesús De Loera, UC Davis February 1, 2012 General Linear Systems of Equations (2.2). Given a system of m equations and n unknowns. Now m n is OK! Apply elementary
More information(a) II and III (b) I (c) I and III (d) I and II and III (e) None are true.
1 Which of the following statements is always true? I The null space of an m n matrix is a subspace of R m II If the set B = {v 1,, v n } spans a vector space V and dimv = n, then B is a basis for V III
More informationMath 1553, Introduction to Linear Algebra
Learning goals articulate what students are expected to be able to do in a course that can be measured. This course has courselevel learning goals that pertain to the entire course, and sectionlevel
More informationLinear Algebra Summary. Based on Linear Algebra and its applications by David C. Lay
Linear Algebra Summary Based on Linear Algebra and its applications by David C. Lay Preface The goal of this summary is to offer a complete overview of all theorems and definitions introduced in the chapters
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 information