Assignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work.

Size: px
Start display at page:

Download "Assignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work."

Transcription

1 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 been a while since you took linear algebra, or because you did not learn some of these concepts Don t worry We will use the Discussion Forum on to ask questions and help one another 1 Suppose A is a real m by n matrix and a What does it mean to say that a vector is a solution to Ax = b? b What does it mean to say that the system Ax = b is consistent? c What does it mean to say that a vector is a least squares solution to Ax = b? 2 Suppose A is a real m by n matrix and a What is the system of normal equations associated with the system Ax = b, and what purpose does this system serve? b Suppose is a least squares solution to Ax = b How is Ax related to the vector b? 3 Suppose A is a real m by n matrix and True or False If the statement is true, then give a reason If the statement is false, then give a counter example a The system Ax = b always has at least one solution b If the system Ax = 0 has a nontrivial solution, then Ax = b is inconsistent c A least squares solution to Ax = b is also a solution to Ax = b d A solution to Ax = b is also a least squares solution to Ax = b 2x1x2 2x3 2 4 Use elementary row operations to solve the linear system x1 2x2 2x3 1 x1 5x2 4x3 1 x1x2 2 5 Show that the system 2x1x2 1 is inconsistent 2x1x2 4 6 Suppose A is a real n by n matrix Describe the process of using elementary row operations to determine if A is invertible, and if it is, finding the inverse of A

2 Use elementary row operations to find the inverse of the matrix Suppose A is a real m by n matrix a What is Col (the column space of sometimes also called the range space of )? b What is Nul (the null space of sometimes also called the kernel of )? c Two vectors, are orthogonal (perpendicular) with respect to the Euclidean dot product if and only if 0 Two subspaces, are orthogonal if and only if 0 for every and In this case, we write Prove that Nul Col d Suppose is a subspace of Describe the orthogonal complement of with respect to the Euclidean dot product e Prove that Nul is the orthogonal complement of Col Let V span,, Give all of the equations that must be satisfied by a vector with Euclidean length 1 that is orthogonal to V (with respect to the Euclidean dot product) 10 State the rank theorem 11 Suppose A is an n by n matrix Give at least 10 equivalent statements to is invertible 12 Suppose A is a 6 by 8 matrix, and dim Col (A T ) = 3 a dim Nul (A T ) = b dim Nul (A) = c Give the size of A T A d Give the size of AA T e dim Col (A) = f Show that Col A T A is a subset of Col (A T )

3 g Is it possible for A to be row equivalent to? Suppose A and B are n by n matrices What does it mean to say that A and B are similar matrices? List some consequences of A and B being similar matrices? 14 True or False If the statement is true, then give a reason If the result is false, then give a counter example a If two matrices have the same trace, then they are similar matrices b Elementary row operations, applied to a matrix, do not change the eigenvalues of the matrix c Elementary row operations, applied to an augmented matrix, do not change the solution set to the corresponding system of linear equations 15 Suppose,,, is a subset of a Describe span,,, b What has to be true for,,, to be an orthonormal subset of? c Suppose,,, is an orthonormal subset of Form the matrix whose columns are,,, What is? 16 Suppose,, a How do you create proj (the projection of onto span)? b How do you create proj, (the projection of onto span, )? c Explain how to find two orthogonal vectors whose span is, d Explain how to find the angle between and, Let v 1 and u a Give projvu (the projection of onto span)

4 b Give the distance from u to 1 span Suppose is a subspace of What does it mean to say that is a subspace of? 19 Show that H 2 3, R is a subspace of 20 P 3 is the set of all polynomials with real coefficients, of a single variable, of degree less 3 R than or equal to 3 Define the function T : P 3 P 3 via T (y) = (2x 2 1)y + y a Show that P 3 is a vector space b Show that T is a linear transformation c Give the matrix representation for T with respect to the standard basis {1, x, x 2, x 3 } of P Find the eigenvalue and associated eigenspaces for the matrix A is a real 3 by 3 matrix with eigenvalues -1/2, 1/2 and 0 Suppose E 1/2 1 span 1, 1 E 1/ /2 1/2 0 span 1 and E0 span 1, and /2 1/2 1 Give the matrix A The matrix A is similar to a Give the eigenvalues of A b Give the trace of A c Give the determinant of A 24 Suppose you have the data,,,,,, a Describe the process of finding the line of best least squares fit for the data 1

5 b Suppose y = mx + b is the line of best least squares fit for the data given in part a How do you find the predicted value of y at a value x = c? 25 Consider the (x,y) data given by 1, 1, 1, 2, 2, 4 a Find the line of best least squares fit for this data b Use this line to predict the value of y when x = 15

Chapter 3. Directions: For questions 1-11 mark each statement True or False. Justify each answer.

