Linear transformations
|
|
- Delphia Adams
- 5 years ago
- Views:
Transcription
1 Linear transformations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T. Linear transformations Differential equations 1 / 36
2 Outline 1 Definition of a linear transformation 2 Kernel and range of a linear transformation 3 The eigenvalue-eigenvector problem 4 General results for eigenvalues and eigenvectors Samy T. Linear transformations Differential equations 2 / 36
3 Outline 1 Definition of a linear transformation 2 Kernel and range of a linear transformation 3 The eigenvalue-eigenvector problem 4 General results for eigenvalues and eigenvectors Samy T. Linear transformations Differential equations 3 / 36
4 Definition of mapping Definition 1. Let Then: V and W two vector spaces 1 A mapping is a rule which assigns to each vector v V a vector w = T (v) W 2 Notation: T : V W Samy T. Linear transformations Differential equations 4 / 36
5 Linear transformation Definition 2. Let V and W two vector spaces A mapping T : V W Then T is a linear transformation if it satisfies: 1 T (u + v) = T (u) + T (v) 2 T (c v) = c T (v) Vocabulary: V is called domain of T W is called codomain of T Samy T. Linear transformations Differential equations 5 / 36
6 Examples of linear transformations Example with matrices: The map T : M n (R) M n (R), T (A) = A T is a linear transformation Example with functions: The map T : C 2 (I) C 0 (I), T (y) = y + y is a linear transformation Samy T. Linear transformations Differential equations 6 / 36
7 Matrix form Theorem 3. Let T : R n R m be a linear transformation (e 1,..., e n ) canonical basis of R n Then T is described by the matrix transformation T (x) = A x, where A is the matrix defined by A = [T (e 1 ), T (e 2 ),..., T (e n )] Samy T. Linear transformations Differential equations 7 / 36
8 Example of matrix form Linear transformation: Defined as T : R 2 R 3 such that T (1, 0) = (2, 3, 1), T (0, 1) = (5, 4, 7) Matrix form: T (x) = A x with A = [T (e 1 ), T (e 2 )] = General expression: 2x 1 + 5x 2 T (x) = A x = 3x 1 4x 2 x 1 + 7x 2 Samy T. Linear transformations Differential equations 8 / 36
9 Matrix form for polynomials Linear transformation: Defined as T : P 1 P 2 such that T (a 0 + a 1 x) = (2a 0 + a 1 ) + (a 0 a 1 )x + 3a 1 x 2. Matrix form: If then we have p(x) = a 0 + a 1 x, and a = [ ] a0 a T (p) = A a, with A = Samy T. Linear transformations Differential equations 9 / 36
10 Outline 1 Definition of a linear transformation 2 Kernel and range of a linear transformation 3 The eigenvalue-eigenvector problem 4 General results for eigenvalues and eigenvectors Samy T. Linear transformations Differential equations 10 / 36
11 Definitions Definition 4. Let T : V W be a linear transformation. Then 1 The kernel of T is defined as a subspace of V : ker(t ) = {v V ; T (v) = 0} 2 The range of T is defined as a subspace of W : Rng(T ) = {w W ; w = T (v) for a v V } Samy T. Linear transformations Differential equations 11 / 36
12 Example Linear transformation: We set T : C 2 (I) C 0 (I), T (y) = y + y Definition of ker(t ): Solution set of the differential equation y + y = 0 Characterization of ker(t ): We have ker(t ) = { y C 2 (I); y(x) = c 1 cos(x) + c 2 sin(x) } Samy T. Linear transformations Differential equations 12 / 36
13 Case of a matrix transformation Proposition 5. Let Then T : R n R m be a linear transformation A matrix of T 1 ker(t ) is the solution set to the system Ax = 0 Otherwise stated, we have ker(t ) = nullspace(a) 2 The range of T is characterized as Rng(T ) = colspace(a) Samy T. Linear transformations Differential equations 13 / 36
14 Application Transformation: We consider T : R 3 R 2 with matrix [ ] A = Reduced row-echelon form: For the augmented matrix we find A ker(t ) and Rng(T ): We find [ ] ker(t ) = { x R 3 ; x = r(2, 1, 0) + s(5, 0, 1) } Rng(T ) = { y R 2 ; y = r(1, 2) } dim[ker(t )] + dim[rng(t )] = 3 = dim[r 3 ] Samy T. Linear transformations Differential equations 14 / 36
