Homework 6 Solutions. Solution. Note {e t, te t, t 2 e t, e 2t } is linearly independent. If β = {e t, te t, t 2 e t, e 2t }, then
|
|
- Joanna Johns
- 5 years ago
- Views:
Transcription
1 Homework 6 Solutions 1 Let V be the real vector space spanned by the functions e t, te t, t 2 e t, e 2t Find a Jordan canonical basis and a Jordan canonical form of T on V dened by T (f) = f Solution Note {e t, te t, t 2 e t, e 2t } is linearly independent If β = {e t, te t, t 2 e t, e 2t }, then 1 1 [T ] β = A = Thus, p T (x) = (x 1) 3 (x 2) We look at N((A I) 3 ) We have 0 1 A I = 2 0 which has rank 3, so V 1 = N((A I)) has dimension d 1 = 1 From this we must have d 2 = 2 and d 3 = 3 (as there is only one Jordan block) We see 2 0 (A I) 2 = and (A I) 3 = Hence, N((A I) 2 ) = {(a, b, c, d) c = d = 0} and N((A I) 3 ) = {(a, b, c, d) d = 0}, so {(0, 0, 1, 0), (A I)(0, 0, 1, 0), (A I) 2 (0, 0, 1, 0)} works as a basis for N((A I)) This is {(0, 0, 1, 0), (0, 2, 0, 0), (2, 0, 0, 0)}, or {t 2 e t, 2te t, 2e t } For N((A 2I)), we see e 2t works, so our Jordan canonical basis is {t 2 e t, 2te t, 2e t, e 2t } with corresponding matrix Find a Jordan canonical basis and a Jordan canonical form of the operator T on P 3 (R) given by T (f(x)) = xf (x) Solution In the basis β = {1, x, x 2, x 3 }, we have A = [T ] β = Thus, p T (x) = x 4 We see A 2 = 2 1
2 and A 3 = 0, so we have d 1 = dim N(A) = 2, d 2 = 3, and d 4 = 4 We see N(A) = {(a, b, c, d) c = d = 0}, N(A 2 ) = {(a, b, c, d) d = 0} Hence, we may choose (0, 0, 0, 1) to start our rst chain We get A(0, 0, 0, 1) = (0, 0, 6, 0) and A 2 (0, 0, 0, 1) = (0, 12, 0, 0) Now, in N(A) we already have (0, 12, 0, 0) Hence, to complete a basis of N(A) we take (1, 0, 0, 0) Thus, our Jordan canonical basis is {(0, 0, 0, 1), (0, 0, 6, 0), (0, 12, 0, 0), (1, 0, 0, 0)} which is {x 3, 6x 2, 12x, 1} with corresponding matrix Find a Jordan canonical basis and a Jordan canonical form of the operator T on P 3 (R) given by T (f) = f + 2f Solution In the basis β = {1, x, x 2, x 3 }, we have A = [T ] β = so p T (x) = (x 2) 4 We have 2 0 (A 2I) = 0 6 and (A 2I) 2 = 0 We see N((A 2I)) = {(a, b, c, d) c = d = 0} and so a basis of N((A 2I) 2 ) relative to N((A 2I)) is given by {(0, 0, 1, 0), (0, 0, 0, 1)} Applying A 2, we get the chains {(0, 0, 1, 0), (2, 0, 0, 0)} and {(0, 0, 0, 1), (0, 6, 0, 0)} Hence, our Jordan basis is {(0, 0, 1, 0), (2, 0, 0, 0), (0, 0, 0, 1), (0, 6, 0, 0)} which is 2 0 {x 2, 2, x 3, 6x} and which has corresponding matrix Find [ a ] Jordan canonical basis and a Jordan canonical form of the operator T on M 2 2 (R) given by 3 1 T (A) = A A 0 3 t Solution We'll use B as the matrix of T in the basis β = {e 1, e 2, e 3, e 4 } where e 1 = ] [ ] [ ], e 3 =, and e = We see 0 1 [ B = [T ] β = [ ] 1 0, e 2 = so p T (x) = (x 2)((x 3)(x 3)(x 2) 1(x 2)) = (x 2) 2 ((x 3) 2 1) This is (x 2) 2 (x 2 6x + 8) = (x 2) 3 (x 4) For λ = 2, we have 1 0 (B 2I) =
