Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of AcI?


 Geraldine Curtis
 1 years ago
 Views:
Transcription
1 Section 5. The Characteristic Polynomial Question: Given an n x n matrix A, how do we find its eigenvalues? Idea: Suppose c is an eigenvalue of A, then what is the determinant of AcI? Property The eigenvalues of a square matrix A are the values of t that satisfy det(a ti n ). Proof v 0 such that (A  ti n )v = 0 det(a  ti n ) = 0.
2 Section 5. The Characteristic Polynomial Definition Example: A = apple 4 6 A ti = apple 4 t 6 t det(a ti )=( 4 t)(6 t) ( ) =(t + )(t 5). So, eigenvalues of A can be  or 5. Question: What can we find eigenvectors corresponding to ? What can we find eigenvectors corresponding to 5?
3 Example: A = apple 4 6 A ti = apple 4 t 6 t det(a ti )=( 4 t)(6 t) ( ) =(t + )(t 5). By solving ( A+ I) x = 0, weget a basis{[ 1] of A corresponding to the eigenvalue. By solving ( A 5I) x = 0, weget a basis{[ 1 of A corresponding to the eigenvalue 5. T ] T }of }of the eigenspace the eigenspace Question: Is the characteristic polynomial of A equal to its reduced row echelon form? Question: What is the order of the characteristic polynomial of an n n matrix A?
4 Properties: 1. In general, a matrix and its reduced row echelon form have different characteristic polynomials. Therefore, elementary operations are not useful in finding the characteristic polynomial of a matrix.. The characteristic polynomial of an n n matrix is indeed a polynomial with degree n, as can be deduced from the complete determinant expansion of det(a  ti n ).. The eigenvalues of an upper triangular matrix are its diagonal entries. Question: Now that elementary operations do not preserve characteristic polynomials, is there any other operations that do?
5 * Property Similar matrices have the same characteristic polynomials. Proof Suppose A and B are n n matrices that are similar. Assume B = P where P is invertible. 1 AP det(b ti n )= =det(a ti n )
6 * Property Similar matrices have the same characteristic polynomials. Proof
7 Definition Example: T : linear operator on R that rotates a vector by 90º. standard matrix is the 90ºrotation matrix characteristic polynomial of T is det apple apple ti =det t 1 1 t apple = t +1 Think: Consider an n x n matrix A. How many eigenvalues does it have? Can we predict the number of eigenvalues of A from its characteristic polynomial?
8 Fact: An n x n matrix A can have less than n eigenvalues due to the following reasons. 1) Complex (not real) solutions of its characteristic polynomial. ) Multiplicity of a real eigenvalue. Definition If is an eigenvalue of an n n matrix M, thenthelargest positive integer k such that (t ) k is a factor of the characteristic polynomial of M (i.e., det(m ti n )), is called the (algebraic) multiplicity of.
9 Example: Find the characteristic polynomials of A = B = and
10 The characteristic polynomials of A = B = and are (t + 1) (t  ) and (t + 1) (t  ), respectively, i.e., eigenvalue 1 eigenvalue 1 A: B: multiplicity 1 multiplicity 1 Then, find the eigenspaces corresponding to each of these eigenvalues.
11 The characteristic polynomials of A = B = and are (t + 1) (t  ) and (t + 1) (t  ), respectively, i.e., eigenvalue 1 eigenvalue 1 A: B: multiplicity 1 multiplicity 1 Then, find the eigenspaces corresponding to each of these eigenvalues.
12 The characteristic polynomials of A = B = and are (t + 1) (t  ) and (t + 1) (t  ), respectively, i.e., eigenvalue 1 eigenvalue 1 A: B: multiplicity 1 multiplicity 1 For A, the eigenspace corresponding to the eigenvalue has a basis 8 < : 1 9 = 5 ; dimension = 1 = multiplicity of the eigenvalue. For B, the eigenspace corresponding to the eigenvalue has a basis 8 < : 0 5, = 5 ; dimension = = multiplicity of the eigenvalue.
13 Facts & Questions: 1) The multiplicity of an eigenvalue λ of A is at least 1. ) When the multiplicity of λ is 1, then must the dimension of its eigenspace be 1? ) When the multiplicity of λ is k (where k > 1), then must the dimension of its eigenspace be k?