15 Rank-nullity theorem Theorem 6. Let V finite dimensional space T : V W linear transformation Then dim[ker(t )] + dim[rng(t )] = dim[v ] Vocabulary: dim[ker(t )] is called nullity of T. Samy T. Linear transformations Differential equations 15 / 36
16 Example Linear transformation: We define T : P 1 P 2 by T (a + b x) = (2a 3b) + (b 5a)x + (a + b)x 2 Matrix form: In the canonical basis (1, x, x 2 ), 2 3 A = Reduced row-echelon form: For the system Ax = 0, we find A Samy T. Linear transformations Differential equations 16 / 36
17 Example (2) Kernel: According to the reduced row-echelon form, ker(t ) = {0} P 1 Range: According to the reduced row-echelon form, Rng(T ) = span { 2 5x + x 2, 3 + x + x 2} Verifying the rank-nullity theorem: We have dim[ker(t )] = 0, dim[rng(t )] = 2, and thus dim[ker(t )] + dim[rng(t )] = 2 = dim[p 1 ] Samy T. Linear transformations Differential equations 17 / 36
18 Outline 1 Definition of a linear transformation 2 Kernel and range of a linear transformation 3 The eigenvalue-eigenvector problem 4 General results for eigenvalues and eigenvectors Samy T. Linear transformations Differential equations 18 / 36
19 Definitions Definition 7. Let Then A be a n n matrix λ R 1 If the the following system has nontrivial solutions: we say that λ is an eigenvalue of A. A v = λ v, (1) 2 If λ is an eigenvalue A vector v satisfying (1) is called eigenvector. Samy T. Linear transformations Differential equations 19 / 36
20 Strategy for the eigenvalue problem Illustration in R 2 : Strategy: 1 The eigenvalues are scalars such that det(a λ I) = 0 2 If the eigenvalues λ 1,..., λ k are given by Step 1, The eigenvectors solve the systems (i = 1,..., k) (A λ i ) v i = 0, Samy T. Linear transformations Differential equations 20 / 36
21 Example with simple eigenvalues Matrix: A = [ ] (2) Characteristic equation: 5 λ λ λ2 + 2λ 3 = 0 Eigenvalues: λ 1 = 3, λ 2 = 1 Samy T. Linear transformations Differential equations 21 / 36
22 Example with simple eigenvalues (2) Computation of A λ 1 I: We get Reduced row-echelon form: Eigenvectors for λ 1 = 3: A + 3I = A + 3I [ ] [ ] {r v 1 ; r R}, where v 1 = (1, 2) Samy T. Linear transformations Differential equations 22 / 36
23 Example with simple eigenvalues (3) Eigenvectors for λ 2 = 1: {r v 2 ; r R}, where v 2 = (1, 1) Remark: The family {v 1, v 2 } forms a basis of R 2. Samy T. Linear transformations Differential equations 23 / 36
24 Example with double eigenvalue Matrix: A = [ ] (3) Characteristic equation: 1 λ λ (λ 1)2 = 0 Eigenvalues: λ 1 = 1, repeated twice Samy T. Linear transformations Differential equations 24 / 36
25 Example with double eigenvalues (2) Computation of A λ 1 I: We get directly a reduced row-echelon form A I = [ ] Eigenvectors for λ 1 = 1: We get only one linearly indep. eigenvector {r v 1 ; r R}, where v 1 = (1, 0) Conclusion: Number of indep. eigenvectors is less than order of eigenvalue The matrix A is said to be defective Samy T. Linear transformations Differential equations 25 / 36
26 Complex eigenvalues Theorem 8. Let A be such that A is an n n matrix A has real-valued elements A admits a complex eigenvalue λ with eigenvector v Then λ is an eigenvalue for A with eigenvector v. Samy T. Linear transformations Differential equations 26 / 36
27 Example with complex eigenvalues Matrix: A = [ ] Characteristic polynomial: 2 λ 6 p(λ) = 3 4 λ = λ2 2λ + 10 Eigenvalues: λ 1 = 1 + 3ı, λ 2 = λ 1 = 1 3ı Samy T. Linear transformations Differential equations 27 / 36
28 Example with complex eigenvalues (2) Computation of A λ 1 I: We get A (1 + 3ı)I = Reduced row-echelon form: (A (1 + 3ı)I) Eigenvectors for λ 1 = 1 + 3ı: [ ] 3 3ı ı [ 1 1 ı ] {r v 1 ; r R}, where v 1 = ( (1 ı), 1) Samy T. Linear transformations Differential equations 28 / 36
29 Example with complex eigenvalues (3) Eigenvectors for λ 2 = 1 3ı: {r v 2 ; r R}, where v 2 = v 1 = ( (1 + ı), 1) Remark: The family {v 1, v 2 } forms a basis of C 2. Samy T. Linear transformations Differential equations 29 / 36
30 David Hilbert Hilbert: Lived in Germany Among top 3 mathematicians of 20th century Foundations of mathematics Infinite dimensional vector spaces Hilbert spaces Number theory Axioms of geometry Coined the term eigenvalue Eigenvalue: In German, own value (or proper value). Samy T. Linear transformations Differential equations 30 / 36
