Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015

Size: px
Start display at page:

Download "Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015"

Transcription

1 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 matrix, a 0 0 A = a 2 a 22 0 a 3 a 32 a 33 we can compute det(a) using an expansion across the first row, so det(a) = a C + 0 C C 3 = a (a 22 a 33 0 a 32 ) = a a 22 a 33. If A is upper triangular, we could compute det(a) using an expansion across the first column, and get the same result OR we could notice that A T is a lower triangular matrix with the same entries as A on the diagonal, so by what we just said, det(a) = det(a T ) is the product of entries on the diagonal. Note that this formula holds for ANY n n triangular matrix, not just a 3 3 matrix. 2. If A is an n n matrix, what is the definition of (a) an eigenvalue of A? An eigenvalue of A is a scalar λ such that the equation Ax = λx has a nonzero solution. (b) an eigenvector of A? An eigenvector of A is a nonzero vector u such that Au = λu for some scalar λ (in other words, a nonzero solution to the above equation). (c) Explain how finding eigenvalues/vectors can be restated in terms of determinants and singular matrices. If we rewrite the equation Ax = λx as Ax λx = 0, and then factor out the x to get (A λi)x = 0, we can see that finding eigenvalues and eigenvectors is the same thing as finding scalars λ such that (A λi)x = 0 has nonzero solutions. But, this has nonzero solutions if and only if the matrix A λi is singular, which occurs if and only if det(a λi) = 0. So, to find eigenvalues, we can first find all the values of λ such that det(a λi) = 0, and then we can find eigenvectors by solving the equation (A λi)x = 0 (which is guaranteed to have a nonzero solution because A λi is singular).

2 3. If B is a basis for R n, explain what is meant by the notation x B. The vector x B is the vector of coordinates of x in the basis B. If B is the basis {u,..., u n }, then the notation means c x B =... c n x = c u + + c n u n. B 4. If B and B 2 are two bases for R n, what is the change of basis matrix from B to B 2? The change of basis matrix is an invertible n n matrix M that allows us to translate between the coordinates of x in one basis to the coordinates of x in another basis. Specifically, the change of basis matrix from B to B 2 is a matrix M such that x B2 = Mx B. There is a formula for the matrix M: if B = {u,..., u n } and B 2 = {v,..., v n }, let U be the matrix U = [u... u n ] and V be the matrix V = [v... v n ]. Then, the matrix M is given by M = V U. 5. Give an example of the following, or explain why such an example does not exist. (a) A nonsingular matrix A such that det(a) = det(a 2 ) Because det(a 2 ) = det(a A) = det(a) det(a) = det(a) 2, if det(a) = det(a) 2, we must have det(a) = or det(a) = 0. In order to be nonsingular, we must have det(a) 0. Therefore, any matrix whose determinant is one will satisfy this, so one example would be the identity matrix, [ ] 0 A = 0 and another would be the matrix A = [ ] 2.

3 (b) A nonsingular matrix A such that det(a) = 0 This is impossible. An nonsingular matrix must be invertible, meaning det(a) 0. (c) A nonsingular matrix A with only one eigenvalue One example of this is the identity matrix, [ ] 0 A =, 0 whose only eigenvalue is one. Any triangular matrix whose entries on the diagonal are equal and nonzero would also work, because the eigenvalues of a triangular matrix are just the numbers on the diagonal. (d) A singular matrix A with only one eigenvalue One example of this is the zero matrix, [ ] 0 0 A =, 0 0 whose only eigenvalue is 0. Any triangular matrix whose entries on the diagonal are all 0 would also work, because the eigenvalues of a triangular matrix are just the numbers on the diagonal. (e) An n n matrix A that has less than n eigenvalues See parts (c) and (d). Any matrix with at least one repeated eigenvalue will have less than n eigenvalues. (f) An n n matrix A with more than n eigenvalues This is impossible. The eigenvalues are the roots of the characteristic polynomial det(a λi). If A is an n n matrix, this is a degree n polynomial, so it can have at most n distinct roots. 6. (This problem is part of a problem in an old exam.) Consider a 4 4 matrix A such that the following eigenvectors of A form a basis of R 4 : ,,, eigenvalues: 2