14 Example: Find The characteristic polynomial of C =
15 Example: The characteristic polynomial of is (t + 1) (t  ), i.e., C: C = eigenvalue 1 multiplicity 1 Now, find the eigenspace corresponding to each of these eigenvalues. 5
16 Example: The characteristic polynomial of is (t + 1) (t  ), i.e., eigenvalue 1 multiplicity 1 For C, the eigenspace corresponding to the eigenvalue has a basis C: C = dimension = 1 < multiplicity of the eigenvalue.
17 Definition Theorem 5.1
18 Theorem 5.1 *Proof Suppose A R n n, the geometric multiplicity of λ is k, and {v 1, v,, v k } is a basis of the eigenspace corresponding to λ. can extend {v 1, v,, v k } to a basis B = {v 1, v,, v n } of R n. Consider the linear operator T A on R n. For 1 j k, [T A (v j ] B =[Av j ] B =[ v j ] B = [v j ] B = e j B matrix representation of T A has the form M = apple I k C O (n k) k D A and M are similar (Theorem 4.1 of Section 4.5). A and M have the same characteristic polynomial, which is
19 apple I det (M ti n ) = det k ti k C O D ti n k apple ( t)ik C = det O D ti n k = (det( t)i k )(det(d ti n k ) = ( t) k (det(d ti n k ) algebraic multiplicity of λ k. Example: T 4 x 1 x x 1 5A = 4 x 1 x 1 x x x standard matrix 5 A = characteristic polynomial det(a ti )=( 1 t)
20 Problem Set for Sections 5.1 and 5. Section 5.1: Problems 1,, 7, 9, 1, 15, 17, 19,, 5, 9, 1, 41, 4, 45, 47, 49, 51, 55, 57 Section 5.: Problems 1,, 5, 7, 9, 1, 15, 17, 1, 5, 7, 9,, 5, 9, 41, 5, 55, 57, 59, 61, 65, 69, 71.
and let s calculate the image of some vectors under the transformation T.
Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =
More informationEigenvalues and Eigenvectors
5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS n n Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),
More informationMath 304 Fall 2018 Exam 3 Solutions 1. (18 Points, 3 Pts each part) Let A, B, C, D be square matrices of the same size such that
Math 304 Fall 2018 Exam 3 Solutions 1. (18 Points, 3 Pts each part) Let A, B, C, D be square matrices of the same size such that det(a) = 2, det(b) = 2, det(c) = 1, det(d) = 4. 2 (a) Compute det(ad)+det((b
More informationCity Suburbs. : population distribution after m years
Section 5.3 Diagonalization of Matrices Definition Example: stochastic matrix To City Suburbs From City Suburbs.85.03 = A.15.97 City.15.85 Suburbs.97.03 probability matrix of a sample person s residence
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 315: Linear Algebra Solutions to Assignment 7
Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are
More informationEigenvalues and Eigenvectors
5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS nn Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),
More informationDiagonalization. Hungyi Lee
Diagonalization Hungyi Lee Review If Av = λv (v is a vector, λ is a scalar) v is an eigenvector of A excluding zero vector λ is an eigenvalue of A that corresponds to v Eigenvectors corresponding to λ
More informationDiagonalization. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics
Diagonalization MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Motivation Today we consider two fundamental questions: Given an n n matrix A, does there exist a basis
More informationRemark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 5 Eigenvectors and Eigenvalues In this chapter, vector means column vector 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
More informationMath 110 Linear Algebra Midterm 2 Review October 28, 2017
Math 11 Linear Algebra Midterm Review October 8, 17 Material Material covered on the midterm includes: All lectures from Thursday, Sept. 1st to Tuesday, Oct. 4th Homeworks 9 to 17 Quizzes 5 to 9 Sections
More informationMath 3191 Applied Linear Algebra
Math 9 Applied Linear Algebra Lecture 9: Diagonalization Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./9 Section. Diagonalization The goal here is to develop a useful
More informationDIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix
DIAGONALIZATION Definition We say that a matrix A of size n n is diagonalizable if there is a basis of R n consisting of eigenvectors of A ie if there are n linearly independent vectors v v n such that
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 information1 Last time: leastsquares problems
MATH Linear algebra (Fall 07) Lecture Last time: leastsquares problems Definition. If A is an m n matrix and b R m, then a leastsquares solution to the linear system Ax = b is a vector x R n such that
More informationTherefore, A and B have the same characteristic polynomial and hence, the same eigenvalues.
