Math 1553 Worksheet 5.3, 5.5

Size: px
Start display at page:

Download "Math 1553 Worksheet 5.3, 5.5"

Transcription

1 Math Worksheet, Answer yes / no / maybe In each case, A is a matrix whose entries are real a) If A is a matrix with characteristic polynomial λ(λ ), then the - eigenspace is -dimensional b) If A is an invertible matrix, then A is diagonalizable c) Can a matrix A have a non-real complex eigenvalue with multiplicity? d) Can a matrix A have eigenvalues,, and + i? a) Maybe The geometric multiplicity of λ = can be or For example, the matrix has a - eigenspace which is -dimensional, whereas the matrix has a -eigenspace which is -dimensional Both matrices have characteristic polynomial λ(λ ) b) Maybe The identity matrix is invertible and diagonalizable, but the matrix is invertible but not diagonalizable c) No If c is a (non-real) complex eigenvalue with multiplicity, then its conjugate c is an eigenvalue with multiplicity since complex eigenvalues always occur in conjugate pairs This would mean A has a characteristic polynomial of degree 4 or more, which is impossible for a matrix d) No If + i is an eigenvalue then so is its conjugate i Let A = The characteristic polynomial for A is λ + 7λ 6λ +, and λ is a factor Decide if A is diagonalizable If it is, find an invertible matrix P and a diagonal matrix D such that A = P DP By polynomial division, λ + 7λ 6λ + = λ + 4λ 4 = (λ ) λ

2 Thus, the characteristic poly factors as (λ )(λ ), so the eigenalues are λ = and λ = For λ =, we row-reduce A I: R =R +6R then R = R R R (New R )/ R =R 6R R =R +6R R =R R 6 6 Therefore, the solutions to A I are x = x, x = x, x = x x x x = x x x = x The -eigenspace has basis For λ =, we row-reduce A I: rref 6 The solutions to A I are x = 6x x, x = x, x = x x x x = 6x x x x 6 The -eigenspace has basis, Therefore, A = P DP where P = 6 6 = x + x D = Note that we arranged the eigenvectors in P in order of the eigenvalues,,, so we had to put the diagonals of D in the same order

3 Let A = a) Find all eigenvalues and eigenvectors of A b) Write A = PC P, where C is a rotation followed by a scale Describe what A does geometrically Draw a picture a) The characteristic polynomial is λ Tr(A)λ + det(a) = λ λ + λ λ + = λ = ± 4 = ± 4i = ± i For the eigenvalue λ = i, we row-reduce A ( i)i i R =R /i i R =R +R i i i So x = i x and x = x A corresponding eigenvector is v = i, and any nonzero complex multiple of v will also be an eigenvector (If we used the trick from the slides, we would have found that an eigenvector is, which is really just i times the eigenvector v above) i From the correspondence between conjugate eigenvalues and their eigenvectors, we know (without doing any additional work!) that for the eigenvalue i λ = + i, a corresponding eigenvector is w = v = i b) We use λ = i and its associated v = A = PC P where P = Re(v) Im(v) = and Re(λ) C = Im(λ) Im(λ) = Re(λ) The scale is by a factor of λ = + i = + = If we factor this out of C we get C = We see cos(θ) = and sin(θ) =, so tan(θ) = and θ = arctan() C is rotation by the angle arctan(), followed by scaling by a factor of See the [interactive] demo for how A acts geometrically

4 ***Note: there are multiple answers possible for part b) For example, the trick from the slides says that if λ is an eigenvalue b of A, then one eigenvector is where a b is the first row of A λi a Row of A λi was i, so as an eigenvector i This would give us P = rather than P = However, it would still be the case that A = PC P since PC P = = = A Supplemental Problems For those who want additional practice problems after completing the worksheet, here are some extra practice problems Let A and B be real matrices Answer yes / no / maybe: a) If A and B have the same eigenvalues, then A is similar to B b) If A and B both have eigenvalues,,, then A is similar to B c) If A is diagonalizable and invertible, then A is diagonalizable a) Maybe For example, and with alg multiplicity ) but are not similar, whereas have the same eigenvalues (λ = is similar to itself b) Yes In this case, A and B are matrices with distinct eigenvalues and thus automatically diagonalizable, and each is similar to D = Since A and D are similar, and B and D are similar, it follows that A and B are similar A = P DP B = QDQ A = P DP = PQ BQP = PQ B(PQ ) c) Yes If A = P DP and A is invertible then its eigenvalues are all nonzero, so the diagonal entries of D are nonzero and thus D is invertible (pivot in every diagonal position) Thus, A = (P DP ) = (P ) D P = P D P