31 Outline 1 Definition of a linear transformation 2 Kernel and range of a linear transformation 3 The eigenvalue-eigenvector problem 4 General results for eigenvalues and eigenvectors Samy T. Linear transformations Differential equations 31 / 36
32 Eigenspace Definition 9. Let A be a n n matrix λ i eigenvalue of A The eigenspace E i is defined as E i {v; A v = λ i v}. Samy T. Linear transformations Differential equations 32 / 36
33 Dimension of an eigenspace Theorem 10. Let A be a n n matrix λ i eigenvalue of A m i multiplicity of λ i Then the eigenspace E i is a subspace of C n such that dim[e i ] m i Samy T. Linear transformations Differential equations 33 / 36
34 Examples Case with simple eigenvalues: For the matrix (2) we have m 1 = dim(e 1 ) = 1, m 2 = dim(e 2 ) = 1 Case with double eigenvalue: For the matrix (3) we have m 1 = 2, dim(e 1 ) = 1 Samy T. Linear transformations Differential equations 34 / 36
35 Defective and nondefective matrices Definition 11. Let A be a n n matrix. If A has n independent eigenvectors A is called nondefective If A has less than n independent eigenvectors A is called defective Remark: If A is nondefective There exists a basis of eigenvectors of A for R n Samy T. Linear transformations Differential equations 35 / 36
36 Criterion of nondefectiveness Theorem 12. Examples of nondefective matrices in M n (R): 1 When A has n distinct eigenvalues, A is nondefective 2 A non defective iff for all i we have dim[e i ] = m i Examples: Matrix (2) is nondefective Matrix (3) is defective Samy T. Linear transformations Differential equations 36 / 36
CHAPTER 5 REVIEW. c 1. c 2 can be considered as the coordinates of v
CHAPTER 5 REVIEW Throughout this note, we assume that V and W are two vector spaces with dimv = n and dimw = m. T : V W is a linear transformation.. A map T : V W is a linear transformation if and only
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 informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Page of 7 Linear Algebra Practice Problems These problems cover Chapters 4, 5, 6, and 7 of Elementary Linear Algebra, 6th ed, by Ron Larson and David Falvo (ISBN-3 = 978--68-78376-2,
More informationMAT Linear Algebra Collection of sample exams
MAT 342 - Linear Algebra Collection of sample exams A-x. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system
More 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 informationThe definition of a vector space (V, +, )
The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element
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 equations in linear algebra
Linear equations in linear algebra Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra Pearson Collections Samy T. Linear
More informationCarleton College, winter 2013 Math 232, Solutions to review problems and practice midterm 2 Prof. Jones 15. T 17. F 38. T 21. F 26. T 22. T 27.
Carleton College, winter 23 Math 232, Solutions to review problems and practice midterm 2 Prof. Jones Solutions to review problems: Chapter 3: 6. F 8. F. T 5. T 23. F 7. T 9. F 4. T 7. F 38. T Chapter
More informationInstructions Please answer the five problems on your own paper. These are essay questions: you should write in complete sentences.
Instructions Please answer the five problems on your own paper. These are essay questions: you should write in complete sentences.. Recall that P 3 denotes the vector space of polynomials of degree less
More informationWe see that this is a linear system with 3 equations in 3 unknowns. equation is A x = b, where
Practice Problems Math 35 Spring 7: Solutions. Write the system of equations as a matrix equation and find all solutions using Gauss elimination: x + y + 4z =, x + 3y + z = 5, x + y + 5z = 3. We see that
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for
More 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 informationMATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix.
MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis.