Similar Matrices and Diagonalization Page 1 Theorem If A and B are n n matrices, which are similar, then they have the same characteristic equation and hence the same eigenvalues. Proof Let A and B be
More informationLinear Algebra 1 Exam 1 Solutions 6/12/3
Linear Algebra 1 Exam 1 Solutions 6/12/3 Question 1 Consider the linear system in the variables (x, y, z, t, u), given by the following matrix, in echelon form: 1 2 1 3 1 2 0 1 1 3 1 4 0 0 0 1 2 3 Reduce
More informationCalculating determinants for larger matrices
Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det
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 informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationRecall : Eigenvalues and Eigenvectors
Recall : Eigenvalues and Eigenvectors Let A be an n n matrix. If a nonzero vector x in R n satisfies Ax λx for a scalar λ, then : The scalar λ is called an eigenvalue of A. The vector x is called an eigenvector
More information(b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).
.(5pts) Let B = 5 5. Compute det(b). (a) (b) (c) 6 (d) (e) 6.(5pts) Determine which statement is not always true for n n matrices A and B. (a) If two rows of A are interchanged to produce B, then det(b)
More information235 Final exam review questions
5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an
More informationComputationally, diagonal matrices are the easiest to work with. With this idea in mind, we introduce similarity:
Diagonalization We have seen that diagonal and triangular matrices are much easier to work with than are most matrices For example, determinants and eigenvalues are easy to compute, and multiplication
More informationProblems for M 10/26:
Math, Lesieutre Problem set # November 4, 25 Problems for M /26: 5 Is λ 2 an eigenvalue of 2? 8 Why or why not? 2 A 2I The determinant is, which means that A 2I has 6 a nullspace, and so there is an eigenvector
More informationDimension. Eigenvalue and eigenvector
Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, ranknullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,
More informationLINEAR ALGEBRA 1, 2012I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS
LINEAR ALGEBRA, I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,
More informationDeterminants by Cofactor Expansion (III)
Determinants by Cofactor Expansion (III) Comment: (Reminder) If A is an n n matrix, then the determinant of A can be computed as a cofactor expansion along the jth column det(a) = a1j C1j + a2j C2j +...
More information1. In this problem, if the statement is always true, circle T; otherwise, circle F.
Math 1553, Extra Practice for Midterm 3 (sections 4565) Solutions 1 In this problem, if the statement is always true, circle T; otherwise, circle F a) T F If A is a square matrix and the homogeneous equation
More informationLinear algebra II Tutorial solutions #1 A = x 1
Linear algebra II Tutorial solutions #. Find the eigenvalues and the eigenvectors of the matrix [ ] 5 2 A =. 4 3 Since tra = 8 and deta = 5 8 = 7, the characteristic polynomial is f(λ) = λ 2 (tra)λ+deta
More informationMath 205, Summer I, Week 4b:
Math 205, Summer I, 2016 Week 4b: Chapter 5, Sections 6, 7 and 8 (5.5 is NOT on the syllabus) 5.6 Eigenvalues and Eigenvectors 5.7 Eigenspaces, nondefective matrices 5.8 Diagonalization [*** See next slide
More informationExamples True or false: 3. Let A be a 3 3 matrix. Then there is a pattern in A with precisely 4 inversions.
The exam will cover Sections 6.6.2 and 7.7.4: True/False 30% Definitions 0% Computational 60% Skip Minors and Laplace Expansion in Section 6.2 and p. 304 (trajectories and phase portraits) in Section
More informationMAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to:
MAC 2 Module Determinants Learning Objectives Upon completing this module, you should be able to:. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor
More informationLinear Algebra Practice Problems
Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a
More 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 informationNo books, notes, any calculator, or electronic devices are allowed on this exam. Show all of your steps in each answer to receive a full credit.