3 which has rank 3 We see which has rank 2 Lastly (B 2I) 2 = (B 2I) 3 = which has rank 1 Thus we have N((B 2I) 2 ) = {(a, b, c, d) b + c = 0 and 2b 2c + d = 0} and N((B 2I) 3 ) = {(a, b, c, d) 2b + 2c d = 0} Hence, a basis for N((B 2I) 3 ) relative to N((B 2I) 2 ) is {(0, 0, 1, 2)} We have (B 2I)(0, 0, 1, 2) = (1, 1, 1, 0) and (B 2I) 2 (0, 0, 1, 2) = (1, 0, 0, 0) We have (B 4I) = so N((B 4I)) = {(a, b, c, d) 2a + c = b c + d = b c = d = 0} which is {(a, b, c, d) 2a + c = b + c = d = 0} Hence, a nonzero vector in N((B 4I)) is (1, 2, 2, 0) Adding this to our basis for N((B 2I) 3 ), we get {(0, 0, 1, 2), (1, 1, 1, 0), (1, 0, 0, 0), (1, 2, 2, 0)} as a Jordan basis with corresponding matrix Find [ a ] Jordan canonical basis and a Jordan canonical form of the operator T on M 2 2 (R) given by 3 1 T (A) = (A A 0 3 t ) Solution With β as in 4 we have B = [T ] β = so p T (x) = x 2 ((x 3) 2 9) = x 2 (x 2 6x) = x 3 (x 6) We see B has rank 1, so dim N(B) = 3 We see N(B) = {(a, b, c, d) b + c = 0}, so a basis for this space is {(0, 1, 1, 0), (1, 0, 0, 0), (0, 0, 0, 1)} Now, B 6I = so N((B 6I) = {(a, b, c, d) 6a b + c = 3b 3c = 6d = 0}, or {(a, b, c, d) 6a b + c = b + c = d = 0} Hence, a nonzero vector in this space is (1, 3, 3, 0) With the other vectors, our Jordan basis is thus {(0, 1, 1, 0), (1, 0, 0, 0), (0, 0, 0, 1), (1, 3, 3, 0)} with corresponding matrix 0 6 3
4 6 Let V be the real vector space of polynomials in 2 variables of degree at most 2 Find a Jordan basis and a Jordan canonical form for the operator T on V given by T (f) = f x + f y (here subscripts denote dierentiation with respect to those variables) Proof Let β = {1, x, y, x 2, xy, y 2 } be a basis for V Then, A = [T ] β = 1 2 so p T (x) = x 6 We see 2 2 A 2 = and so A 3 = 0 Thus, d 1 = 3, d 2 = 5, and d 3 = 6 For a basis of N(A 3 )/N(A 2 ), we may take any vector (a, b, c, d, e, f) with d + e + f 0, such as (0, 0, 0, 0, 1, 1) Then, A(0, 0, 0, 0, 1, 1) = (0, 1, 3, 0, 0, 0) and A 2 (0, 0, 0, 0, 1, 1) = (4, 0, 0, 0, 0, 0) Now, in N(A 2 )/N(A), we have (0, 1, 3, 0, 0, 0) + N(A) We wish to nd a basis relative to N(A), so rst we nd a basis of N(A) We have N(A) = {(a, b, c, d, e, f) b + c = 2d+e = e+2f = 0}, so a basis for this space is {(1, 0, 0, 0, 0, 0), (0, 1, 1, 0, 0, 0), (0, 0, 0, 1, 2, 1)} We extend this to a basis of N(A 2 ) by including (0, 1, 3, 0, 0, 0) from our chain and some other vector (a, b, c, d, e, f) with d + e + f = 0 We see (0, 0, 0, 0, 1, 1) works, and we use it to get a chain with A(0, 0, 0, 0, 1, 1) = (0, 1, 1, 0, 0, 0) Now, in N(A), we have (4, 0, 0, 0, 0, 0) and (0, 1, 1, 0, 0, 0) To extend this to a basis of N(A), we include (0, 0, 0, 1, 2, 1) Hence, our Jordan basis is {(0, 0, 0, 0, 1, 1), (0, 1, 3, 0, 0, 0), (4, 0, 0, 0, 0, 0), (0, 0, 0, 0, 1, 1), (0, 1, 1, 0, 0, 0), (0, 0, 0, 1, 2, 1)} which is {xy + y 2, x + 3y, 4, xy + y 2, x + y, x 2 2xy + y 2 } with corresponding matrix Let A M n n (F ) be such that p A splits Prove that A and A t are similar Proof 1 Let J A be a Jordan canonical form for A, which exists as p A splits We note A J A We claim J A JA t Indeed, suppose β = m k=1 β k is a Jordan canonical basis for which each block in J A corresponds to some β k For β k = {x 1, x 2,, x l }, let γ k = {x l,, x 1 } Then, for W k = span(β k ), we see that W k is T -invariant with λ λ 0 [T ] βk = 0 0 λ 4