5 Give an example of a non-diagonal matrix which is diagonalizable but not invertible Justify your answer is not invertible (row of zeros) but is diagonalizable since its has two distinct eigenvalues and (it is triangular, so its diagonals are its eigenvalues) Suppose A is a 7 7 matrix with four distinct eigenvalues One eigenspace has dimension, while another eigenspace has dimension Is it possible that A is not diagonalizable? A must be diagonalizable It is a general fact that every eigenvalue of a matrix has a corresponding eigenspace which is at least -dimensional Given this and the fact that A has four total eigenvalues, we see the sum of dimensions of the eigenspaces of A is at least = 7, and in fact must equal 7 since that is the max possible for a 7 7 matrix Therefore, A has 7 linearly independent eigenvectors and is therefore diagonalizable 4 Let A = 4 4 a) Find all (complex) eigenvalues and eigenvectors of A b) Write A = PC P, where C is a block diagonal matrix, as in the slides near the end of section c) What does A do geometrically? Draw a picture a) First we compute the characteristic polynomial by expanding cofactors along the third row: 4 λ 4 λ f (λ) = det 4 λ = ( λ) det 4 λ λ = ( λ) (4 λ) + 9 = ( λ)(λ 8λ + ) Using the quadratic equation on the second factor, we find the eigenvalues λ = λ = 4 i λ = 4 + i Next compute an eigenvector with eigenvalue λ = : RREF A I =

6 The parametric form is x =, y = z, so the parametric vector form of the solution is x eigenvector y = z v = z Now we compute an eigenvector with eigenvalue λ = 4 i: i i A = (4 i)i = i R R i i i R =R ir row replacements i + i i i R =R R =R (+i) i i i The parametric form of the solution is x = i y, z =, so the parametric vector form is x i eigenvector i y = y v = z An eigenvector for the complex conjugate eigenvalue λ = 4+i is the complex conjugate eigenvector v = i b) According to the block-diagonalization theorem, we have A = PC P where P = Re v Im v v = and Re λ Im λ 4 C = Im λ Re λ = 4 λ (I ve ordered the eigenvalues in this way to make the picture look nicer in my z is up coordinate system) c) The matrix C scales by in the z-direction, and rotates by arg( λ ) = arctan(/4) 64 radians and scales by λ = 4 + = in the x y-directions The matrix A does the same thing, with respect to the basis of columns of P [interactive],,

MATH 1553, C. JANKOWSKI MIDTERM 3

MATH 1553, C. JANKOWSKI MIDTERM 3 MATH 1553, C JANKOWSKI MIDTERM 3 Name GT Email @gatechedu Write your section number (E6-E9) here: Please read all instructions carefully before beginning Please leave your GT ID card on your desk until

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

PROBLEM SET. Problems on Eigenvalues and Diagonalization. Math 3351, Fall Oct. 20, 2010 ANSWERS

PROBLEM SET. Problems on Eigenvalues and Diagonalization. Math 3351, Fall Oct. 20, 2010 ANSWERS PROBLEM SET Problems on Eigenvalues and Diagonalization Math 335, Fall 2 Oct. 2, 2 ANSWERS i Problem. In each part, find the characteristic polynomial of the matrix and the eigenvalues of the matrix by

More information

MATH 1553-C MIDTERM EXAMINATION 3

MATH 1553-C MIDTERM EXAMINATION 3 MATH 553-C 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 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

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

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

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

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

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

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

MATH 1553 PRACTICE MIDTERM 3 (VERSION A)

MATH 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 information

Math 205, Summer I, Week 4b:

Math 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 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

(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

Problems for M 11/2: A =

Problems for M 11/2: A = Math 30 Lesieutre Problem set # November 0 Problems for M /: 4 Let B be the basis given by b b Find the B-matrix for the transformation T : R R given by x Ax where 3 4 A (This just means the matrix for

More information

Math 205, Summer I, Week 4b: Continued. Chapter 5, Section 8

Math 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 information

MAT2342 : Introduction to Linear Algebra Mike Newman, 5 October assignment 1

MAT2342 : Introduction to Linear Algebra Mike Newman, 5 October assignment 1 [/8 MAT4 : Introduction to Linear Algebra Mike Newman, 5 October 07 assignment You must show your work. You may use software or solvers of some sort to check calculation of eigenvalues, but you should

More information

Math 3191 Applied Linear Algebra

Math 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 information

Announcements Monday, November 06

Announcements Monday, November 06 Announcements Monday, November 06 This week s quiz: covers Sections 5 and 52 Midterm 3, on November 7th (next Friday) Exam covers: Sections 3,32,5,52,53 and 55 Section 53 Diagonalization Motivation: Difference

More information

Section 5.5. Complex Eigenvalues

Section 5.5. Complex Eigenvalues Section 55 Complex Eigenvalues A Matrix with No Eigenvectors In recitation you discussed the linear transformation for rotation by π/4 in the plane The matrix is: A = 1 ( ) 1 1 2 1 1 This matrix has no

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

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

HW2 - Due 01/30. Each answer must be mathematically justified. Don t forget your name.

HW2 - 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 information

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

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 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 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

DIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix

DIAGONALIZATION. 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

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

Announcements Wednesday, November 7

Announcements Wednesday, November 7 Announcements Wednesday, November 7 The third midterm is on Friday, November 6 That is one week from this Friday The exam covers 45, 5, 52 53, 6, 62, 64, 65 (through today s material) WeBWorK 6, 62 are

More information

MATH 310, REVIEW SHEET 2

MATH 310, REVIEW SHEET 2 MATH 310, REVIEW SHEET 2 These notes are a very short summary of the key topics in the book (and follow the book pretty closely). You should be familiar with everything on here, but it s not comprehensive,

More information

Math 215 HW #9 Solutions

Math 215 HW #9 Solutions Math 5 HW #9 Solutions. Problem 4.4.. If A is a 5 by 5 matrix with all a ij, then det A. Volumes or the big formula or pivots should give some upper bound on the determinant. Answer: Let v i be the ith

More information

Lecture 12: Diagonalization

Lecture 12: Diagonalization Lecture : Diagonalization A square matrix D is called diagonal if all but diagonal entries are zero: a a D a n 5 n n. () Diagonal matrices are the simplest matrices that are basically equivalent to vectors

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

Eigenvalues and Eigenvectors

Eigenvalues 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 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

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

MAT1302F Mathematical Methods II Lecture 19

MAT1302F Mathematical Methods II Lecture 19 MAT302F Mathematical Methods II Lecture 9 Aaron Christie 2 April 205 Eigenvectors, Eigenvalues, and Diagonalization Now that the basic theory of eigenvalues and eigenvectors is in place most importantly

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

MAT 1302B Mathematical Methods II

MAT 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 information

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

Chapters 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 information

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

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

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

Diagonalization. Hung-yi Lee

Diagonalization. Hung-yi Lee Diagonalization Hung-yi 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 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

Section 5.5. Complex Eigenvalues

Section 5.5. Complex Eigenvalues Section 5.5 Complex Eigenvalues A Matrix with No Eigenvectors Consider the matrix for the linear transformation for rotation by π/4 in the plane. The matrix is: A = 1 ( ) 1 1. 2 1 1 This matrix has no

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

Eigenvalues, Eigenvectors, and Diagonalization

Eigenvalues, Eigenvectors, and Diagonalization Math 240 TA: Shuyi Weng Winter 207 February 23, 207 Eigenvalues, Eigenvectors, and Diagonalization The concepts of eigenvalues, eigenvectors, and diagonalization are best studied with examples. We will

