MATH10212 Linear Algebra B Homework 7

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "MATH10212 Linear Algebra B Homework 7"

Transcription

1 MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments will consist of some odd numbered exercises from the Textbook The Textbook contains answers to most odd numbered exercises The Student Handbook 2 (f) says: As a rough guide you should be spending approximately twice the number of instruction hours in private study, mainly working through the examples sheets and reading your lecture notes and the recommended text books In respect of MATH22 Linear Algebra B this means that students are expected to spend 8 (eight!) hours a week in private study of Linear Algebra The homework is set as an approximately two hours task of written work, plus oral questions where workload is harder to quantify these questions serve mostly for self-control of understanding of lecture material Be prepared to answer the following oral questions if asked in the supervision class: (Mostly 3227, 28) Answer True or False The determinant of a zero matrix is zero 2 A row replacement operation does not affect the determinant of a matrix 2 3 If the columns of A are linearly dependent, then 3 det A = 4 det(a + B) = det A + det B 4 5 If two row interchanges are made in succession then the new determinant equals the old determinant 5 6 The determinant of A is the product of diagonal entries of A 6 7 If det A = then two rows or two columns are the same, or a row or a column is zero 7 8 det A T = ( ) det A 8

2 MATH22 Linear Algebra B Homework If A is an n n matrix then 9 det(a + A) = 2 n det A If A is an n n matrix then det A T = ( ) n det A Submit for marking: 2 (*) What can you say about the shape of an m n matrix A when the columns of A form a basis of R m? 3 (*) Find the determinant by row reduction to echelon form det (*) Combine the methods of row reduction and cofactor expansion to compute the determinant 2 3 det (*) Let A and P be n n matrices, with P invertible Show that det(p AP ) = det A Solve the following exercises but do not submit them for marking 6 (285) Let v = 3, v 2 = 5, and w = Determine if w is in subspace if R 3 generated by v and v 2 7 (287, 9) Let v = 8, v 2 = 8, v 3 = 6, p =, and A = [ v v 2 v 3 ] (a) How many vectors are in { v, v 2, v 3 }? (b) How many vectors are in Col A? (c) Is p in Col A? Why or why not? (d) Determine if p is in Nul A 8 (28, 3) Give integers p and q such that Nul A is a subspace of R p and Col A is a subspace of R q : A = Find a nonzero vector in Nul A and a nonzero vector in Col A 9 (285, 7, 9) Determine which sets are bases for R 2 or R 3 : {[ ] [ ]} 5, , 7, , 2 5

3 MATH22 Linear Algebra B Homework 7 3 (2823) Shown are a matrix A and an echelon form for A: A = Find a basis for Col A and a basis for Nul A (2827, 28, 29) Construct a 3 3 matrix A and a nonzero vector b such that (a) b is in Col A, but b is not the same as any one of the columns of A (b) b is not in Col A (c) b is in Nul A 2 (283 36) Respond as comprehensively as possible, and justify your answer (a) Suppose F is a 5 5 matrix whose column space is not equal to R 5 What can you say about Nul F? (b) If R is a 6 6 matrix and Nul R is not the zero subspace, what can you say about Col R? (c) If Q is a 4 4 matrix and Col Q = R 4, what can you say about solutions of equations of the form for b R 4? Qx = b (d) If P is a 5 5 matrix and Nul P is the zero subspace, what can you say about solutions of equations of the form P x = b for b R 5? (e) What can you say about Nul B when B is a 5 4 matrix with linearly independent columns? 3 (29) Find the vector x determined [ ] by the given coordinate vector x and the given basis B: B B = {[ ] [ ]} 2,, [ ] [ ] 3 x = B 2 4 (293, 5) The vector x is in subspace H with a basis B = {b, b 2 } Find the B-coordinate vector of x b = [ ], b 4 2 = [ ] 2, x = 7 [ ] b = 5, b 2 = 7, x = The following exercises use the following notation for determinants: etc a b [ ] a b c d = det c d a b c a b c d e f = det d e f, g h i g h i 5 (3, 3, 5) Compute the determinants using a cofactor expansion (a) across the first row, and (b) across the second column Solution:

4 MATH22 Linear Algebra B Homework (39,, 3) Compute the determinants by cofactor expansion At each step, choose a row or column that involves the least amount of computation (329) Compute the determinant by row reduction to echelon form: (b) (c) (d) (e) (f) (g) a c b d f e g i h a b c g h i d e f a d g b e h c f i a b c 2d + a 2e + b 2f + c g h i a b c d e f g h i 2a 2b 2c 2d 2e 2f 2g 2h 2i 8 (323) Combine the methods of row reduction and cofactor expansion to compute the determinant: (Mostly ) Given that a b c d e f g h i = 7, find the following determinants: (a) a b c d e f 5g 5h 5i 2 (3232) Find a formula for det(ra) where A is a n n matrix and r is a scalar 2 (3239) Let A and B be 3 3 matrices, with det A = 4 and det B = 3 Use properties of determinants to compute: (a) det AB (b) det 5A (c) det B T (d) det A (e) det A 3 22 (333, 5) Use Cramer s rule to compute the solutions of the following systems:

5 MATH22 Linear Algebra B Homework 7 5 (a) (b) 3x 2x 2 = 7 5x + 6x 2 = 5 which the system has a unique solution, and describe the solution: 6sx + 4x 2 = 5 9x + 2sx 2 = 2 2x + x 2 = 7 3x + x 3 = 8 x 2 + 2x 3 = 3 23 (337) Use Cramer s rule to determine the values of the parameter s for 24 (33) Compute the adjugate of the given matrix, and then use it to give the inverse of the matrix: 2 3 Answers to True/False questions True 2 True 3 True 4 False 5 True 6 False 7 False 8 False 9 True False Solutions to non-starred exercises 6 Solution: No The system is inconsistent x v + x 2 v 2 = w 7 Solution: (a) Three vectors: v, v 2, and v 3 (b) Infinitely many vectors (c) Yes, because Ax = p has a solution (d) No, because Ap 8 Solution: p = 4 and q = 3 For a nonzero vector in Nul A chose, for example, [ 2 ] T or any other nontrivial solution x of the homogeneous system Ax = For a nonzero vector in Col A select, for example, any column of A 9 Solution: The first two sets The third set cannot be a basis of R 3 because it contains only two vectors, while each basis of R 3 contains three linearly independent vectors Solution: For a basis of Col A, you can take the two pivoted columns of A: 4 5 6, Indeed, non-pivoted columns are linear combinations of pivoted columns (see the solution to Problem 26(b) in Homework 4) On the other hand, it is obvious that pivoted columns are linearly independent Therefore pivoted columns form a basis of Col A (it was discussed in the lectures) One of possible bases for Nul A is 4 7 5, 6

6 MATH22 Linear Algebra B Homework 7 6 Indeed, to find a basis for Nul A is the same as to solve the homogeneous system of equations AX = Let us complete reduction of A to its reduced echelon form: Variables x 3 and x 4 are free, denote them x 3 = s and x 4 = t while x = 4x 3 5x 4 = 4s 5t x 2 = 5x 3 + 6x 4 = 5s + 6t We see now that an arbitrary solution is uniquely written as a linear combination x 4s 5t 4 7 x 2 x 3 = 5s + 6t s = s 5 +t 6 x 5 t which means that 4 5, 7 6 is a basis of the solution space of the solution space of Ax =, that is, of Nul A Solution: The problem has infinitely many possible answers One of them is A =, then the vector belongs to Col A but is not equal to any of the columns of A, while the vector b = simultaneously does not belong to Col A (because the system Ax = b is inconsistent) and belongs to Nul A (because Ab = 2 Solution: (a) Nul F is not the zero subspace (b) Col R R 6 (c) Solutions exist and are unique for each b R 4 (d) Solutions exist and are unique for each b R 5 (e) Nul B is the zero subspace of R 5 3 Solution: 4 Solution: [ ] 7, 5 x = 3v + 2v 2 = [ ] /4 5/4 5 Solution: Solution: 3 8 Solution: 6 [ ] 7 9 Solution: (a) 35 (b) 7 (c) 7 (d) 7 (e) 4 (f) 7 (g) 56 2 Solution: det(ra) = r n det A 2 Solution: (a) 2 (b) 5 (c) 3 (d) /4 (e) Solution: (a) (b) [ ] 4 5/2 3/2 4 7/2