Chapter 3. Directions: For questions 1-11 mark each statement True or False. Justify each answer. Chapter 3 Directions: For questions 1-11 mark each statement True or False. Justify each answer. 1. (True False) Asking whether the linear system corresponding to an augmented matrix [ a 1 a 2 a 3 b ]

More information

MATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)

MATH 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 information

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions

YORK UNIVERSITY. Faculty of Science Department of Mathematics and Statistics MATH M Test #1. July 11, 2013 Solutions YORK UNIVERSITY Faculty of Science Department of Mathematics and Statistics MATH 222 3. M Test # July, 23 Solutions. For each statement indicate whether it is always TRUE or sometimes FALSE. Note: For

More information

MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL

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

More information

1. 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

1. 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 information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

More information

Test 3, Linear Algebra

Test 3, Linear Algebra Test 3, Linear Algebra Dr. Adam Graham-Squire, Fall 2017 Name: I pledge that I have neither given nor received any unauthorized assistance on this exam. (signature) DIRECTIONS 1. Don t panic. 2. Show all

More information

Linear Algebra Practice Problems

Linear 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 information

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET

IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each

More information

Linear Algebra Final Exam Study Guide Solutions Fall 2012

Linear 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 information

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

2. Every linear system with the same number of equations as unknowns has a unique solution. 1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations

More information

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

Math 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system

More information

MATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.

MATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible. MATH 2331 Linear Algebra Section 2.1 Matrix Operations Definition: A : m n, B : n p ( 1 2 p ) ( 1 2 p ) AB = A b b b = Ab Ab Ab Example: Compute AB, if possible. 1 Row-column rule: i-j-th entry of AB:

More information

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

Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017 Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...

More information

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

MATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether

More information

MATH 240 Spring, Chapter 1: Linear Equations and Matrices

MATH 240 Spring, Chapter 1: Linear Equations and Matrices MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear

More information

Math 3191 Applied Linear Algebra

Math 3191 Applied Linear Algebra Math 191 Applied Linear Algebra Lecture 16: Change of Basis Stephen Billups University of Colorado at Denver Math 191Applied Linear Algebra p.1/0 Rank The rank of A is the dimension of the column space

More information

spring, math 204 (mitchell) list of theorems 1 Linear Systems Linear Transformations Matrix Algebra

spring, 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 information

1. Let m 1 and n 1 be two natural numbers such that m > n. Which of the following is/are true?

1. 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 information

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

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

More information

Math 54 HW 4 solutions

Math 54 HW 4 solutions Math 54 HW 4 solutions 2.2. Section 2.2 (a) False: Recall that performing a series of elementary row operations A is equivalent to multiplying A by a series of elementary matrices. Suppose that E,...,

More information

MATH 1553 PRACTICE FINAL EXAMINATION

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

More information

Answer Key for Exam #2

Answer Key for Exam #2 . Use elimination on an augmented matrix: Answer Key for Exam # 4 4 8 4 4 4 The fourth column has no pivot, so x 4 is a free variable. The corresponding system is x + x 4 =, x =, x x 4 = which we solve

More information

Glossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

Glossary 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 information

LINEAR ALGEBRA QUESTION BANK

LINEAR 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 information

Linear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4

Linear Systems. Class 27. c 2008 Ron Buckmire. TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5.4 Linear Systems Math Spring 8 c 8 Ron Buckmire Fowler 9 MWF 9: am - :5 am http://faculty.oxy.edu/ron/math//8/ Class 7 TITLE Projection Matrices and Orthogonal Diagonalization CURRENT READING Poole 5. Summary

More information

LINEAR ALGEBRA SUMMARY SHEET.