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 informationLinear Algebra (Math-324) Lecture Notes
Linear Algebra (Math-324) Lecture Notes Dr. Ali Koam and Dr. Azeem Haider September 24, 2017 c 2017,, Jazan All Rights Reserved 1 Contents 1 Real Vector Spaces 6 2 Subspaces 11 3 Linear Combination and
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 informationChapter 2. General Vector Spaces. 2.1 Real Vector Spaces
Chapter 2 General Vector Spaces Outline : Real vector spaces Subspaces Linear independence Basis and dimension Row Space, Column Space, and Nullspace 2 Real Vector Spaces 2 Example () Let u and v be vectors
More informationWhat is on this week. 1 Vector spaces (continued) 1.1 Null space and Column Space of a matrix
Professor Joana Amorim, jamorim@bu.edu What is on this week Vector spaces (continued). Null space and Column Space of a matrix............................. Null Space...........................................2
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 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 250B Midterm II Review Session Spring 2019 SOLUTIONS
Math 250B Midterm II Review Session Spring 2019 SOLUTIONS [ Problem #1: Find a spanning set for nullspace 1 2 0 2 3 4 8 0 8 12 1 2 0 2 3 SOLUTION: The row-reduced form of this matrix is Setting 0 0 0 0
More information1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?
. Let m and n be two natural numbers such that m > n. Which of the following is/are true? (i) A linear system of m equations in n variables is always consistent. (ii) A linear system of n equations in
More informationMATH 2360 REVIEW PROBLEMS
MATH 2360 REVIEW PROBLEMS Problem 1: In (a) (d) below, either compute the matrix product or indicate why it does not exist: ( )( ) 1 2 2 1 (a) 0 1 1 2 ( ) 0 1 2 (b) 0 3 1 4 3 4 5 2 5 (c) 0 3 ) 1 4 ( 1
More informationStudy Guide for Linear Algebra Exam 2
Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real
More informationMath 205, Summer I, Week 3a (continued): Chapter 4, Sections 5 and 6. Week 3b. Chapter 4, [Sections 7], 8 and 9
Math 205, Summer I, 2016 Week 3a (continued): Chapter 4, Sections 5 and 6. Week 3b Chapter 4, [Sections 7], 8 and 9 4.5 Linear Dependence, Linear Independence 4.6 Bases and Dimension 4.7 Change of Basis,
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 informationLinear Algebra Practice Problems
Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a
More 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 informationMath 113 Midterm Exam Solutions
Math 113 Midterm Exam Solutions Held Thursday, May 7, 2013, 7-9 pm. 1. (10 points) Let V be a vector space over F and T : V V be a linear operator. Suppose that there is a non-zero vector v V such that
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 informationDefinition Suppose S R n, V R m are subspaces. A map U : S V is linear if
.6. Restriction of Linear Maps In this section, we restrict linear maps to subspaces. We observe that the notion of linearity still makes sense for maps whose domain and codomain are subspaces of R n,
More informationEigenvalues and Eigenvectors 7.1 Eigenvalues and Eigenvecto
7.1 November 6 7.1 Eigenvalues and Eigenvecto Goals Suppose A is square matrix of order n. Eigenvalues of A will be defined. Eigenvectors of A, corresponding to each eigenvalue, will be defined. Eigenspaces
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 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 informationMath 369 Exam #2 Practice Problem Solutions
Math 369 Exam #2 Practice Problem Solutions 2 5. Is { 2, 3, 8 } a basis for R 3? Answer: No, it is not. To show that it is not a basis, it suffices to show that this is not a linearly independent set.
More informationAdvanced Linear Algebra Math 4377 / 6308 (Spring 2015) March 5, 2015
Midterm 1 Advanced Linear Algebra Math 4377 / 638 (Spring 215) March 5, 215 2 points 1. Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain
More informationLINEAR ALGEBRA MICHAEL PENKAVA
LINEAR ALGEBRA MICHAEL PENKAVA 1. Linear Maps Definition 1.1. If V and W are vector spaces over the same field K, then a map λ : V W is called a linear map if it satisfies the two conditions below: (1)
More informationContents. 6 Systems of First-Order Linear Dierential Equations. 6.1 General Theory of (First-Order) Linear Systems
Dierential Equations (part 3): Systems of First-Order Dierential Equations (by Evan Dummit, 26, v 2) Contents 6 Systems of First-Order Linear Dierential Equations 6 General Theory of (First-Order) Linear
More informationRemark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.