7 MATH22 Linear Algebra B Homework Solution: s ± 3 x = 5s + 4 6(s 2 3) x 2 = 4s 5 4(s 2 3) 24 Solution: adj A = 3 3, A =

MATH10212 Linear Algebra B Homework 6. Be prepared to answer the following oral questions if asked in the supervision class:

MATH10212 Linear Algebra B Homework 6. Be prepared to answer the following oral questions if asked in the supervision class: MATH0 Linear Algebra B Homework 6 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra its Applications Pearson, 006 (or other editions) Normally, homework assignments

More information

MATH10212 Linear Algebra B Homework Week 5

MATH10212 Linear Algebra B Homework Week 5 MATH Linear Algebra B Homework Week 5 Students are strongly advised to acquire a copy of the Textbook: D C Lay Linear Algebra its Applications Pearson 6 (or other editions) Normally homework assignments

More information

MATH10212 Linear Algebra B Homework Week 3. Be prepared to answer the following oral questions if asked in the supervision class

MATH10212 Linear Algebra B Homework Week 3. Be prepared to answer the following oral questions if asked in the supervision class MATH10212 Linear Algebra B Homework Week Students are strongly advised to acquire a copy of the Textbook: D. C. Lay Linear Algebra its Applications. Pearson, 2006. ISBN 0-521-2871-4. Normally, homework

More information

MATH10212 Linear Algebra B Homework Week 4

MATH10212 Linear Algebra B Homework Week 4 MATH22 Linear Algebra B Homework Week 4 Students are strongly advised to acquire a copy of the Textbook: D. C. Lay Linear Algebra and its Applications. Pearson, 26. ISBN -52-2873-4. Normally, homework

More information

Homework Set #8 Solutions

Homework Set #8 Solutions Exercises.2 (p. 19) Homework Set #8 Solutions Assignment: Do #6, 8, 12, 14, 2, 24, 26, 29, 0, 2, 4, 5, 6, 9, 40, 42 6. Reducing the matrix to echelon form: 1 5 2 1 R2 R2 R1 1 5 0 18 12 2 1 R R 2R1 1 5

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 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 313 (Linear Algebra) Exam 2 - Practice Exam

Math 313 (Linear Algebra) Exam 2 - Practice Exam Name: Student ID: Section: Instructor: Math 313 (Linear Algebra) Exam 2 - Practice Exam Instructions: For questions which require a written answer, show all your work. Full credit will be given only if

More information

Math 314H EXAM I. 1. (28 points) The row reduced echelon form of the augmented matrix for the system. is the matrix

Math 314H EXAM I. 1. (28 points) The row reduced echelon form of the augmented matrix for the system. is the matrix Math 34H EXAM I Do all of the problems below. Point values for each of the problems are adjacent to the problem number. Calculators may be used to check your answer but not to arrive at your answer. That

More information

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

MATH10212 Linear Algebra Lecture Notes

MATH10212 Linear Algebra Lecture Notes MATH1212 Linear Algebra Lecture Notes Textbook Students are strongly advised to acquire a copy of the Textbook: D. C. Lay. Linear Algebra and its Applications. Pearson, 26. ISBN -521-28713-4. Other editions

More information

MATH10212 Linear Algebra Lecture Notes

MATH10212 Linear Algebra Lecture Notes MATH10212 Linear Algebra Lecture Notes Last change: 23 Apr 2018 Textbook Students are strongly advised to acquire a copy of the Textbook: D. C. Lay. Linear Algebra and its Applications. Pearson, 2006.