LINEAR 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 information

Study Guide for Linear Algebra Exam 2

Study Guide for Linear Algebra Exam 2 Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real

More information

March 27 Math 3260 sec. 56 Spring 2018

March 27 Math 3260 sec. 56 Spring 2018 March 27 Math 3260 sec. 56 Spring 2018 Section 4.6: Rank Definition: The row space, denoted Row A, of an m n matrix A is the subspace of R n spanned by the rows of A. We now have three vector spaces associated

More information

Overview. Motivation for the inner product. Question. Definition

Overview. Motivation for the inner product. Question. Definition Overview Last time we studied the evolution of a discrete linear dynamical system, and today we begin the final topic of the course (loosely speaking) Today we ll recall the definition and properties of

More information

MATH 260 LINEAR ALGEBRA EXAM III Fall 2014

MATH 260 LINEAR ALGEBRA EXAM III Fall 2014 MAH 60 LINEAR ALGEBRA EXAM III Fall 0 Instructions: the use of built-in functions of your calculator such as det( ) or RREF is permitted ) Consider the table and the vectors and matrices given below Fill

More information

MA 265 FINAL EXAM Fall 2012

MA 265 FINAL EXAM Fall 2012 MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators

More information

Practice Final Exam. Solutions.

Practice Final Exam. Solutions. MATH Applied Linear Algebra December 6, 8 Practice Final Exam Solutions Find the standard matrix f the linear transfmation T : R R such that T, T, T Solution: Easy to see that the transfmation T can be

More information

No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question.

No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question. Math 304 Final Exam (May 8) Spring 206 No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question. Name: Section: Question Points

More information

Miderm II Solutions To find the inverse we row-reduce the augumented matrix [I A]. In our case, we row reduce

Miderm II Solutions To find the inverse we row-reduce the augumented matrix [I A]. In our case, we row reduce Miderm II Solutions Problem. [8 points] (i) [4] Find the inverse of the matrix A = To find the inverse we row-reduce the augumented matrix [I A]. In our case, we row reduce We have A = 2 2 (ii) [2] Possibly

More information

The 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.

The 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 information

Spring 2014 Math 272 Final Exam Review Sheet

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

More information

Dimension. Eigenvalue and eigenvector

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

More information

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

LINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,

More information

Review Notes for Linear Algebra True or False Last Updated: February 22, 2010

Review Notes for Linear Algebra True or False Last Updated: February 22, 2010 Review Notes for Linear Algebra True or False Last Updated: February 22, 2010 Chapter 4 [ Vector Spaces 4.1 If {v 1,v 2,,v n } and {w 1,w 2,,w n } are linearly independent, then {v 1 +w 1,v 2 +w 2,,v n

More information

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

(a) II and III (b) I (c) I and III (d) I and II and III (e) None are true. 1 Which of the following statements is always true? I The null space of an m n matrix is a subspace of R m II If the set B = {v 1,, v n } spans a vector space V and dimv = n, then B is a basis for V III

More information

Projections and Least Square Solutions. Recall that given an inner product space V with subspace W and orthogonal basis for

Projections and Least Square Solutions. Recall that given an inner product space V with subspace W and orthogonal basis for Math 57 Spring 18 Projections and Least Square Solutions Recall that given an inner product space V with subspace W and orthogonal basis for W, B {v 1, v,..., v k }, the orthogonal projection of V onto

More information

PRACTICE PROBLEMS FOR THE FINAL

PRACTICE PROBLEMS FOR THE FINAL PRACTICE PROBLEMS FOR THE FINAL Here are a slew of practice problems for the final culled from old exams:. Let P be the vector space of polynomials of degree at most. Let B = {, (t ), t + t }. (a) Show

More information

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

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

More information

Math 3191 Applied Linear Algebra

Math 3191 Applied Linear Algebra Math 191 Applied Linear Algebra Lecture 1: Inner Products, Length, Orthogonality Stephen Billups University of Colorado at Denver Math 191Applied Linear Algebra p.1/ Motivation Not all linear systems have

More information

Final Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2

Final 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 information

5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.

