Linear Algebra
|
|
- Erick Mosley
- 5 years ago
- Views:
Transcription
1 . Linear Algebra Final Solutions. (pts) Let A =. (a) Find an orthogonal matrix S a diagonal matrix D such that S AS = D. (b) Find a formula for the entries of A t,wheret is a positive integer. Also find the vector lim A t. t (a) Let f A (λ) =det(a λi )=det λ λ = λ λ +λ += (λ ) (λ +) =.Wehavetheeigenvalues are with multiplicity with multiplicity. For λ =, the eigenspace E = ker = ker = ker = span.
2 Therefore, forms an orthonormal basis of E. For λ =, the eigenspace E = ker ( ) ( ) ( ) = ker = ker = span,. Therefore, v = v = form a eigenbasis of E. Using Gram-schmidt process on v v to get an orthonormal eigenbasis w w for E,wehave w = v k v k =, with u = v ( w v ) w = = on h, we have k u k = w = v ( w v ) w k v ( w v ) w k = u k u k = =.
3 Set S = D =. We have S is orthogonal D is diagonal. Therefore, we have D = S AS where S is orthogonal D is diagonal. (b)from(a),wehavea = SDS. Since S is orthogonal, we have S = S T. Hence, for a positive integer t, A t = SD t S = SD t S T t = ( ) t ( ) t t +( ) t t ( ) t t ( ) t = t ( ) t t +( ) t t ( ) t t ( ) t t ( ) t t +( ) t. Therefore, lim t At t +( ) t t ( ) t t ( ) t = lim t t ( ) t t +( ) t t ( ) t t ( ) t t ( ) t t +( ) t = lim ( )t. t ( ) t That means the limit does not exist since you have two different limit points.. (pts) Let A =. (a) Find a singular value decomposition for A. (b) Describe the image of the unit circle under the linear transformation T ( x) =A x.
4 4 (a) The singular values are the square roots of the eigenvalues of A T A =. Let f A T A (λ) = det A T A λi = µ λ det =(λ ) (λ ) =. We have the λ eigenvalues of A T A are λ = λ =. Therefore, the singular values of A are σ = p λ = σ = p λ =. For σ = µ, the eigenspace E =ker µ ker unit vector =span ½ v = = ¾. Therefore, the nonzero forms an µ orthonormal basis of E.Forσ µ =, the eigenspace ½ E =ker =ker =span Therefore, the nonzero unit vector v = ¾. forms an orthonormal basis of E.Let u = A v = σ = µ
5 5 u = A v = σ =. µ Since A is a matrix, we know that U will be a matrix. So, we need to exp { u, u } into a orthonormal basis of R. That means we need to find a u such that u, u u form an orthonormal basis of R.Choose w =. Obviously, w is not a linear combination of u u.so, u, u w form a basis of R.Since u u are orthogonal unit vectors already. We use Gram-Schmidt process on w to get u = w ( u w ) u ( u w ) u k w ( u w ) u ( u w ) u k = And, we have u, u u form an orthonormal basis of R. Let V = v v,. U = u u u = Σ = σ σ. Therefore, we get a singular value decomposition(svd) for A = UΣV T.
6 (b) From (a), we have T ( v ) = A v = = µ T ( v ) = A v = =. µ The unit circle in R consists of all vectors of the form x = c v + c v where c + c =. Therefore,theimageoftheunitcircleunderT consists of the vectors T ( x) =c T ( v )+c T ( v ) where c + c =. That means T ( x) =c + c where c + c =. This is an ellipse with the semimajor axe is the line generated by with the length σ = the semiminor axe is the line generated by. (pts) Let q be a quadratic form q (x,x )=9x 4x x +x. (a) Determine the definiteness of q. with the length σ =.