5 and so λ λ 0 [T ] γk = 1 0 λ Thus, J A J t A We claim J A t J t A Indeed, if J A = C 1 AC, then J t A = Ct A t (C 1 ) t and (C 1 ) t = (C t ) 1, so J t A At Proof 2 We note that for arbitrary λ F and k N 0, dim N((A t λi) k ) = dim N(((A λi) k ) t ) = dim V columnrank(((a λi) k ) t ) which, as column rank = row rank, is dim V rowrank(((a λi) k ) t ) = dim V columnrank((a λi) k ) = dim N((A λi) k ) As J A is determined by the values dim N((A λi) k ), we conclude J A = J A t up to rearrangement Hence, A J A = J A t A t 8 Let T be a linear operator on a nite dimensional vector space such that p T splits and let λ 1,, λ k be the eigenvalues of T Dene S : V V by S(v) = k j=1 λ jv j where v = k j=1 v j is the (unique) expansion of v as a sum of generalized eigenvectors Claim S is diagonalizable and T S is nilpotent Proof Let β be a Jordan basis for T, so [T ] β = J where J is a Jordan block matrix Then, if v β, we have v W λj (the generalized eigenspace corresponding to λ j ) for some 1 j k Hence, S(v) = λ j v, so [S] β is a diagonal matrix Moreover, [S] β [T ] β is a matrix consisting of all zeros except possibly some ones immediately below the diagonal Thus, [S T ] β is strictly lower triangular and hence nilpotent (eg by (S T ) t being strinctly upper triangular and hence nilpotent and ((S T ) t ) m = ((S T ) m ) t for all m) 9 Let P 1, P 2 be two projectors in the vector space V such that P 1 P 2 = P 2 P 1 Show that P 1 P 2 is also a projector and that R(P 1 P 2 ) = R(P 1 ) R(P 2 ) Proof We see (P 1 P 2 ) 2 = P 1 P 2 P 1 P 2 = P 1 P 1 P 2 P 2 = P 1 P 2, so P 1 P 2 is a projector Now, suppose x R(P 1 P 2 ), say x = P 1 P 2 (v) Then, x = P 1 (P 2 (v)), so x R(P 1 ) Moreover, x = P 2 (P 1 (v)) by hypothesis, so x R(P 2 ) Thus, x R(P 1 ) R(P 2 ) Now, let x R(P 1 ) R(P 2 ), so x = P 1 (v) = P 2 (u) for some v, u V Then, P 1 (x) = P 1 (P 1 (v)) = P 1 (v) = x and P 2 (x) = P 2 (P 2 (u)) = P 2 (u) = x Thus, P 1 (P 2 (x)) = P 1 (x) = x, so x R(P 1 P 2 ) 10 Let P [ : V V ] be a projector in a vector space V of dimension n Prove there is a basis β of V such Ik 0 that [P ] β = n k Proof P 2 P = 0 so if f(x) = x(x 1), we see m p f Thus, m P is a product of distinct linear factors, so P is diagonalizable As 0 and 1 are the only zeros of f, they are the only eigenvalues of P, so by choosing a basis β of eigenvectors in which those with eigenvalue 1 come rst, we get the desired result 5
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 informationDefinition (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 informationThe Jordan Canonical Form
The Jordan Canonical Form The Jordan canonical form describes the structure of an arbitrary linear transformation on a finite-dimensional vector space over an algebraically closed field. Here we develop
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
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 informationBare-bones outline of eigenvalue theory and the Jordan canonical form
Bare-bones outline of eigenvalue theory and the Jordan canonical form April 3, 2007 N.B.: You should also consult the text/class notes for worked examples. Let F be a field, let V be a finite-dimensional
More informationMath 113 Homework 5. Bowei Liu, Chao Li. Fall 2013
Math 113 Homework 5 Bowei Liu, Chao Li Fall 2013 This homework is due Thursday November 7th at the start of class. Remember to write clearly, and justify your solutions. Please make sure to put your name