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

Eigenvalues for Triangular Matrices. ENGI 7825: Linear Algebra Review Finding Eigenvalues and Diagonalization

Eigenvalues 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 information

EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants

EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. Determinants Ex... Let A = 0 4 4 2 0 and B = 0 3 0. (a) Compute 0 0 0 0 A. (b) Compute det(2a 2 B), det(4a + B), det(2(a 3 B 2 )). 0 t Ex..2. For

More information

Practice Exam. 2x 1 + 4x 2 + 2x 3 = 4 x 1 + 2x 2 + 3x 3 = 1 2x 1 + 3x 2 + 4x 3 = 5

Practice Exam. 2x 1 + 4x 2 + 2x 3 = 4 x 1 + 2x 2 + 3x 3 = 1 2x 1 + 3x 2 + 4x 3 = 5 Practice Exam. Solve the linear system using an augmented matrix. State whether the solution is unique, there are no solutions or whether there are infinitely many solutions. If the solution is unique,

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

LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS

LINEAR ALGEBRA 1, 2012-I 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 information

Math Matrix Algebra

Math 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 information

Eigenvalues and Eigenvectors

Eigenvalues 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 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

MATH 1553 SAMPLE FINAL EXAM, SPRING 2018

MATH 1553 SAMPLE FINAL EXAM, SPRING 2018 MATH 1553 SAMPLE FINAL EXAM, SPRING 2018 Name Circle the name of your instructor below: Fathi Jankowski Kordek Strenner Yan Please read all instructions carefully before beginning Each problem is worth

More information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT 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 information

Diagonalization of Matrix

Diagonalization 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 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

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

Practice problems for Exam 3 A =

Practice problems for Exam 3 A = Practice problems for Exam 3. Let A = 2 (a) Determine whether A is diagonalizable. If so, find a matrix S such that S AS is diagonal. If not, explain why not. (b) What are the eigenvalues of A? Is A diagonalizable?

More information

Solutions Problem Set 8 Math 240, Fall

Solutions 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 information

Calculating determinants for larger matrices

Calculating 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 information

ft-uiowa-math2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST

ft-uiowa-math2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST me me ft-uiowa-math2550 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 information

City Suburbs. : population distribution after m years

City 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 information

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MATH 3 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left

More information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT 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 information

MATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION

MATH 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 information

Homework sheet 4: EIGENVALUES AND EIGENVECTORS. DIAGONALIZATION (with solutions) Year ? Why or why not? 6 9

Homework sheet 4: EIGENVALUES AND EIGENVECTORS. DIAGONALIZATION (with solutions) Year ? Why or why not? 6 9 Bachelor in Statistics and Business Universidad Carlos III de Madrid Mathematical Methods II María Barbero Liñán Homework sheet 4: EIGENVALUES AND EIGENVECTORS DIAGONALIZATION (with solutions) Year - Is

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

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

Practice Final Exam Solutions

Practice Final Exam Solutions MAT 242 CLASS 90205 FALL 206 Practice Final Exam Solutions The final exam will be cumulative However, the following problems are only from the material covered since the second exam For the material prior

More information

4. Linear transformations as a vector space 17

4. 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 information

Computationally, diagonal matrices are the easiest to work with. With this idea in mind, we introduce similarity:

Computationally, 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 information

EIGENVALUES AND EIGENVECTORS

EIGENVALUES AND EIGENVECTORS EIGENVALUES AND EIGENVECTORS Diagonalizable linear transformations and matrices Recall, a matrix, D, is diagonal if it is square and the only non-zero entries are on the diagonal This is equivalent to

More information

Solving a system by back-substitution, checking consistency of a system (no rows of the form

Solving a system by back-substitution, checking consistency of a system (no rows of the form MATH 520 LEARNING OBJECTIVES SPRING 2017 BROWN UNIVERSITY SAMUEL S. WATSON Week 1 (23 Jan through 27 Jan) Definition of a system of linear equations, definition of a solution of a linear system, elementary

More information

MATH 304 Linear Algebra Lecture 33: Bases of eigenvectors. Diagonalization.