7 (b) Sketch the curve defined by q (x,x )=. Draw label the principal axes, label the intercepts of the curve with the principal axes, give the formula of the curve in the coordinate system defined by the principal axes. 7 9 (a) Let A = [Method ] Calculate.Wehaveq ( x) = x A x. A () =det([9])=9> µ 9 A () =det =5>. By the theorem in the textbook, we have q is positive definite. [Method ] Let λ λ be the two eigenvalues of A with associated eigenvectors v v, respectively. We have λ λ =det(a) = 54 4=5> λ + λ =tr(a) =9+=5>. This implies that λ > λ >. Moreover, we have q ( x) =λ c + λ c where x = c v + c v. That means q ( x) > for all x =. By the definition of definiteness, A is positive definite. 9 (b) Let A =. We have q ( x) = x A x. Set = µ 9 λ f A (λ) = det(a λi ) = det = λ λ 5λ +5=(λ ) (λ 5). We get the eigenvalues λ = λ =5.Forλ =, the eigenspace µ 9 E = ker µ = ker 4 =span ½ ¾.
8 8 ½ ¾ Therefore, 5 forms an orthonormal basis of E. For λ =5, the eigenspace µ 9 5 E 5 = ker 5 µ ½ ¾ 4 = ker =span. ½ ¾ Therefore, 5 forms an orthonormal basis of E 5. Hencewehaveanorthonormaleigenbasis 5 5 v = v =. 5 5 Moreover, the curve can be re-written as c +5c =. Thatmeansthecurveisaellipseinc c coordinates system where the principal axes are E E 5 which are generated by v v, respectively. The graph is
9 4. (pts) Decide whether the matrix A = is diagonalizable. If possible, find an invertible S a diagonal D such that S AS = D. Using f A (λ) =det(a λi )=det λ λ = λ ( λ) λ =, we have eigenvalues are,. For λ =,the eigenspace E = ker(a I )=ker = ker =span. For λ =, the eigenspace E = ker(a I )=ker = ker =span. Since dim (E )+dim(e )=+=<, we have no eigenbasis of R in this case. This implies that A is not diagonalizable. 5. (pts) Find the trigonometric function of the form f (t) =c + c sin (t)+c cos (t) that best fits the data points (, ), π,, (π, ) π,, using lease squares. We want to find a f (t) =c + c sin (t)+c cos (t) such that f () =, f π =, f (π) = f π =. These conditions give the 9
10 system of linear equations c + c sin () + c cos () = f () = c + c sin π + c cos π = f π = c + c sin (π) + c cos (π) = f (π) = c + c sin π + c cos, π = f π = or, c + c = c + c =. c c = c c = Let A =, x = c c b = c write the system as A x = b. Since rref (A) = ker (A) ={}. So, the unique least-squares solution of A x = b is x = A T A A T b = = 4. We can,wehave =. Moreover, the trigonometric function, f (t) =+ sin (t) cos (t), best fits the data points (, ), π,, (π, ) π, in the leastsquares sense.
11 . (pts) Given a matrix A = (a) Find a basis of kernel of A dim (ker (A)). (b) Find a basis of image of A dim (im (A)). By Gauss-Jordan elimination, we have A = = rref (A). (a) Assume x = x x x x 4 ker A. Then we have A x =. By Gauss-Jordan elimination(which implies ker (A) = ker rref (A)), we have x x x =. x 4 Assume x = s x 4 = t for all s, t R. Wehavesolutions of the system, x s + t x x = s + t s = s + t. x 4 t
12 Assume v = v = span { v, v }. Moreover, we check that A ( v )= A ( v )= Thatmeansker (A) = = This tells us that { v, v } ker (A) which implies ker (A) = span { v, v }.Tocheck v v are linearly independent, we assume that c v + c v =,or =c + c = c + c c + c c c.. This implies that c = c =. That proves that v v are linearly independent. Now, we can say v, v form a basis of ker (A). And,dim (ker (A)) equals the number of vectors in a basis. So, dim (ker (A)) =. (b) Set v, v, v, v 4 be the column vectors of A. Set w, w, w, w 4 bethecolumnvectorsofrref A. Weknowthat im (A) =span{ v, v, v, v 4 }. From our reduced row-echlon form, we can read two relations of w, w, w, w 4, w = w + w w 4 = w w. This implies we have the same relation of v, v, v, v 4 which are v = v + v
13 v 4 = v v. Therefore, we can conclude that v, v are linearly independent im (A) =span{ v, v, v, v 4 } =span{ v, v }. Now, we can say that v =, v = 7 form a basis of im (A). And,dim (im (A)) equals the number of vectors in a basis. So, dim (im (A)) =. 7. (pts) Let T from R to R be the reflection in the plane given by the equation x +x +x =. (a) Find the matrix B of this transformation with respect to the basis v =, v =, v =. (b) Use your answer in part (a) to find the stard matrix A of T. (a) Let P be the plane given by the equation x +x +x =. We observe that v v arebothontheplanep.sincet is areflection, it keeps v v unchanged, that is, T ( v )= v T ( v )= v. And, v is the normal vector of P. That means v is perpendicular to the plane P. Therefore, since T is a reflection in P,wehaveT ( v )= v. Withrespect to the basis B = { v, v, v },wehave[t ( v )] B = [T ( v )] B = [T ( v )] B =.So B B B = [T ( v )] B [T ( v )] B [T ( v )] B =. B,