5.) 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 information

Announcements Monday, November 26

Announcements Monday, November 26 Announcements Monday, November 26 Please fill out your CIOS survey! WeBWorK 6.6, 7.1, 7.2 are due on Wednesday. No quiz on Friday! But this is the only recitation on chapter 7. My office is Skiles 244

More information

I. Multiple Choice Questions (Answer any eight)

I. Multiple Choice Questions (Answer any eight) Name of the student : Roll No : CS65: Linear Algebra and Random Processes Exam - Course Instructor : Prashanth L.A. Date : Sep-24, 27 Duration : 5 minutes INSTRUCTIONS: The test will be evaluated ONLY

More information

Announcements Wednesday, November 01

Announcements Wednesday, November 01 Announcements Wednesday, November 01 WeBWorK 3.1, 3.2 are due today at 11:59pm. The quiz on Friday covers 3.1, 3.2. My office is Skiles 244. Rabinoffice hours are Monday, 1 3pm and Tuesday, 9 11am. Section

More information

Math 4377/6308 Advanced Linear Algebra

Math 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 information

MATH. 20F SAMPLE FINAL (WINTER 2010)

MATH. 20F SAMPLE FINAL (WINTER 2010) MATH. 20F SAMPLE FINAL (WINTER 2010) You have 3 hours for this exam. Please write legibly and show all working. No calculators are allowed. Write your name, ID number and your TA s name below. The total

More information

Solutions to Final Exam

Solutions to Final Exam Solutions to Final Exam. Let A be a 3 5 matrix. Let b be a nonzero 5-vector. Assume that the nullity of A is. (a) What is the rank of A? 3 (b) Are the rows of A linearly independent? (c) Are the columns

More information

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

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

More information

Math 2331 Linear Algebra

Math 2331 Linear Algebra 6.1 Inner Product, Length & Orthogonality Math 2331 Linear Algebra 6.1 Inner Product, Length & Orthogonality Shang-Huan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/

More information

SOLUTION KEY TO THE LINEAR ALGEBRA FINAL EXAM 1 2 ( 2) ( 1) c a = 1 0

SOLUTION 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 information

The set of all solutions to the homogeneous equation Ax = 0 is a subspace of R n if A is m n.

The set of all solutions to the homogeneous equation Ax = 0 is a subspace of R n if A is m n. 0 Subspaces (Now, we are ready to start the course....) Definitions: A linear combination of the vectors v, v,..., v m is any vector of the form c v + c v +... + c m v m, where c,..., c m R. A subset V

More information

Problem 1: Solving a linear equation

Problem 1: Solving a linear equation Math 38 Practice Final Exam ANSWERS Page Problem : Solving a linear equation Given matrix A = 2 2 3 7 4 and vector y = 5 8 9. (a) Solve Ax = y (if the equation is consistent) and write the general solution

More information

Announcements Monday, November 26

Announcements Monday, November 26 Announcements Monday, November 26 Please fill out your CIOS survey! WeBWorK 6.6, 7.1, 7.2 are due on Wednesday. No quiz on Friday! But this is the only recitation on chapter 7. My office is Skiles 244

More information

PRACTICE FINAL EXAM. why. If they are dependent, exhibit a linear dependence relation among them.

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 information

This is a closed book exam. No notes or calculators are permitted. We will drop your lowest scoring question for you.

This is a closed book exam. No notes or calculators are permitted. We will drop your lowest scoring question for you. Math 54 Fall 2017 Practice Exam 2 Exam date: 10/31/17 Time Limit: 80 Minutes Name: Student ID: GSI or Section: This exam contains 7 pages (including this cover page) and 7 problems. Problems are printed

More information

Math 1553, Introduction to Linear Algebra

Math 1553, Introduction to Linear Algebra Learning goals articulate what students are expected to be able to do in a course that can be measured. This course has course-level learning goals that pertain to the entire course, and section-level

More information

2018 Fall 2210Q Section 013 Midterm Exam II Solution

2018 Fall 2210Q Section 013 Midterm Exam II Solution 08 Fall 0Q Section 0 Midterm Exam II Solution True or False questions points 0 0 points) ) Let A be an n n matrix. If the equation Ax b has at least one solution for each b R n, then the solution is unique

More information

MATH 304 Linear Algebra Lecture 34: Review for Test 2.