More informationLecture 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 informationThe Jordan Normal Form and its Applications
The and its Applications Jeremy IMPACT Brigham Young University A square matrix A is a linear operator on {R, C} n. A is diagonalizable if and only if it has n linearly independent eigenvectors. What happens
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 informationSUPPLEMENT TO CHAPTERS VII/VIII
SUPPLEMENT TO CHAPTERS VII/VIII The characteristic polynomial of an operator Let A M n,n (F ) be an n n-matrix Then the characteristic polynomial of A is defined by: C A (x) = det(xi A) where I denotes
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 informationMath 4153 Exam 3 Review. The syllabus for Exam 3 is Chapter 6 (pages ), Chapter 7 through page 137, and Chapter 8 through page 182 in Axler.
Math 453 Exam 3 Review The syllabus for Exam 3 is Chapter 6 (pages -2), Chapter 7 through page 37, and Chapter 8 through page 82 in Axler.. You should be sure to know precise definition of the terms we
More informationLinear 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 information4.1 Eigenvalues, Eigenvectors, and The Characteristic Polynomial
Linear Algebra (part 4): Eigenvalues, Diagonalization, and the Jordan Form (by Evan Dummit, 27, v ) Contents 4 Eigenvalues, Diagonalization, and the Jordan Canonical Form 4 Eigenvalues, Eigenvectors, and
More informationMath 24 Spring 2012 Sample Homework Solutions Week 8
Math 4 Spring Sample Homework Solutions Week 8 Section 5. (.) Test A M (R) for diagonalizability, and if possible find an invertible matrix Q and a diagonal matrix D such that Q AQ = D. ( ) 4 (c) A =.
More informationMATH 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 informationGeneralized Eigenvectors and Jordan Form
Generalized Eigenvectors and Jordan Form We have seen that an n n matrix A is diagonalizable precisely when the dimensions of its eigenspaces sum to n. So if A is not diagonalizable, there is at least
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 informationLecture 21: The decomposition theorem into generalized eigenspaces; multiplicity of eigenvalues and upper-triangular matrices (1)
Lecture 21: The decomposition theorem into generalized eigenspaces; multiplicity of eigenvalues and upper-triangular matrices (1) Travis Schedler Tue, Nov 29, 2011 (version: Tue, Nov 29, 1:00 PM) Goals
More informationHomework For each of the following matrices, find the minimal polynomial and determine whether the matrix is diagonalizable.
Math 5327 Fall 2018 Homework 7 1. For each of the following matrices, find the minimal polynomial and determine whether the matrix is diagonalizable. 3 1 0 (a) A = 1 2 0 1 1 0 x 3 1 0 Solution: 1 x 2 0
More informationHomework Set 5 Solutions
MATH 672-010 Vector Spaces Prof. D. A. Edwards Due: Oct. 7, 2014 Homework Set 5 Solutions 1. Let S, T L(V, V finite-dimensional. (a (5 points Prove that ST and T S have the same eigenvalues. Solution.
More information(VI.D) Generalized Eigenspaces
(VI.D) Generalized Eigenspaces Let T : C n C n be a f ixed linear transformation. For this section and the next, all vector spaces are assumed to be over C ; in particular, we will often write V for C
More informationMATH 304 Linear Algebra Lecture 34: Review for Test 2.
MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1
More informationand 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 informationLinear Algebra Final Exam Solutions, December 13, 2008
Linear Algebra Final Exam Solutions, December 13, 2008 Write clearly, with complete sentences, explaining your work. You will be graded on clarity, style, and brevity. If you add false statements to a
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationMath 113 Homework 5 Solutions (Starred problems) Solutions by Guanyang Wang, with edits by Tom Church.
Math 113 Homework 5 Solutions (Starred problems) Solutions by Guanyang Wang, with edits by Tom Church. Exercise 5.C.1 Suppose T L(V ) is diagonalizable. Prove that V = null T range T. Proof. Let v 1,...,
More information5. Diagonalization. plan given T : V V Does there exist a basis β of V such that [T] β is diagonal if so, how can it be found
5. Diagonalization plan given T : V V Does there exist a basis β of V such that [T] β is diagonal if so, how can it be found eigenvalues EV, eigenvectors, eigenspaces 5.. Eigenvalues and eigenvectors.
More informationSolution. That ϕ W is a linear map W W follows from the definition of subspace. The map ϕ is ϕ(v + W ) = ϕ(v) + W, which is well-defined since
MAS 5312 Section 2779 Introduction to Algebra 2 Solutions to Selected Problems, Chapters 11 13 11.2.9 Given a linear ϕ : V V such that ϕ(w ) W, show ϕ induces linear ϕ W : W W and ϕ : V/W V/W : Solution.
More informationEconomics 204 Fall 2013 Problem Set 5 Suggested Solutions
Economics 204 Fall 2013 Problem Set 5 Suggested Solutions 1. Let A and B be n n matrices such that A 2 = A and B 2 = B. Suppose that A and B have the same rank. Prove that A and B are similar. Solution.
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 information1.4 Solvable Lie algebras
1.4. SOLVABLE LIE ALGEBRAS 17 1.4 Solvable Lie algebras 1.4.1 Derived series and solvable Lie algebras The derived series of a Lie algebra L is given by: L (0) = L, L (1) = [L, L],, L (2) = [L (1), L (1)
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 informationEigenvalue and Eigenvector Homework
Eigenvalue and Eigenvector Homework Olena Bormashenko November 4, 2 For each of the matrices A below, do the following:. Find the characteristic polynomial of A, and use it to find all the eigenvalues
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 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 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 informationspring, math 204 (mitchell) list of theorems 1 Linear Systems Linear Transformations Matrix Algebra
spring, 2016. math 204 (mitchell) list of theorems 1 Linear Systems THEOREM 1.0.1 (Theorem 1.1). Uniqueness of Reduced Row-Echelon Form THEOREM 1.0.2 (Theorem 1.2). Existence and Uniqueness Theorem THEOREM
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 informationLinear Algebra II Lecture 22
Linear Algebra II Lecture 22 Xi Chen University of Alberta March 4, 24 Outline Characteristic Polynomial, Eigenvalue, Eigenvector and Eigenvalue, Eigenvector and Let T : V V be a linear endomorphism. We
More information5 Eigenvalues and Diagonalization
Linear Algebra (part 5): Eigenvalues and Diagonalization (by Evan Dummit, 27, v 5) Contents 5 Eigenvalues and Diagonalization 5 Eigenvalues, Eigenvectors, and The Characteristic Polynomial 5 Eigenvalues
More informationMATH 315 Linear Algebra Homework #1 Assigned: August 20, 2018
Homework #1 Assigned: August 20, 2018 Review the following subjects involving systems of equations and matrices from Calculus II. Linear systems of equations Converting systems to matrix form Pivot entry
More informationMath 113 Winter 2013 Prof. Church Midterm Solutions
Math 113 Winter 2013 Prof. Church Midterm Solutions Name: Student ID: Signature: Question 1 (20 points). Let V be a finite-dimensional vector space, and let T L(V, W ). Assume that v 1,..., v n is a basis
More informationSolution: a) Let us consider the matrix form of the system, focusing on the augmented matrix: 0 k k + 1. k 2 1 = 0. k = 1
Exercise. Given the system of equations 8 < : x + y + z x + y + z k x + k y + z k where k R. a) Study the system in terms of the values of k. b) Find the solutions when k, using the reduced row echelon
More informationHomework 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 informationAnswer Keys For Math 225 Final Review Problem
Answer Keys For Math Final Review Problem () For each of the following maps T, Determine whether T is a linear transformation. If T is a linear transformation, determine whether T is -, onto and/or bijective.