MATH 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 information

Eigenvalues and Eigenvectors: An Introduction

Eigenvalues and Eigenvectors: An Introduction Eigenvalues and Eigenvectors: An Introduction The eigenvalue problem is a problem of considerable theoretical interest and wide-ranging application. For example, this problem is crucial in solving systems

More information

Linear Algebra Practice Final

Linear 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 information

Lecture 15, 16: Diagonalization

Lecture 15, 16: Diagonalization Lecture 15, 16: Diagonalization Motivation: Eigenvalues and Eigenvectors are easy to compute for diagonal matrices. Hence, we would like (if possible) to convert matrix A into a diagonal matrix. Suppose

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

Jordan Canonical Form Homework Solutions

Jordan Canonical Form Homework Solutions Jordan Canonical Form Homework Solutions For each of the following, put the matrix in Jordan canonical form and find the matrix S such that S AS = J. [ ]. A = A λi = λ λ = ( λ) = λ λ = λ =, Since we have

More information

Announcements Wednesday, November 01

Announcements 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 information

Math 217: Eigenspaces and Characteristic Polynomials Professor Karen Smith

Math 217: Eigenspaces and Characteristic Polynomials Professor Karen Smith Math 217: Eigenspaces and Characteristic Polynomials Professor Karen Smith (c)2015 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. Definition: Let V T V be a linear transformation.

More information

Linear Algebra. Rekha Santhanam. April 3, Johns Hopkins Univ. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, / 7

Linear Algebra. Rekha Santhanam. April 3, Johns Hopkins Univ. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, / 7 Linear Algebra Rekha Santhanam Johns Hopkins Univ. April 3, 2009 Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 1 / 7 Dynamical Systems Denote owl and wood rat populations at time k

More information

AMS10 HW7 Solutions. All credit is given for effort. (-5 pts for any missing sections) Problem 1 (20 pts) Consider the following matrix 2 A =

AMS10 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 information

Third Midterm Exam Name: Practice Problems November 11, Find a basis for the subspace spanned by the following vectors.

Third Midterm Exam Name: Practice Problems November 11, Find a basis for the subspace spanned by the following vectors. Math 7 Treibergs Third Midterm Exam Name: Practice Problems November, Find a basis for the subspace spanned by the following vectors,,, We put the vectors in as columns Then row reduce and choose the pivot

More information

MATH 310, REVIEW SHEET

MATH 310, REVIEW SHEET MATH 310, REVIEW SHEET These notes are a summary of the key topics in the book (and follow the book pretty closely). You should be familiar with everything on here, but it s not comprehensive, so please

More information

Announcements Wednesday, November 01

Announcements 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 information

Question 7. Consider a linear system A x = b with 4 unknown. x = [x 1, x 2, x 3, x 4 ] T. The augmented

Question 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 information

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

More information

Eigenvalues for Triangular Matrices. ENGI 7825: Linear Algebra Review Finding Eigenvalues and Diagonalization

Eigenvalues 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 information

MATH 220 FINAL EXAMINATION December 13, Name ID # Section #

MATH 220 FINAL EXAMINATION December 13, Name ID # Section # MATH 22 FINAL EXAMINATION December 3, 2 Name ID # Section # There are??multiple choice questions. Each problem is worth 5 points. Four possible answers are given for each problem, only one of which is

More information

22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices

22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices m:33 Notes: 7. Diagonalization of Symmetric Matrices Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman May 3, Symmetric matrices Definition. A symmetric matrix is a matrix

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

MATH 1553 PRACTICE FINAL EXAMINATION

MATH 1553 PRACTICE FINAL EXAMINATION MATH 553 PRACTICE FINAL EXAMINATION Name Section 2 3 4 5 6 7 8 9 0 Total Please read all instructions carefully before beginning. The final exam is cumulative, covering all sections and topics on the master

More 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

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors Sec. 6.1 Eigenvalues and Eigenvectors Linear transformations L : V V that go from a vector space to itself are often called linear operators. Many linear operators can be understood geometrically by identifying

More information