14 4 (b) Write S = v v v =. Bythetheorem in the textbook, we have A = SBS. To find S,weuse Gauss-Jordan elimination on Hence, S = A = SBS = (pts) Consider a linear transformation T from V to W. (a) For f,f,,f n V,ifT(f ),T (f ),,T (f n ) are linearly independent, show that f,f,,f n are linearly independent. (b) Assume that f,f,,f n form a basis of V. If T is an isomorphism, show that T (f ),T (f ),,T (f n ) is a basis of W.
15 (a) Assume a f + a f + + a n f n =. By applying T on both sides, we have ³ T (a f + a f + + a n f n )=T =. Since T is a linear transformation, a T (f )+ + a n T (f n )=T (a f + + a n f n )=. Since T (f ),T (f ),,T (f n ) are linearly independent, we have a = a = = a n =. That tells us that f,f,,f n are linearly independent. (b) To show that T (f ),T (f ),,T (f n ) are linearly independent, we assume c T (f )+c T (f )+ + c n T (f n ) =. Hence, we have T (c f + + c n f n )=since T is a linear transformation. Since T is an isomorphism from V to W,we have T is invertible, that means, ker (T )={}. This implies c f + + c n f n =. Moreover, f,f,,f n form a basis of V. This condition forces that c = c = = c n =. So, T (f ),T (f ),,T (f n ) are linearly independent. Since T is an isomorphism from V to W,wehaveT is an isomorphism from W to V such that T (T (g)) = g for all g W. For all g W, we have T (g) V. Since f,f,,f n form a basis of V,thereexistt,t,,t n such that T (g) =t f + + t n f n. Therefore, g = T (T (g)) = T (t f + + t n f n )=t T (f )+ t n T (f n ). That means g is a linear combination of T (f ),T (f ),,T (f n ). And, obviously, T (f ),T (f ),,T (f n ) are all in W. Thus, we can conclude that W =span{t (f ),T (f ),,T (f n )}. So, T (f ),T (f ),,T (f n ) form a basis of W 5
PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them.
Prof A Suciu MTH U37 LINEAR ALGEBRA Spring 2005 PRACTICE FINAL EXAM Are the following vectors independent or dependent? If they are independent, say why If they are dependent, exhibit a linear dependence
More 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 informationTBP MATH33A Review Sheet. November 24, 2018
TBP MATH33A Review Sheet November 24, 2018 General Transformation Matrices: Function Scaling by k Orthogonal projection onto line L Implementation If we want to scale I 2 by k, we use the following: [
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 =
. (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 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 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 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 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 informationConceptual 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 informationPractice 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 informationDefinitions for Quizzes
Definitions for Quizzes Italicized text (or something close to it) will be given to you. Plain text is (an example of) what you should write as a definition. [Bracketed text will not be given, nor does
More informationMATH 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 informationAPPLICATIONS The eigenvalues are λ = 5, 5. An orthonormal basis of eigenvectors consists of
CHAPTER III APPLICATIONS The eigenvalues are λ =, An orthonormal basis of eigenvectors consists of, The eigenvalues are λ =, A basis of eigenvectors consists of, 4 which are not perpendicular However,
More informationGlossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the
More informationEigenvalues and Eigenvectors A =
Eigenvalues and Eigenvectors Definition 0 Let A R n n be an n n real matrix A number λ R is a real eigenvalue of A if there exists a nonzero vector v R n such that A v = λ v The vector v is called an eigenvector
More informationUNIT 6: The singular value decomposition.