4 (a) Find a basis for null(a 2I). Explain why your answer is correct. The eigenspace of 2 is precisely the nullspace of A 2I, and the eigenvalue 2 has multiplicity, meaning the dimension of the eigenspace must actually be. Therefore, a basis for null(a 2I) is the eigenvector corresponding to 2, 0. (b) What is rank(a I)? Justify your answer. The eigenspace of has dimension 2 because the vectors corresponding to the eigenvalue are linearly independent. Therefore, nullity(a I) = 2, so by the rank-nullity theorem, rank(a I) = 2. (c) If x = 3, find A7 x. 3 We can write x as a linear combination of eigenvectors: 0 x = so A 7 x = A A = ( )7 = 0 2 = For each statement below, determine if it is True or False. Justify your answer. The justification is the most important part! (a) If A and B are n n matrices, then AB is singular if and only if A or B is singular. TRUE. AB is singular if and only if det(ab) = 0. But, det(ab) = det(a) det(b), so this is zero if any only if det(a) = 0 or det(b) = 0, which holds if and only if A or B is singular.

5 (b) If T : R n R n is a linear transformation with ker(t ) {0}, then the matrix A such that T (x) = Ax has det(a) = 0. TRUE. This is part of the Big Theorem. If ker(t ) {0}, then null(a) {0}, so there is some nonzero solution to the equation Ax = 0. But, that means the columns of A are linearly dependent, so A is singular, so det(a) = 0. (c) If A and B are row equivalent matrices, then det(a) = det(b). FALSE. If A = [ ] 0, B = 0 [ ] then A and B are row equivalent, but det(a) = det(b) = 2. (d) If det(a) = det(b) and A is a nonsingular matrix, then A and B are row equivalent matrices. TRUE. If A is nonsingular, det(a) 0 and A row reduces to the identity matrix. But, because 0 det(a) = det(b), we know B is also nonsingular, so B also row reduces to the identity matrix. Therefore, A and B are row equivalent (they reduce to the same matrix). (e) If det(a) = 0, then one of the columns of A can be written as a linear combination of the others. TRUE. This is part of the Big Theorem. If det(a) = 0, then the columns of A are not linearly independent, so one must be a linear combination of the other columns. (f) If det(a) = 0, then 0 is an eigenvalue of A. TRUE. If 0 = det(a) = det(a 0 I), then 0 is an eigenvalue of A because is satisfies the equation det(a λi) = 0. We could also justify this with the Big Theorem: if det(a) = 0, then A is singular, so the Big Theorem tells us 0 must be an eigenvalue of A. (g) If 0 is an eigenvalue of A, then the linear transformation T (x) = Ax is not onto. TRUE. If 0 is an eigenvalue of A, by the Big Theorem, the columns of A do not span R n. Therefore, the linear transformation T is not onto. (h) If λ is an eigenvalue of A, the eigenspace of λ is equal to null(a λi). TRUE. The eigenspace of λ is defined to be all vectors x that satisfy the equation Ax = λx (eigenspace = all eigenvectors plus the zero vector). But,

6 we can rewrite this equation as (A λi)x = 0 (see problem 2(c)), and the set of all solutions to this equation is exactly null(a λi). (i) If λ is an eigenvalue of A, then any vector u such that Au = λu is an eigenvector. FALSE. The zero vector satisfies A0 = λ0, but eigenvectors are required to be nonzero. (j) If B and B 2 are two bases for R n, then the change of basis matrix from B to B 2 is invertible. TRUE. The change of basis matrix is ALWAYS invertible. We could see this from the formula: if B = {u,..., u n } and B 2 = {v,..., v n }, let U be the matrix U = [u... u n ] and V be the matrix V = [v... v n ]. Then, the matrix change of basis matrix is given by V U, which has an inverse U V, so is invertible. Note that each of U and V are invertible because their columns form a basis for R n, so are linearly independent and spanning. (k) If M is the change of basis matrix from B to B 2 and N is the change of basis matrix from B 2 and B 3, then MN is the change of basis matrix from B and B 3. FALSE. The problem says that x B2 = Mx B and x B3 = Nx B2. Substituting the first equation in to the second, we see that x B3 = N(Mx B ) = NMx B so the change of basis matrix from B and B 3 is NM.