MATH 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 information

Reduction to the associated homogeneous system via a particular solution

Reduction 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 information

Problem # Max points possible Actual score Total 120

Problem # 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 information

Lecture 3: Linear Algebra Review, Part II

Lecture 3: Linear Algebra Review, Part II Lecture 3: Linear Algebra Review, Part II Brian Borchers January 4, Linear Independence Definition The vectors v, v,..., v n are linearly independent if the system of equations c v + c v +...+ c n v n

More information

Linear Algebra Massoud Malek

Linear Algebra Massoud Malek CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product

More information

6. Orthogonality and Least-Squares

6. Orthogonality and Least-Squares Linear Algebra 6. Orthogonality and Least-Squares CSIE NCU 1 6. Orthogonality and Least-Squares 6.1 Inner product, length, and orthogonality. 2 6.2 Orthogonal sets... 8 6.3 Orthogonal projections... 13

More information

MTH 2032 SemesterII

MTH 2032 SemesterII MTH 202 SemesterII 2010-11 Linear Algebra Worked Examples Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education December 28, 2011 ii Contents Table of Contents

More information

Conceptual Questions for Review

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

More information

1. Determine by inspection which of the following sets of vectors is linearly independent. 3 3.

1. Determine by inspection which of the following sets of vectors is linearly independent. 3 3. 1. Determine by inspection which of the following sets of vectors is linearly independent. (a) (d) 1, 3 4, 1 { [ [,, 1 1] 3]} (b) 1, 4 5, (c) 3 6 (e) 1, 3, 4 4 3 1 4 Solution. The answer is (a): v 1 is

More information

Math 20F Final Exam(ver. c)

Math 20F Final Exam(ver. c) Name: Solutions Student ID No.: Discussion Section: Math F Final Exam(ver. c) Winter 6 Problem Score /48 /6 /7 4 /4 5 /4 6 /4 7 /7 otal / . (48 Points.) he following are rue/false questions. For this problem

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)

(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 information

YORK 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 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 information

Example Linear Algebra Competency Test

Example Linear Algebra Competency Test Example Linear Algebra Competency Test The 4 questions below are a combination of True or False, multiple choice, fill in the blank, and computations involving matrices and vectors. In the latter case,

More information

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

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

More information

Homework 5. (due Wednesday 8 th Nov midnight)

Homework 5. (due Wednesday 8 th Nov midnight) Homework (due Wednesday 8 th Nov midnight) Use this definition for Column Space of a Matrix Column Space of a matrix A is the set ColA of all linear combinations of the columns of A. In other words, if

More information

FINAL EXAM Ma (Eakin) Fall 2015 December 16, 2015

FINAL EXAM Ma (Eakin) Fall 2015 December 16, 2015 FINAL EXAM Ma-00 Eakin Fall 05 December 6, 05 Please make sure that your name and GUID are on every page. This exam is designed to be done with pencil-and-paper calculations. You may use your calculator

More information

Columbus State Community College Mathematics Department Public Syllabus

Columbus State Community College Mathematics Department Public Syllabus Columbus State Community College Mathematics Department Public Syllabus Course and Number: MATH 2568 Elementary Linear Algebra Credits: 4 Class Hours Per Week: 4 Prerequisites: MATH 2153 with a C or higher

More information

GRE 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. 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 information

Transpose & Dot Product

Transpose & Dot Product Transpose & Dot Product Def: The transpose of an m n matrix A is the n m matrix A T whose columns are the rows of A. So: The columns of A T are the rows of A. The rows of A T are the columns of A. Example:

More information

Quizzes for Math 304

Quizzes for Math 304 Quizzes for Math 304 QUIZ. A system of linear equations has augmented matrix 2 4 4 A = 2 0 2 4 3 5 2 a) Write down this system of equations; b) Find the reduced row-echelon form of A; c) What are the pivot

More information

Math 54. Selected Solutions for Week 5

Math 54. Selected Solutions for Week 5 Math 54. Selected Solutions for Week 5 Section 4. (Page 94) 8. Consider the following two systems of equations: 5x + x 3x 3 = 5x + x 3x 3 = 9x + x + 5x 3 = 4x + x 6x 3 = 9 9x + x + 5x 3 = 5 4x + x 6x 3