More informationA proof of the Jordan normal form theorem
A proof of the Jordan normal form theorem Jordan normal form theorem states that any matrix is similar to a blockdiagonal matrix with Jordan blocks on the diagonal. To prove it, we first reformulate it
More informationMath 108b: Notes on the Spectral Theorem
Math 108b: Notes on the Spectral Theorem From section 6.3, we know that every linear operator T on a finite dimensional inner product space V has an adjoint. (T is defined as the unique linear operator
More informationMATH 115A: SAMPLE FINAL SOLUTIONS
MATH A: SAMPLE FINAL SOLUTIONS JOE HUGHES. Let V be the set of all functions f : R R such that f( x) = f(x) for all x R. Show that V is a vector space over R under the usual addition and scalar multiplication
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 informationMATH JORDAN FORM
MATH 53 JORDAN FORM Let A,, A k be square matrices of size n,, n k, respectively with entries in a field F We define the matrix A A k of size n = n + + n k as the block matrix A 0 0 0 0 A 0 0 0 0 A k It
More informationNotes on the matrix exponential
Notes on the matrix exponential Erik Wahlén erik.wahlen@math.lu.se February 14, 212 1 Introduction The purpose of these notes is to describe how one can compute the matrix exponential e A when A is not
More informationLecture 11: Diagonalization
Lecture 11: Elif Tan Ankara University Elif Tan (Ankara University) Lecture 11 1 / 11 Definition The n n matrix A is diagonalizableif there exits nonsingular matrix P d 1 0 0. such that P 1 AP = D, where
More informationFinal Review Written by Victoria Kala SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015
Final Review Written by Victoria Kala vtkala@mathucsbedu SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015 Summary This review contains notes on sections 44 47, 51 53, 61, 62, 65 For your final,
More 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 informationMATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS
MATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS There will be eight problems on the final. The following are sample problems. Problem 1. Let F be the vector space of all real valued functions on
More informationJordan Normal Form and Singular Decomposition
University of Debrecen Diagonalization and eigenvalues Diagonalization We have seen that if A is an n n square matrix, then A is diagonalizable if and only if for all λ eigenvalues of A we have dim(u λ
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 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 informationNONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction
NONCOMMUTATIVE POLYNOMIAL EQUATIONS Edward S Letzter Introduction My aim in these notes is twofold: First, to briefly review some linear algebra Second, to provide you with some new tools and techniques
More informationDefinition: An n x n matrix, "A", is said to be diagonalizable if there exists a nonsingular matrix "X" and a diagonal matrix "D" such that X 1 A X
DIGONLIZTION Definition: n n x n matrix, "", is said to be diagonalizable if there exists a nonsingular matrix "X" and a diagonal matrix "D" such that X X D. Theorem: n n x n matrix, "", is diagonalizable
More informationBasic Theorems about Independence, Spanning, Basis, and Dimension
Basic Theorems about Independence, Spanning, Basis, and Dimension Theorem 1 If v 1 ; : : : ; v m span V; then every set of vectors in V with more than m vectors must be linearly dependent. Proof. Let w
More informationSolutions 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 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 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 informationDimension. 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 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 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 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 informationLinear Algebra Review
Chapter 1 Linear Algebra Review It is assumed that you have had a course in linear algebra, and are familiar with matrix multiplication, eigenvectors, etc. I will review some of these terms here, but quite
More informationMath 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 informationLinear Algebra 2 Spectral Notes
Linear Algebra 2 Spectral Notes In what follows, V is an inner product vector space over F, where F = R or C. We will use results seen so far; in particular that every linear operator T L(V ) has a complex
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 informationJordan Canonical Form
Jordan Canonical Form Massoud Malek Jordan normal form or Jordan canonical form (named in honor of Camille Jordan) shows that by changing the basis, a given square matrix M can be transformed into a certain
More informationThe Cayley-Hamilton Theorem and the Jordan Decomposition
LECTURE 19 The Cayley-Hamilton Theorem and the Jordan Decomposition Let me begin by summarizing the main results of the last lecture Suppose T is a endomorphism of a vector space V Then T has a minimal
More informationA NOTE ON THE JORDAN CANONICAL FORM
A NOTE ON THE JORDAN CANONICAL FORM H. Azad Department of Mathematics and Statistics King Fahd University of Petroleum & Minerals Dhahran, Saudi Arabia hassanaz@kfupm.edu.sa Abstract A proof of the Jordan
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 informationTopics in linear algebra
Chapter 6 Topics in linear algebra 6.1 Change of basis I want to remind you of one of the basic ideas in linear algebra: change of basis. Let F be a field, V and W be finite dimensional vector spaces over
More information2 Eigenvectors and Eigenvalues in abstract spaces.