Warm-up. True or false? Baby proof. 2. The system of normal equations for A x = y has solutions iff A x = y has solutions

Warm-up. True or false? Baby proof. 2. The system of normal equations for A x = y has solutions iff A x = y has solutions Warm-up 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 information

Final 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 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 information

Recall : Eigenvalues and Eigenvectors

Recall : 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

Remark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Remark 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 information

1. Select the unique answer (choice) for each problem. Write only the answer.

1. Select the unique answer (choice) for each problem. Write only the answer. MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +

More information

Math 308 Practice Test for Final Exam Winter 2015

Math 308 Practice Test for Final Exam Winter 2015 Math 38 Practice Test for Final Exam Winter 25 No books are allowed during the exam. But you are allowed one sheet ( x 8) of handwritten notes (back and front). You may use a calculator. For TRUE/FALSE

More information

LINEAR ALGEBRA REVIEW

LINEAR ALGEBRA REVIEW LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN 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 information

Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Remark 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 information

1. In this problem, if the statement is always true, circle T; otherwise, circle F.

1. In this problem, if the statement is always true, circle T; otherwise, circle F. Math 1553, Extra Practice for Midterm 3 (sections 45-65) 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 information

In Class Peer Review Assignment 2

In Class Peer Review Assignment 2 Name: Due Date: Tues. Dec. 5th In Class Peer Review Assignment 2 D.M. 1 : 7 (7pts) Short Answer 8 : 14 (32pts) T/F and Multiple Choice 15 : 30 (15pts) Total out of (54pts) Directions: Put only your answers

More information

Chapter 3: Theory Review: Solutions Math 308 F Spring 2015

Chapter 3: Theory Review: Solutions Math 308 F Spring 2015 Chapter : Theory Review: Solutions Math 08 F Spring 05. What two properties must a function T : R m R n satisfy to be a linear transformation? (a) For all vectors u and v in R m, T (u + v) T (u) + T (v)

More information

MATH. 20F SAMPLE FINAL (WINTER 2010)

MATH. 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 information

Solutions to Final Practice Problems Written by Victoria Kala Last updated 12/5/2015

Solutions 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 information

2018 Fall 2210Q Section 013 Midterm Exam II Solution

2018 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 information

MATH 240 Spring, Chapter 1: Linear Equations and Matrices

MATH 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 information

2. Every linear system with the same number of equations as unknowns has a unique solution.

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 information

Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017

Math 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 information

ELE/MCE 503 Linear Algebra Facts Fall 2018

ELE/MCE 503 Linear Algebra Facts Fall 2018 ELE/MCE 503 Linear Algebra Facts Fall 2018 Fact N.1 A set of vectors is linearly independent if and only if none of the vectors in the set can be written as a linear combination of the others. Fact N.2

More information

MATH 2360 REVIEW PROBLEMS

MATH 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 information

Math 323 Exam 2 Sample Problems Solution Guide October 31, 2013

Math 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 information

Problems for M 10/26:

Problems 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 information

MA 265 FINAL EXAM Fall 2012

MA 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 information

(b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).

(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 information

Announcements Monday, October 29

Announcements 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 information

MATH 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. 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 information

Math Linear Algebra Final Exam Review Sheet

Math Linear Algebra Final Exam Review Sheet Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of

More information

Math Final December 2006 C. Robinson

Math Final December 2006 C. Robinson Math 285-1 Final December 2006 C. Robinson 2 5 8 5 1 2 0-1 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 information

Diagonalization. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics

Diagonalization. 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 information