More information

Math 21b: Linear Algebra Spring 2018

Math 21b: Linear Algebra Spring 2018 Math b: Linear Algebra Spring 08 Homework 8: Basis This homework is due on Wednesday, February 4, respectively on Thursday, February 5, 08. Which of the following sets are linear spaces? Check in each

More information

MATH 1553, SPRING 2018 SAMPLE MIDTERM 2 (VERSION B), 1.7 THROUGH 2.9

MATH 1553, SPRING 2018 SAMPLE MIDTERM 2 (VERSION B), 1.7 THROUGH 2.9 MATH 155, SPRING 218 SAMPLE MIDTERM 2 (VERSION B), 1.7 THROUGH 2.9 Name Section 1 2 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 1 points. The maximum score

More information

Answer Keys For Math 225 Final Review Problem

Answer 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 information

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

ft-uiowa-math2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST me me ft-uiowa-math2550 Assignment NOTRequiredJustHWformatOfQuizReviewForExam3part2 due 12/31/2014 at 07:10pm CST 1. (1 pt) local/library/ui/eigentf.pg A is n n an matrices.. There are an infinite number

More information

Math 2331 Linear Algebra

Math 2331 Linear Algebra 4.5 The Dimension of a Vector Space Math 233 Linear Algebra 4.5 The Dimension of a Vector Space Shang-Huan Chiu Department of Mathematics, University of Houston schiu@math.uh.edu math.uh.edu/ schiu/ Shang-Huan

More information

Transpose & Dot Product

Transpose & Dot Product Transpose & Dot Product Def: The transpose of an m n matrix A is the n m matrix A T whose columns are the rows of A. So: The columns of A T are the rows of A. The rows of A T are the columns of A. Example:

More information

MATH 2360 REVIEW PROBLEMS

MATH 2360 REVIEW PROBLEMS MATH 2360 REVIEW PROBLEMS Problem 1: In (a) (d) below, either compute the matrix product or indicate why it does not exist: ( )( ) 1 2 2 1 (a) 0 1 1 2 ( ) 0 1 2 (b) 0 3 1 4 3 4 5 2 5 (c) 0 3 ) 1 4 ( 1

More information

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

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

More information

Math 2114 Common Final Exam May 13, 2015 Form A

Math 2114 Common Final Exam May 13, 2015 Form A Math 4 Common Final Exam May 3, 5 Form A Instructions: Using a # pencil only, write your name and your instructor s name in the blanks provided. Write your student ID number and your CRN in the blanks

More information

MODULE 8 Topics: Null space, range, column space, row space and rank of a matrix

MODULE 8 Topics: Null space, range, column space, row space and rank of a matrix MODULE 8 Topics: Null space, range, column space, row space and rank of a matrix Definition: Let L : V 1 V 2 be a linear operator. The null space N (L) of L is the subspace of V 1 defined by N (L) = {x

More information

MAT2342 : Introduction to Applied Linear Algebra Mike Newman, fall Projections. introduction

MAT2342 : Introduction to Applied Linear Algebra Mike Newman, fall Projections. introduction MAT4 : Introduction to Applied Linear Algebra Mike Newman fall 7 9. Projections introduction One reason to consider projections is to understand approximate solutions to linear systems. A common example

More information

Vector space and subspace

Vector space and subspace Vector space and subspace Math 112, week 8 Goals: Vector space, subspace. Linear combination and span. Kernel and range (null space and column space). Suggested Textbook Readings: Sections 4.1, 4.2 Week

More information

Answer Key for Exam #2

Answer Key for Exam #2 Answer Key for Exam #. Use elimination on an augmented matrix: 8 6 7 7. The corresponding system is x 7x + x, x + x + x, x x which we solve for the pivot variables x, x x : x +7x x x x x x x x x x x Therefore

More information

Review problems for MA 54, Fall 2004.

Review problems for MA 54, Fall 2004. Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on

More information

Mid-term Exam #2 MATH 205, Fall 2014

Mid-term Exam #2 MATH 205, Fall 2014 Mid-term Exam # MATH 05, Fall 04 Name: Instructions: Please answer as many of the following questions as possible Show all of your work and give complete explanations when requested Write your final answer

More information

Math 21b. Review for Final Exam

Math 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 information