UNIT 6: The singular value decomposition. María Barbero Liñán Universidad Carlos III de Madrid Bachelor in Statistics and Business Mathematical methods II 2011-2012 A square matrix is symmetric if A T
More informationChapter 7: Symmetric Matrices and Quadratic Forms
Chapter 7: Symmetric Matrices and Quadratic Forms (Last Updated: December, 06) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). A few theorems have been moved
More informationMath 118, Fall 2014 Final Exam
Math 8, Fall 4 Final Exam True or false Please circle your choice; no explanation is necessary True There is a linear transformation T such that T e ) = e and T e ) = e Solution Since T is linear, if T
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 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 informationLINEAR ALGEBRA QUESTION BANK
LINEAR ALGEBRA QUESTION BANK () ( points total) Circle True or False: TRUE / FALSE: If A is any n n matrix, and I n is the n n identity matrix, then I n A = AI n = A. TRUE / FALSE: If A, B are n n matrices,
More informationProblem # Max points possible Actual score Total 120
FINAL EXAMINATION - MATH 2121, FALL 2017. Name: ID#: Email: Lecture & Tutorial: Problem # Max points possible Actual score 1 15 2 15 3 10 4 15 5 15 6 15 7 10 8 10 9 15 Total 120 You have 180 minutes to
More informationMATH 369 Linear Algebra
Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine
More informationSOLUTION KEY TO THE LINEAR ALGEBRA FINAL EXAM 1 2 ( 2) ( 1) c a = 1 0
SOLUTION KEY TO THE LINEAR ALGEBRA FINAL EXAM () We find a least squares solution to ( ) ( ) A x = y or 0 0 a b = c 4 0 0. 0 The normal equation is A T A x = A T y = y or 5 0 0 0 0 0 a b = 5 9. 0 0 4 7
More informationMath 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 informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More 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 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 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 informationAfter we have found an eigenvalue λ of an n n matrix A, we have to find the vectors v in R n such that
7.3 FINDING THE EIGENVECTORS OF A MATRIX After we have found an eigenvalue λ of an n n matrix A, we have to find the vectors v in R n such that A v = λ v or (λi n A) v = 0 In other words, we have to find
More informationYORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #2 Solutions
YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 3. M Test # Solutions. (8 pts) For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For this
More informationLINEAR ALGEBRA KNOWLEDGE SURVEY
LINEAR ALGEBRA KNOWLEDGE SURVEY Instructions: This is a Knowledge Survey. For this assignment, I am only interested in your level of confidence about your ability to do the tasks on the following pages.
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More informationDiagonalizing Matrices
Diagonalizing Matrices Massoud Malek A A Let A = A k be an n n non-singular matrix and let B = A = [B, B,, B k,, B n ] Then A n A B = A A 0 0 A k [B, B,, B k,, B n ] = 0 0 = I n 0 A n Notice that A i B
More informationEIGENVALUES 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(a) II and III (b) I (c) I and III (d) I and II and III (e) None are true.
1 Which of the following statements is always true? I The null space of an m n matrix is a subspace of R m II If the set B = {v 1,, v n } spans a vector space V and dimv = n, then B is a basis for V III
More information1. 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 informationTopic 2 Quiz 2. choice C implies B and B implies C. correct-choice C implies B, but B does not imply C
Topic 1 Quiz 1 text A reduced row-echelon form of a 3 by 4 matrix can have how many leading one s? choice must have 3 choice may have 1, 2, or 3 correct-choice may have 0, 1, 2, or 3 choice may have 0,
More informationLinear 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 informationMath 21b Final Exam Thursday, May 15, 2003 Solutions
Math 2b Final Exam Thursday, May 5, 2003 Solutions. (20 points) True or False. No justification is necessary, simply circle T or F for each statement. T F (a) If W is a subspace of R n and x is not in
More information7. Symmetric Matrices and Quadratic Forms
Linear Algebra 7. Symmetric Matrices and Quadratic Forms CSIE NCU 1 7. Symmetric Matrices and Quadratic Forms 7.1 Diagonalization of symmetric matrices 2 7.2 Quadratic forms.. 9 7.4 The singular value
More informationhomogeneous 71 hyperplane 10 hyperplane 34 hyperplane 69 identity map 171 identity map 186 identity map 206 identity matrix 110 identity matrix 45
address 12 adjoint matrix 118 alternating 112 alternating 203 angle 159 angle 33 angle 60 area 120 associative 180 augmented matrix 11 axes 5 Axiom of Choice 153 basis 178 basis 210 basis 74 basis test
More informationMATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2.
MATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2. Diagonalization Let L be a linear operator on a finite-dimensional vector space V. Then the following conditions are equivalent:
More informationLINEAR ALGEBRA SUMMARY SHEET.
LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linear-algebra-summary-sheet/ This document is a concise collection of many of the important theorems of linear algebra, organized
More informationChapter 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 information18.06SC Final Exam Solutions
18.06SC Final Exam Solutions 1 (4+7=11 pts.) Suppose A is 3 by 4, and Ax = 0 has exactly 2 special solutions: 1 2 x 1 = 1 and x 2 = 1 1 0 0 1 (a) Remembering that A is 3 by 4, find its row reduced echelon
More informationExamples True or false: 3. Let A be a 3 3 matrix. Then there is a pattern in A with precisely 4 inversions.
The exam will cover Sections 6.-6.2 and 7.-7.4: True/False 30% Definitions 0% Computational 60% Skip Minors and Laplace Expansion in Section 6.2 and p. 304 (trajectories and phase portraits) in Section
More informationFinal Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2
Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch
More informationReduction to the associated homogeneous system via a particular solution
June PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 5) Linear Algebra This study guide describes briefly the course materials to be covered in MA 5. In order to be qualified for the credit, one
More informationMath 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam
Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system
More 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 informationMath 315: Linear Algebra Solutions to Assignment 7
Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are
More 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 informationx 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7
Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)
More informationMath 314/ Exam 2 Blue Exam Solutions December 4, 2008 Instructor: Dr. S. Cooper. Name:
Math 34/84 - Exam Blue Exam Solutions December 4, 8 Instructor: Dr. S. Cooper Name: Read each question carefully. Be sure to show all of your work and not just your final conclusion. You may not use your
More informationMATH Spring 2011 Sample problems for Test 2: Solutions
MATH 304 505 Spring 011 Sample problems for Test : Solutions Any problem may be altered or replaced by a different one! Problem 1 (15 pts) Let M, (R) denote the vector space of matrices with real entries
More informationPractice 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 informationMath 21b. Review for Final Exam
Math 21b. Review for Final Exam Thomas W. Judson Spring 2003 General Information The exam is on Thursday, May 15 from 2:15 am to 5:15 pm in Jefferson 250. Please check with the registrar if you have a
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 informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More informationSection 7.3: SYMMETRIC MATRICES AND ORTHOGONAL DIAGONALIZATION
Section 7.3: SYMMETRIC MATRICES AND ORTHOGONAL DIAGONALIZATION When you are done with your homework you should be able to Recognize, and apply properties of, symmetric matrices Recognize, and apply properties
More informationLinear Algebra Final Exam Study Guide Solutions Fall 2012
. Let A = Given that v = 7 7 67 5 75 78 Linear Algebra Final Exam Study Guide Solutions Fall 5 explain why it is not possible to diagonalize A. is an eigenvector for A and λ = is an eigenvalue for A diagonalize
More informationGRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.
GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,
More informationFinal A. Problem Points Score Total 100. Math115A Nadja Hempel 03/23/2017
Final A Math115A Nadja Hempel 03/23/2017 nadja@math.ucla.edu Name: UID: Problem Points Score 1 10 2 20 3 5 4 5 5 9 6 5 7 7 8 13 9 16 10 10 Total 100 1 2 Exercise 1. (10pt) Let T : V V be a linear transformation.