Study Guide for Linear Algebra Exam 2

Study 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 information

Math 2174: Practice Midterm 1

Math 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 information

Math 369 Exam #2 Practice Problem Solutions

Math 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 information

Dimension. Eigenvalue and eigenvector

Dimension. Eigenvalue and eigenvector Dimension. Eigenvalue and eigenvector Math 112, week 9 Goals: Bases, dimension, rank-nullity theorem. Eigenvalue and eigenvector. Suggested Textbook Readings: Sections 4.5, 4.6, 5.1, 5.2 Week 9: Dimension,

More information

Math 314/ Exam 2 Blue Exam Solutions December 4, 2008 Instructor: Dr. S. Cooper. Name:

Math 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 information

Linear Algebra (MATH ) Spring 2011 Final Exam Practice Problem Solutions

Linear Algebra (MATH ) Spring 2011 Final Exam Practice Problem Solutions Linear Algebra (MATH 4) Spring 2 Final Exam Practice Problem Solutions Instructions: Try the following on your own, then use the book and notes where you need help. Afterwards, check your solutions with

More information

Lecture Summaries for Linear Algebra M51A

Lecture 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 information

1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued)

1. Linear systems of equations. Chapters 7-8: Linear Algebra. Solution(s) of a linear system of equations (continued) 1 A linear system of equations of the form Sections 75, 78 & 81 a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a m1 x 1 + a m2 x 2 + + a mn x n = b m can be written in matrix

More information

ft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST

ft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST me me ft-uiowa-math255 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 2/3/2 at :3pm CST. ( pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following

More information

(a) II and III (b) I (c) I and III (d) I and II and III (e) None are true.

(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 information

1. General Vector Spaces

1. General Vector Spaces 1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule

More information

c c c c c c c c c c a 3x3 matrix C= has a determinant determined by

c 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 information

Math 18.6, Spring 213 Problem Set #6 April 5, 213 Problem 1 ( 5.2, 4). Identify all the nonzero terms in the big formula for the determinants of the following matrices: 1 1 1 2 A = 1 1 1 1 1 1, B = 3 4

More information

REVIEW FOR EXAM II. The exam covers sections , the part of 3.7 on Markov chains, and

REVIEW FOR EXAM II. The exam covers sections , the part of 3.7 on Markov chains, and REVIEW FOR EXAM II The exam covers sections 3.4 3.6, the part of 3.7 on Markov chains, and 4.1 4.3. 1. The LU factorization: An n n matrix A has an LU factorization if A = LU, where L is lower triangular

More information

and let s calculate the image of some vectors under the transformation T.

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 information

ICS 6N Computational Linear Algebra Eigenvalues and Eigenvectors

ICS 6N Computational Linear Algebra Eigenvalues and Eigenvectors ICS 6N Computational Linear Algebra Eigenvalues and Eigenvectors Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 34 The powers of matrix Consider the following dynamic

More information

235 Final exam review questions

235 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 information

(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =

(a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax = . (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)

More information

Chapter 5. Eigenvalues and Eigenvectors

Chapter 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 information

Math 315: Linear Algebra Solutions to Assignment 7

Math 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 information

MATH 221, Spring Homework 10 Solutions

MATH 221, Spring Homework 10 Solutions MATH 22, Spring 28 - Homework Solutions Due Tuesday, May Section 52 Page 279, Problem 2: 4 λ A λi = and the characteristic polynomial is det(a λi) = ( 4 λ)( λ) ( )(6) = λ 6 λ 2 +λ+2 The solutions to the

More information

MATH 235. Final ANSWERS May 5, 2015

MATH 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 information

MTH 464: Computational Linear Algebra

MTH 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 information

Name: Final Exam MATH 3320

Name: Final Exam MATH 3320 Name: Final Exam MATH 3320 Directions: Make sure to show all necessary work to receive full credit. If you need extra space please use the back of the sheet with appropriate labeling. (1) State the following

More information

Spring 2014 Math 272 Final Exam Review Sheet

Spring 2014 Math 272 Final Exam Review Sheet Spring 2014 Math 272 Final Exam Review Sheet You will not be allowed use of a calculator or any other device other than your pencil or pen and some scratch paper. Notes are also not allowed. In kindness

More information

CS 143 Linear Algebra Review

CS 143 Linear Algebra Review CS 143 Linear Algebra Review Stefan Roth September 29, 2003 Introductory Remarks This review does not aim at mathematical rigor very much, but instead at ease of understanding and conciseness. Please see

More information

Therefore, A and B have the same characteristic polynomial and hence, the same eigenvalues.