Sec 5 Eigenvectors and Eigenvalues In this chapter, vector means column vector Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called
More informationMath 250B Midterm II Information Spring 2019 SOLUTIONS TO PRACTICE PROBLEMS
Math 50B Midterm II Information Spring 019 SOLUTIONS TO PRACTICE PROBLEMS Problem 1. Determine whether each set S below forms a subspace of the given vector space V. Show carefully that your answer is
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 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 information5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.
Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the
More informationMath 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 informationLinear Algebra Highlights
Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to
More 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 informationJune 2011 PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 262) Linear Algebra and Differential Equations
June 20 PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 262) Linear Algebra and Differential Equations The topics covered in this exam can be found in An introduction to differential equations
More informationQuizzes for Math 304
Quizzes for Math 304 QUIZ. A system of linear equations has augmented matrix 2 4 4 A = 2 0 2 4 3 5 2 a) Write down this system of equations; b) Find the reduced row-echelon form of A; c) What are the pivot
More informationControl Systems. Linear Algebra topics. L. Lanari
Control Systems Linear Algebra topics L Lanari outline basic facts about matrices eigenvalues - eigenvectors - characteristic polynomial - algebraic multiplicity eigenvalues invariance under similarity
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 informationMath 250B Final Exam Review Session Spring 2015 SOLUTIONS
Math 5B Final Exam Review Session Spring 5 SOLUTIONS Problem Solve x x + y + 54te 3t and y x + 4y + 9e 3t λ SOLUTION: We have det(a λi) if and only if if and 4 λ only if λ 3λ This means that the eigenvalues
More information4.9 The Rank-Nullity Theorem
For Problems 7 10, use the ideas in this section to determine a basis for the subspace of R n spanned by the given set of vectors. 7. {(1, 1, 2), (5, 4, 1), (7, 5, 4)}. 8. {(1, 3, 3), (1, 5, 1), (2, 7,
More information8 General Linear Transformations
8 General Linear Transformations 8.1 Basic Properties Definition 8.1 If T : V W is a function from a vector space V into a vector space W, then T is called a linear transformation from V to W if, for all
More informationLecture 7: Positive Semidefinite Matrices
Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.
More informationPractice 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 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 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 informationQuestion 7. Consider a linear system A x = b with 4 unknown. x = [x 1, x 2, x 3, x 4 ] T. The augmented
Question. How many solutions does x 6 = 4 + i have Practice Problems 6 d) 5 Question. Which of the following is a cubed root of the complex number i. 6 e i arctan() e i(arctan() π) e i(arctan() π)/3 6
More information(a) only (ii) and (iv) (b) only (ii) and (iii) (c) only (i) and (ii) (d) only (iv) (e) only (i) and (iii)
. Which of the following are Vector Spaces? (i) V = { polynomials of the form q(t) = t 3 + at 2 + bt + c : a b c are real numbers} (ii) V = {at { 2 + b : a b are real numbers} } a (iii) V = : a 0 b is
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
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 informationMatrices and systems of linear equations
Matrices and systems of linear equations Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T.
More information2.3. VECTOR SPACES 25
2.3. VECTOR SPACES 25 2.3 Vector Spaces MATH 294 FALL 982 PRELIM # 3a 2.3. Let C[, ] denote the space of continuous functions defined on the interval [,] (i.e. f(x) is a member of C[, ] if f(x) is continuous
More informationMath 110: Worksheet 3
Math 110: Worksheet 3 September 13 Thursday Sept. 7: 2.1 1. Fix A M n n (F ) and define T : M n n (F ) M n n (F ) by T (B) = AB BA. (a) Show that T is a linear transformation. Let B, C M n n (F ) and a
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 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 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 informationOnline Exercises for Linear Algebra XM511
This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2
More informationComps Study Guide for Linear Algebra
Comps Study Guide for Linear Algebra Department of Mathematics and Statistics Amherst College September, 207 This study guide was written to help you prepare for the linear algebra portion of the Comprehensive
More informationAbstract Vector Spaces and Concrete Examples
LECTURE 18 Abstract Vector Spaces and Concrete Examples Our discussion of linear algebra so far has been devoted to discussing the relations between systems of linear equations, matrices, and vectors.
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 informationx y + z = 3 2y z = 1 4x + y = 0
MA 253: Practice Exam Solutions You may not use a graphing calculator, computer, textbook, notes, or refer to other people (except the instructor). Show all of your work; your work is your answer. Problem
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 314H Solutions to Homework # 3
Math 34H Solutions to Homework # 3 Complete the exercises from the second maple assignment which can be downloaded from my linear algebra course web page Attach printouts of your work on this problem to
More informationThis paper is not to be removed from the Examination Halls
~~MT1173 ZA d0 This paper is not to be removed from the Examination Halls UNIVERSITY OF LONDON MT1173 ZA BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences,
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 informationAssignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work.