MA322 Sathaye Notes on Eigenvalues Spring 27 Introduction In these notes, we start with the definition of eigenvectors in abstract vector spaces and follow with the more common definition of eigenvectors
More informationINTRODUCTION TO LIE ALGEBRAS. LECTURE 2.
INTRODUCTION TO LIE ALGEBRAS. LECTURE 2. 2. More examples. Ideals. Direct products. 2.1. More examples. 2.1.1. Let k = R, L = R 3. Define [x, y] = x y the cross-product. Recall that the latter is defined
More informationMath 121 Practice Final Solutions
Math Practice Final Solutions December 9, 04 Email me at odorney@college.harvard.edu with any typos.. True or False. (a) If B is a 6 6 matrix with characteristic polynomial λ (λ ) (λ + ), then rank(b)
More informationWarm-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 informationMath 4A Notes. Written by Victoria Kala Last updated June 11, 2017
Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...
More informationTMA 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 informationGroup Theory. 1. Show that Φ maps a conjugacy class of G into a conjugacy class of G.
Group Theory Jan 2012 #6 Prove that if G is a nonabelian group, then G/Z(G) is not cyclic. Aug 2011 #9 (Jan 2010 #5) Prove that any group of order p 2 is an abelian group. Jan 2012 #7 G is nonabelian nite
More informationMath Camp Notes: Linear Algebra II
Math Camp Notes: Linear Algebra II Eigenvalues Let A be a square matrix. An eigenvalue is a number λ which when subtracted from the diagonal elements of the matrix A creates a singular matrix. In other
More informationAdditional Homework Problems
Math 216 2016-2017 Fall Additional Homework Problems 1 In parts (a) and (b) assume that the given system is consistent For each system determine all possibilities for the numbers r and n r where r is the
More informationChapter 5 Eigenvalues and Eigenvectors
Chapter 5 Eigenvalues and Eigenvectors Outline 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 2 5.1 Eigenvalues and Eigenvectors Eigenvalue and Eigenvector If A is a n n
More informationSolutions to the Calculus and Linear Algebra problems on the Comprehensive Examination of January 28, 2011
Solutions to the Calculus and Linear Algebra problems on the Comprehensive Examination of January 8, Solutions to Problems 5 are omitted since they involve topics no longer covered on the Comprehensive
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 informationLinear Algebra problems
Linear Algebra problems 1. Show that the set F = ({1, 0}, +,.) is a field where + and. are defined as 1+1=0, 0+0=0, 0+1=1+0=1, 0.0=0.1=1.0=0, 1.1=1.. Let X be a non-empty set and F be any field. Let X
More informationLINEAR 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 informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
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 informationAlgorithms to Compute Bases and the Rank of a Matrix
Algorithms to Compute Bases and the Rank of a Matrix Subspaces associated to a matrix Suppose that A is an m n matrix The row space of A is the subspace of R n spanned by the rows of A The column space
More informationACM 104. Homework Set 4 Solutions February 14, 2001
ACM 04 Homework Set 4 Solutions February 4, 00 Franklin Chapter, Problem 4, page 55 Suppose that we feel that some observations are more important or reliable than others Redefine the function to be minimized
More informationLinear Algebra. Workbook
Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx
More informationChapter 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 informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More information