Therefore, 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 information

Chapter 3. Determinants and Eigenvalues

Chapter 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 information

Eigenvalues and Eigenvectors

Eigenvalues 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 information

Chapter 4 & 5: Vector Spaces & Linear Transformations

Chapter 4 & 5: Vector Spaces & Linear Transformations Chapter 4 & 5: Vector Spaces & Linear Transformations Philip Gressman University of Pennsylvania Philip Gressman Math 240 002 2014C: Chapters 4 & 5 1 / 40 Objective The purpose of Chapter 4 is to think

More information

Econ Slides from Lecture 7

Econ Slides from Lecture 7 Econ 205 Sobel Econ 205 - Slides from Lecture 7 Joel Sobel August 31, 2010 Linear Algebra: Main Theory A linear combination of a collection of vectors {x 1,..., x k } is a vector of the form k λ ix i for

More information

Definition (T -invariant subspace) Example. Example

Definition (T -invariant subspace) Example. Example Eigenvalues, Eigenvectors, Similarity, and Diagonalization We now turn our attention to linear transformations of the form T : V V. To better understand the effect of T on the vector space V, we begin

More information

Solutions to Final Exam

Solutions to Final Exam Solutions to Final Exam. Let A be a 3 5 matrix. Let b be a nonzero 5-vector. 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 information

Numerical Linear Algebra Homework Assignment - Week 2

Numerical 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 information

Math 20F Practice Final Solutions. Jor-el Briones

Math 20F Practice Final Solutions. Jor-el Briones Math 2F Practice Final Solutions Jor-el Briones Jor-el Briones / Math 2F Practice Problems for Final Page 2 of 6 NOTE: For the solutions to these problems, I skip all the row reduction calculations. Please

More information

Quizzes for Math 304

Quizzes 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 information

Conceptual Questions for Review

Conceptual Questions for Review Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.

More information

Linear Algebra Highlights

Linear Algebra Highlights Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to

More information

MATH 1553 PRACTICE MIDTERM 3 (VERSION B)

MATH 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 information

YORK 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 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 information

Math 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam

Math 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 information

MAT Linear Algebra Collection of sample exams

MAT Linear Algebra Collection of sample exams MAT 342 - Linear Algebra Collection of sample exams A-x. (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 information

Linear Algebra: Sample Questions for Exam 2

Linear 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

Math 265 Linear Algebra Sample Spring 2002., rref (A) =

Math 265 Linear Algebra Sample Spring 2002., rref (A) = Math 265 Linear Algebra Sample Spring 22. It is given that A = rref (A T )= 2 3 5 3 2 6, rref (A) = 2 3 and (a) Find the rank of A. (b) Find the nullityof A. (c) Find a basis for the column space of A.

More information

Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Exam 2 will be held on Tuesday, April 8, 7-8pm in 117 MacMillan What will be covered The exam will cover material from the lectures

More information

Math 313 Chapter 5 Review

Math 313 Chapter 5 Review Math 313 Chapter 5 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 5.1 Real Vector Spaces 2 2 5.2 Subspaces 3 3 5.3 Linear Independence 4 4 5.4 Basis and Dimension 5 5 5.5 Row

More information

Determinants by Cofactor Expansion (III)

Determinants 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 information

Problem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show

Problem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show MTH 0: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology - Kanpur Problem Set Problems marked (T) are for discussions in Tutorial sessions (T) If A is an m n matrix,

More information

Chapter 3 Transformations

Chapter 3 Transformations Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases

More information

TMA Calculus 3. Lecture 21, April 3. Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013

TMA Calculus 3. Lecture 21, April 3. Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013 TMA4115 - Calculus 3 Lecture 21, April 3 Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013 www.ntnu.no TMA4115 - Calculus 3, Lecture 21 Review of last week s lecture Last week

More information

Review problems for MA 54, Fall 2004.