MTH 309001 Fall 2016 Exam 1 10/05/16 Name (Print): PID: READ CAREFULLY THE FOLLOWING INSTRUCTION Do not open your exam until told to do so. This exam contains 7 pages (including this cover page) and 7
More informationEigenvalues and Eigenvectors: An Introduction
Eigenvalues and Eigenvectors: An Introduction The eigenvalue problem is a problem of considerable theoretical interest and wideranging application. For example, this problem is crucial in solving systems
More informationMATH 1553 PRACTICE MIDTERM 3 (VERSION B)
MATH 1553 PRACTICE MIDTERM 3 (VERSION B) Name Section 1 2 3 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 10 points. The maximum score on this exam is 50 points.
More informationEigenvalues for Triangular Matrices. ENGI 7825: Linear Algebra Review Finding Eigenvalues and Diagonalization
Eigenvalues for Triangular Matrices ENGI 78: Linear Algebra Review Finding Eigenvalues and Diagonalization Adapted from Notes Developed by Martin Scharlemann The eigenvalues for a triangular matrix are
More informationMath 240 Calculus III
The Calculus III Summer 2015, Session II Wednesday, July 8, 2015 Agenda 1. of the determinant 2. determinants 3. of determinants What is the determinant? Yesterday: Ax = b has a unique solution when A
More information4. Determinants.
4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.
More informationSolutions Problem Set 8 Math 240, Fall
Solutions Problem Set 8 Math 240, Fall 2012 5.6 T/F.2. True. If A is upper or lower diagonal, to make det(a λi) 0, we need product of the main diagonal elements of A λi to be 0, which means λ is one of
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 informationEvaluating Determinants by Row Reduction
Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives Reduce a matrix to row echelon form and evaluate its determinant.
More informationMATH 1553C MIDTERM EXAMINATION 3
MATH 553C MIDTERM EXAMINATION 3 Name GT Email @gatech.edu Please read all instructions carefully before beginning. Please leave your GT ID card on your desk until your TA scans your exam. Each problem
More informationftuiowamath2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST
me me ftuiowamath2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST 1. (1 pt) local/library/ui/eigentf.pg A is n n an matrices.. There are an infinite number
More informationAMS10 HW7 Solutions. All credit is given for effort. (5 pts for any missing sections) Problem 1 (20 pts) Consider the following matrix 2 A =
AMS1 HW Solutions All credit is given for effort. ( pts for any missing sections) Problem 1 ( pts) Consider the following matrix 1 1 9 a. Calculate the eigenvalues of A. Eigenvalues are 1 1.1, 9.81,.1
More informationChapter 3. Determinants and Eigenvalues
Chapter 3. Determinants and Eigenvalues 3.1. Determinants With each square matrix we can associate a real number called the determinant of the matrix. Determinants have important applications to the theory
More informationDiagonalization of Matrix
of Matrix King Saud University August 29, 2018 of Matrix Table of contents 1 2 of Matrix Definition If A M n (R) and λ R. We say that λ is an eigenvalue of the matrix A if there is X R n \ {0} such that
More informationChapter 2:Determinants. Section 2.1: Determinants by cofactor expansion
Chapter 2:Determinants Section 2.1: Determinants by cofactor expansion [ ] a b Recall: The 2 2 matrix is invertible if ad bc 0. The c d ([ ]) a b function f = ad bc is called the determinant and it associates
More informationLecture Notes: Eigenvalues and Eigenvectors. 1 Definitions. 2 Finding All Eigenvalues
Lecture Notes: Eigenvalues and Eigenvectors Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk 1 Definitions Let A be an n n matrix. If there
More informationReview for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions.
Review for Exam. Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. x + y z = 2 x + 2y + z = 3 x + y + (a 2 5)z = a 2 The augmented matrix for
More informationProperties of Linear Transformations from R n to R m
Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation
More informationc c c c c c c c c c a 3x3 matrix C= has a determinant determined by
Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.
More informationElementary Linear Algebra Review for Exam 3 Exam is Friday, December 11th from 1:153:15
Elementary Linear Algebra Review for Exam 3 Exam is Friday, December th from :53:5 The exam will cover sections: 6., 6.2, 7. 7.4, and the class notes on dynamical systems. You absolutely must be able
More informationMATH 1210 Assignment 4 Solutions 16RT1
MATH 1210 Assignment 4 Solutions 16RT1 Attempt all questions and show all your work. Due November 13, 2015. 1. Prove using mathematical induction that for any n 2, and collection of n m m matrices A 1,
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 informationMATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization.
MATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization. Eigenvalues and eigenvectors of an operator Definition. Let V be a vector space and L : V V be a linear operator. A number λ
More informationLU Factorization. A m x n matrix A admits an LU factorization if it can be written in the form of A = LU
LU Factorization A m n matri A admits an LU factorization if it can be written in the form of Where, A = LU L : is a m m lower triangular matri with s on the diagonal. The matri L is invertible and is
More informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors Definition 0 Let A R n n be an n n real matrix A number λ R is a real eigenvalue of A if there exists a nonzero vector v R n such that A v = λ v The vector v is called an eigenvector
More informationEigenvalues and Eigenvectors A =
Eigenvalues and Eigenvectors Definition 0 Let A R n n be an n n real matrix A number λ R is a real eigenvalue of A if there exists a nonzero vector v R n such that A v = λ v The vector v is called an eigenvector
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 informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
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 informationLinear Algebra Practice Final
. Let (a) First, Linear Algebra Practice Final Summer 3 3 A = 5 3 3 rref([a ) = 5 so if we let x 5 = t, then x 4 = t, x 3 =, x = t, and x = t, so that t t x = t = t t whence ker A = span(,,,, ) and a basis
More informationMATH 1553, C. JANKOWSKI MIDTERM 3
MATH 1553, C JANKOWSKI MIDTERM 3 Name GT Email @gatechedu Write your section number (E6E9) here: Please read all instructions carefully before beginning Please leave your GT ID card on your desk until
More informationSolutions to Final Exam
Solutions to Final Exam. Let A be a 3 5 matrix. Let b be a nonzero 5vector. Assume that the nullity of A is. (a) What is the rank of A? 3 (b) Are the rows of A linearly independent? (c) Are the columns
More informationMATH 1553 PRACTICE MIDTERM 3 (VERSION A)
MATH 1553 PRACTICE MIDTERM 3 (VERSION A) Name Section 1 2 3 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 10 points. The maximum score on this exam is 50 points.
More informationQuestion 7. Consider a linear system A x = b with 4 unknown. x = [x 1, x 2, x 3, x 4 ] T. The augmented
Question. How many solutions does x 6 = 4 + i have Practice Problems 6 d) 5 Question. Which of the following is a cubed root of the complex number i. 6 e i arctan() e i(arctan() π) e i(arctan() π)/3 6
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 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 informationANSWERS. E k E 2 E 1 A = B
MATH 7 Final Exam Spring ANSWERS Essay Questions points Define an Elementary Matrix Display the fundamental matrix multiply equation which summarizes a sequence of swap, combination and multiply operations,
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 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 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 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 informationBASIC ALGORITHMS IN LINEAR ALGEBRA. Matrices and Applications of Gaussian Elimination. A 2 x. A T m x. A 1 x A T 1. A m x
BASIC ALGORITHMS IN LINEAR ALGEBRA STEVEN DALE CUTKOSKY Matrices and Applications of Gaussian Elimination Systems of Equations Suppose that A is an n n matrix with coefficents in a field F, and x = (x,,
More informationCAAM 335 Matrix Analysis
CAAM 335 Matrix Analysis Solutions to Homework 8 Problem (5+5+5=5 points The partial fraction expansion of the resolvent for the matrix B = is given by (si B = s } {{ } =P + s + } {{ } =P + (s (5 points
More informationMATH 235. Final ANSWERS May 5, 2015
MATH 235 Final ANSWERS May 5, 25. ( points) Fix positive integers m, n and consider the vector space V of all m n matrices with entries in the real numbers R. (a) Find the dimension of V and prove your
More informationEigenvalues and Eigenvectors
Eigenvalues and Eigenvectors week 2 Fall 26 Eigenvalues and eigenvectors The most simple linear transformation from R n to R n may be the transformation of the form: T (x,,, x n ) (λ x, λ 2,, λ n x n
More informationDeterminants Chapter 3 of Lay
Determinants Chapter of Lay Dr. Doreen De Leon Math 152, Fall 201 1 Introduction to Determinants Section.1 of Lay Given a square matrix A = [a ij, the determinant of A is denoted by det A or a 11 a 1j
More informationMATH. 20F SAMPLE FINAL (WINTER 2010)
MATH. 20F SAMPLE FINAL (WINTER 2010) You have 3 hours for this exam. Please write legibly and show all working. No calculators are allowed. Write your name, ID number and your TA s name below. The total
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 informationEigenvalues for Triangular Matrices. ENGI 7825: Linear Algebra Review Finding Eigenvalues and Diagonalization
Eigenvalues for Triangular Matrices ENGI 78: Linear Algebra Review Finding Eigenvalues and Diagonalization Adapted from Notes Developed by Martin Scharlemann June 7, 04 The eigenvalues for a triangular
More informationNumerical Linear Algebra Homework Assignment  Week 2
Numerical Linear Algebra Homework Assignment  Week 2 Đoàn Trần Nguyên Tùng Student ID: 1411352 8th October 2016 Exercise 2.1: Show that if a matrix A is both triangular and unitary, then it is diagonal.