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 information1. 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 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 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 informationBASIC ALGORITHMS IN LINEAR ALGEBRA. Matrices and Applications of Gaussian Elimination. A 2 x. A T m x. A 1 x A T 1. A m x
BASIC ALGORITHMS IN LINEAR ALGEBRA STEVEN DALE CUTKOSKY Matrices and Applications of Gaussian Elimination Systems of Equations Suppose that A is an n n matrix with coefficents in a field F, and x = (x,,
More informationLinear Algebra- Final Exam Review
Linear Algebra- Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.
More informationMath Computation Test 1 September 26 th, 2016 Debate: Computation vs. Theory Whatever wins, it ll be Huuuge!
Math 5- Computation Test September 6 th, 6 Debate: Computation vs. Theory Whatever wins, it ll be Huuuge! Name: Answer Key: Making Math Great Again Be sure to show your work!. (8 points) Consider the following
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 informationMATH 15a: Linear Algebra Practice Exam 2
MATH 5a: Linear Algebra Practice Exam 2 Write all answers in your exam booklet. Remember that you must show all work and justify your answers for credit. No calculators are allowed. Good luck!. Compute
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 informationANSWERS. E k E 2 E 1 A = B
MATH 7- Final Exam Spring ANSWERS Essay Questions points Define an Elementary Matrix Display the fundamental matrix multiply equation which summarizes a sequence of swap, combination and multiply operations,
More informationProblem 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 informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationFinal Exam Practice Problems Answers Math 24 Winter 2012
Final Exam Practice Problems Answers Math 4 Winter 0 () The Jordan product of two n n matrices is defined as A B = (AB + BA), where the products inside the parentheses are standard matrix product. Is the
More informationMATH 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 informationThen x 1,..., x n is a basis as desired. Indeed, it suffices to verify that it spans V, since n = dim(v ). We may write any v V as r
Practice final solutions. I did not include definitions which you can find in Axler or in the course notes. These solutions are on the terse side, but would be acceptable in the final. However, if you
More information33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM
33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM (UPDATED MARCH 17, 2018) The final exam will be cumulative, with a bit more weight on more recent material. This outline covers the what we ve done since the
More informationThe value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver I.N.
Math 410 Homework Problems In the following pages you will find all of the homework problems for the semester. Homework should be written out neatly and stapled and turned in at the beginning of class
More informationMath 1553 Worksheet 5.3, 5.5
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
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 information(b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).
.(5pts) Let B = 5 5. Compute det(b). (a) (b) (c) 6 (d) (e) 6.(5pts) Determine which statement is not always true for n n matrices A and B. (a) If two rows of A are interchanged to produce B, then det(b)
More informationMATH 1553 PRACTICE MIDTERM 3 (VERSION B)
MATH 1553 PRACTICE MIDTERM 3 (VERSION B) Name Section 1 2 3 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 10 points. The maximum score on this exam is 50 points.
More informationLinear Algebra Primer
Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................
More informationLINEAR ALGEBRA BOOT CAMP WEEK 2: LINEAR OPERATORS
LINEAR ALGEBRA BOOT CAMP WEEK 2: LINEAR OPERATORS Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F has characteristic zero. The following are facts
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 informationMathematics I: Algebra
Mathematics I: lgera Prolem. Solve the following system of equations y using the reduced row echelon form of its augmented matrix: x + x x x x + x ; x x + 6x Solution: The augmented matrix is 6 F FF F;
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Math 24 Calculus III Summer 25, Session II. Determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless otherwise stated,
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 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003
MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003 1. True or False (28 points, 2 each) T or F If V is a vector space
More informationJordan 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 informationEcon 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 informationFinal Exam, Linear Algebra, Fall, 2003, W. Stephen Wilson
Final Exam, Linear Algebra, Fall, 2003, W. Stephen Wilson Name: TA Name and section: NO CALCULATORS, SHOW ALL WORK, NO OTHER PAPERS ON DESK. There is very little actual work to be done on this exam if
More information1. 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 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 information2. Every linear system with the same number of equations as unknowns has a unique solution.
1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations
More informationNATIONAL 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