Review 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 information

EXAM. Exam #3. Math 2360, Spring April 24, 2001 ANSWERS

EXAM. Exam #3. Math 2360, Spring April 24, 2001 ANSWERS EXAM Exam #3 Math 2360, Spring 200 April 24, 200 ANSWERS i 40 pts Problem In this problem, we will work in the vectorspace P 3 = { ax 2 + bx + c a, b, c R }, the space of polynomials of degree less than

More information

4. Determinants.

4. 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 information

MATH 1553, Intro to Linear Algebra FINAL EXAM STUDY GUIDE

MATH 1553, Intro to Linear Algebra FINAL EXAM STUDY GUIDE MATH 553, Intro to Linear Algebra FINAL EXAM STUDY GUIDE In studying for the final exam, you should FIRST study all tests andquizzeswehave had this semester (solutions can be found on Canvas). Then go

More information

80 min. 65 points in total. The raw score will be normalized according to the course policy to count into the final score.

80 min. 65 points in total. The raw score will be normalized according to the course policy to count into the final score. This is a closed book, closed notes exam You need to justify every one of your answers unless you are asked not to do so Completely correct answers given without justification will receive little credit

More information

Examples True or false: 3. Let A be a 3 3 matrix. Then there is a pattern in A with precisely 4 inversions.

Examples 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 information

Practice Final Exam. Solutions.

Practice 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 information

Solution Set 7, Fall '12

Solution Set 7, Fall '12 Solution Set 7, 18.06 Fall '12 1. Do Problem 26 from 5.1. (It might take a while but when you see it, it's easy) Solution. Let n 3, and let A be an n n matrix whose i, j entry is i + j. To show that det

More information

Math 2331 Linear Algebra

Math 2331 Linear Algebra 5. Eigenvectors & Eigenvalues Math 233 Linear Algebra 5. Eigenvectors & Eigenvalues Shang-Huan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/ schiu/ Shang-Huan Chiu,

More information

Elementary Linear Algebra Review for Exam 2 Exam is Monday, November 16th.

Elementary 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 information

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column

More information

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 A-cI?

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 A-cI? 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 A-cI? Property The eigenvalues

More information

Online Exercises for Linear Algebra XM511

Online Exercises for Linear Algebra XM511 This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2

More information

ANSWERS. E k E 2 E 1 A = B

ANSWERS. 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 information

Mathematical Methods for Engineers 1 (AMS10/10A)

Mathematical Methods for Engineers 1 (AMS10/10A) Mathematical Methods for Engineers 1 (AMS10/10A) Quiz 5 - Friday May 27th (2016) 2:00-3:10 PM AMS 10 AMS 10A Name: Student ID: Multiple Choice Questions (3 points each; only one correct answer per question)

More information

Family Feud Review. Linear Algebra. October 22, 2013

Family 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 information

Cheat Sheet for MATH461

Cheat Sheet for MATH461 Cheat Sheet for MATH46 Here is the stuff you really need to remember for the exams Linear systems Ax = b Problem: We consider a linear system of m equations for n unknowns x,,x n : For a given matrix A

More information

Math 224, Fall 2007 Exam 3 Thursday, December 6, 2007

Math 224, Fall 2007 Exam 3 Thursday, December 6, 2007 Math 224, Fall 2007 Exam 3 Thursday, December 6, 2007 You have 1 hour and 20 minutes. No notes, books, or other references. You are permitted to use Maple during this exam, but you must start with a blank

More information

Properties of Linear Transformations from R n to R m

Properties 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 information

NATIONAL UNIVERSITY OF SINGAPORE MA1101R

NATIONAL UNIVERSITY OF SINGAPORE MA1101R Student Number: NATIONAL UNIVERSITY OF SINGAPORE - Linear Algebra I (Semester 2 : AY25/26) Time allowed : 2 hours INSTRUCTIONS TO CANDIDATES. Write down your matriculation/student number clearly in the

More information

Linear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.

Linear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0. Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the

More information