More informationMath Matrix Algebra
Math 44  Matrix Algebra Review notes  4 (Alberto Bressan, Spring 27) Review of complex numbers In this chapter we shall need to work with complex numbers z C These can be written in the form z = a+ib,
More informationChapter 5. Eigenvalues and Eigenvectors
Chapter 5 Eigenvalues and Eigenvectors Section 5. Eigenvectors and Eigenvalues Motivation: Difference equations A Biology Question How to predict a population of rabbits with given dynamics:. half of the
More informationMath 323 Exam 2 Sample Problems Solution Guide October 31, 2013
Math Exam Sample Problems Solution Guide October, Note that the following provides a guide to the solutions on the sample problems, but in some cases the complete solution would require more work or justification
More information2 b 3 b 4. c c 2 c 3 c 4
OHSx XM511 Linear Algebra: Multiple Choice Questions for Chapter 4 a a 2 a 3 a 4 b b 1. What is the determinant of 2 b 3 b 4 c c 2 c 3 c 4? d d 2 d 3 d 4 (a) abcd (b) abcd(a b)(b c)(c d)(d a) (c) abcd(a
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 informationHW2  Due 01/30. Each answer must be mathematically justified. Don t forget your name.
HW2  Due 0/30 Each answer must be mathematically justified. Don t forget your name. Problem. Use the row reduction algorithm to find the inverse of the matrix 0 0, 2 3 5 if it exists. Double check your
More informationPRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them.
Prof A Suciu MTH U37 LINEAR ALGEBRA Spring 2005 PRACTICE FINAL EXAM Are the following vectors independent or dependent? If they are independent, say why If they are dependent, exhibit a linear dependence
More informationLinear Algebra M1  FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8.
Linear Algebra M1  FIB Contents: 5 Matrices, systems of linear equations and determinants 6 Vector space 7 Linear maps 8 Diagonalization Anna de Mier Montserrat Maureso Dept Matemàtica Aplicada II Translation:
More information1. Let A = (a) 2 (b) 3 (c) 0 (d) 4 (e) 1
. Let A =. The rank of A is (a) (b) (c) (d) (e). Let P = {a +a t+a t } where {a,a,a } range over all real numbers, and let T : P P be a linear transformation dedifined by T (a + a t + a t )=a +9a t If
More informationMath 314/ Exam 2 Blue Exam Solutions December 4, 2008 Instructor: Dr. S. Cooper. Name:
Math 34/84  Exam Blue Exam Solutions December 4, 8 Instructor: Dr. S. Cooper Name: Read each question carefully. Be sure to show all of your work and not just your final conclusion. You may not use your
More informationMath 205, Summer I, Week 4b: Continued. Chapter 5, Section 8
Math 205, Summer I, 2016 Week 4b: Continued Chapter 5, Section 8 2 5.8 Diagonalization [reprint, week04: Eigenvalues and Eigenvectors] + diagonaliization 1. 5.8 Eigenspaces, Diagonalization A vector v
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More informationMAT 1302B Mathematical Methods II
MAT 1302B Mathematical Methods II Alistair Savage Mathematics and Statistics University of Ottawa Winter 2015 Lecture 19 Alistair Savage (uottawa) MAT 1302B Mathematical Methods II Winter 2015 Lecture
More informationSymmetric and anti symmetric matrices
Symmetric and anti symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, matrix A is symmetric if. A = A Because equal matrices have equal
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 information