Assignment 1 Math 5341 Linear Algebra Review Give complete answers to each of the following questions Show all of your work Note: You might struggle with some of these questions, either because it has
More informationMath 4377/6308 Advanced Linear Algebra
2. Linear Transformations Math 4377/638 Advanced Linear Algebra 2. Linear Transformations, Null Spaces and Ranges Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/
More informationLinear Algebra and Matrix Theory
Linear Algebra and Matrix Theory Part 3 - Linear Transformations 1 References (1) S Friedberg, A Insel and L Spence, Linear Algebra, Prentice-Hall (2) M Golubitsky and M Dellnitz, Linear Algebra and Differential
More informationDeterminants. Samy Tindel. Purdue University. Differential equations and linear algebra - MA 262
Determinants Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra by Goode and Annin Samy T. Determinants Differential equations
More informationAnnouncements Wednesday, November 01
Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section
More informationMATH 3321 Sample Questions for Exam 3. 3y y, C = Perform the indicated operations, if possible: (a) AC (b) AB (c) B + AC (d) CBA
MATH 33 Sample Questions for Exam 3. Find x and y so that x 4 3 5x 3y + y = 5 5. x = 3/7, y = 49/7. Let A = 3 4, B = 3 5, C = 3 Perform the indicated operations, if possible: a AC b AB c B + AC d CBA AB
More informationLINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS
LINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F has characteristic zero. The following are facts (in
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 informationFall TMA4145 Linear Methods. Exercise set Given the matrix 1 2
Norwegian University of Science and Technology Department of Mathematical Sciences TMA445 Linear Methods Fall 07 Exercise set Please justify your answers! The most important part is how you arrive at an
More informationEigenvalues, Eigenvectors. Eigenvalues and eigenvector will be fundamentally related to the nature of the solutions of state space systems.
Chapter 3 Linear Algebra In this Chapter we provide a review of some basic concepts from Linear Algebra which will be required in order to compute solutions of LTI systems in state space form, discuss
More informationMTH 362: Advanced Engineering Mathematics
MTH 362: Advanced Engineering Mathematics Lecture 5 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 26, 2017 1 Linear Independence and Dependence of Vectors
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 informationft-uiowa-math2550 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 12/31/2014 at 10:36pm CST
me me ft-uiowa-math255 Assignment OptionalFinalExamReviewMultChoiceMEDIUMlengthForm due 2/3/2 at :3pm CST. ( pt) Library/TCNJ/TCNJ LinearSystems/problem3.pg Give a geometric description of the following
More 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 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 250B Midterm III Information Fall 2018
Math 250B Midterm III Information Fall 2018 WHEN: Monday, December 10, in class (no notes, books, calculators I will supply a table of annihilators and integrals.) EXTRA OFFICE HOURS : Friday, December
More informationMATH 225 Summer 2005 Linear Algebra II Solutions to Assignment 1 Due: Wednesday July 13, 2005
MATH 225 Summer 25 Linear Algebra II Solutions to Assignment 1 Due: Wednesday July 13, 25 Department of Mathematical and Statistical Sciences University of Alberta Question 1. [p 224. #2] The set of all
More informationVector Spaces 4.5 Basis and Dimension
Vector Spaces 4.5 and Dimension Summer 2017 Vector Spaces 4.5 and Dimension Goals Discuss two related important concepts: Define of a Vectors Space V. Define Dimension dim(v ) of a Vectors Space V. Vector
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 informationDr. Abdulla Eid. Section 4.2 Subspaces. Dr. Abdulla Eid. MATHS 211: Linear Algebra. College of Science
Section 4.2 Subspaces College of Science MATHS 211: Linear Algebra (University of Bahrain) Subspaces 1 / 42 Goal: 1 Define subspaces. 2 Subspace test. 3 Linear Combination of elements. 4 Subspace generated
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 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 informationChapter 7. Canonical Forms. 7.1 Eigenvalues and Eigenvectors
Chapter 7 Canonical Forms 7.1 Eigenvalues and Eigenvectors Definition 7.1.1. Let V be a vector space over the field F and let T be a linear operator on V. An eigenvalue of T is a scalar λ F such that there
More information