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

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 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

MAT 1341A Final Exam, 2011

MAT 1341A Final Exam, 2011 MAT 1341A Final Exam, 2011 16-December, 2011. Instructor - Barry Jessup 1 Family Name: First Name: Student number: Some Advice Take 5 minutes to read the entire paper before you begin to write, and read

More information

(i) [7 points] Compute the determinant of the following matrix using cofactor expansion.

(i) [7 points] Compute the determinant of the following matrix using cofactor expansion. Question (i) 7 points] Compute the determinant of the following matrix using cofactor expansion 2 4 2 4 2 Solution: Expand down the second column, since it has the most zeros We get 2 4 determinant = +det

More information

MTH501- Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~

MTH501- Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~ MTH501- Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~ Question No: 1 (Marks: 1) If for a linear transformation the equation T(x) =0 has only the trivial solution then T is One-to-one Onto Question

More information

Homework 11/Solutions. (Section 6.8 Exercise 3). Which pairs of the following vector spaces are isomorphic?

Homework 11/Solutions. (Section 6.8 Exercise 3). Which pairs of the following vector spaces are isomorphic? MTH 9-4 Linear Algebra I F Section Exercises 6.8,4,5 7.,b 7.,, Homework /Solutions (Section 6.8 Exercise ). Which pairs of the following vector spaces are isomorphic? R 7, R, M(, ), M(, 4), M(4, ), P 6,

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

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

1 Last time: inverses

1 Last time: inverses MATH Linear algebra (Fall 8) Lecture 8 Last time: inverses The following all mean the same thing for a function f : X Y : f is invertible f is one-to-one and onto 3 For each b Y there is exactly one a

More information

No books, notes, any calculator, or electronic devices are allowed on this exam. Show all of your steps in each answer to receive a full credit.

No books, notes, any calculator, or electronic devices are allowed on this exam. Show all of your steps in each answer to receive a full credit. MTH 309-001 Fall 2016 Exam 1 10/05/16 Name (Print): PID: READ CAREFULLY THE FOLLOWING INSTRUCTION Do not open your exam until told to do so. This exam contains 7 pages (including this cover page) and 7

More information

MTH 464: Computational Linear Algebra

MTH 464: Computational Linear Algebra MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)

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

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2 MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

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

Evaluating Determinants by Row Reduction

Evaluating Determinants by Row Reduction Evaluating Determinants by Row Reduction MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Objectives Reduce a matrix to row echelon form and evaluate its determinant.

More information

Choose three of: Choose three of: Choose three of:

Choose three of: Choose three of: Choose three of: MATH Final Exam (Version ) Solutions July 8, 8 S. F. Ellermeyer Name Instructions. Remember to include all important details of your work. You will not get full credit (or perhaps even any partial credit)

More information

Section 1.1 System of Linear Equations. Dr. Abdulla Eid. College of Science. MATHS 211: Linear Algebra

Section 1.1 System of Linear Equations. Dr. Abdulla Eid. College of Science. MATHS 211: Linear Algebra Section 1.1 System of Linear Equations College of Science MATHS 211: Linear Algebra (University of Bahrain) Linear System 1 / 33 Goals:. 1 Define system of linear equations and their solutions. 2 To represent

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

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

(b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).

(b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a). .(5pts) Let B = 5 5. Compute det(b). (a) (b) (c) 6 (d) (e) 6.(5pts) Determine which statement is not always true for n n matrices A and B. (a) If two rows of A are interchanged to produce B, then det(b)

More information

MATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian.

MATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian. MATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian. Spanning set Let S be a subset of a vector space V. Definition. The span of the set S is the smallest subspace W V that contains S. If

More information

Linear Algebra: Sample Questions for Exam 2

Linear Algebra: Sample Questions for Exam 2 Linear Algebra: Sample Questions for Exam 2 Instructions: This is not a comprehensive review: there are concepts you need to know that are not included. Be sure you study all the sections of the book and

More information

Find the solution set of 2x 3y = 5. Answer: We solve for x = (5 + 3y)/2. Hence the solution space consists of all vectors of the form

Find the solution set of 2x 3y = 5. Answer: We solve for x = (5 + 3y)/2. Hence the solution space consists of all vectors of the form Math 2 Homework #7 March 4, 2 7.3.3. Find the solution set of 2x 3y = 5. Answer: We solve for x = (5 + 3y/2. Hence the solution space consists of all vectors of the form ( ( ( ( x (5 + 3y/2 5/2 3/2 x =

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

MTH Linear Algebra. Study Guide. Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education

MTH Linear Algebra. Study Guide. Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education MTH 3 Linear Algebra Study Guide Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education June 3, ii Contents Table of Contents iii Matrix Algebra. Real Life

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

ECON 186 Class Notes: Linear Algebra

ECON 186 Class Notes: Linear Algebra ECON 86 Class Notes: Linear Algebra Jijian Fan Jijian Fan ECON 86 / 27 Singularity and Rank As discussed previously, squareness is a necessary condition for a matrix to be nonsingular (have an inverse).

More information

Midterm 1 Review. Written by Victoria Kala SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015

Midterm 1 Review. Written by Victoria Kala SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015 Midterm 1 Review Written by Victoria Kala vtkala@math.ucsb.edu SH 6432u Office Hours: R 12:30 1:30 pm Last updated 10/10/2015 Summary This Midterm Review contains notes on sections 1.1 1.5 and 1.7 in your

More information

Determinants Chapter 3 of Lay

Determinants Chapter 3 of Lay Determinants Chapter of Lay Dr. Doreen De Leon Math 152, Fall 201 1 Introduction to Determinants Section.1 of Lay Given a square matrix A = [a ij, the determinant of A is denoted by det A or a 11 a 1j

More information

Methods for Solving Linear Systems Part 2

Methods for Solving Linear Systems Part 2 Methods for Solving Linear Systems Part 2 We have studied the properties of matrices and found out that there are more ways that we can solve Linear Systems. In Section 7.3, we learned that we can use

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

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

MA 242 LINEAR ALGEBRA C1, Solutions to First Midterm Exam

MA 242 LINEAR ALGEBRA C1, Solutions to First Midterm Exam MA 242 LINEAR ALGEBRA C Solutions to First Midterm Exam Prof Nikola Popovic October 2 9:am - :am Problem ( points) Determine h and k such that the solution set of x + = k 4x + h = 8 (a) is empty (b) contains

More information

Row Space, Column Space, and Nullspace

Row Space, Column Space, and Nullspace Row Space, Column Space, and Nullspace MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Every matrix has associated with it three vector spaces: row space

More information

ANSWERS. E k E 2 E 1 A = B

ANSWERS. 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 information

Linear Independence x

Linear Independence x Linear Independence A consistent system of linear equations with matrix equation Ax = b, where A is an m n matrix, has a solution set whose graph in R n is a linear object, that is, has one of only n +

More information

MATH 2210Q MIDTERM EXAM I PRACTICE PROBLEMS

MATH 2210Q MIDTERM EXAM I PRACTICE PROBLEMS MATH Q MIDTERM EXAM I PRACTICE PROBLEMS Date and place: Thursday, November, 8, in-class exam Section : : :5pm at MONT Section : 9: :5pm at MONT 5 Material: Sections,, 7 Lecture 9 8, Quiz, Worksheet 9 8,

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

Summer Session Practice Final Exam

Summer Session Practice Final Exam Math 2F Summer Session 25 Practice Final Exam Time Limit: Hours Name (Print): Teaching Assistant This exam contains pages (including this cover page) and 9 problems. Check to see if any pages are missing.

More information

Linear Algebra (MATH ) Spring 2011 Final Exam Practice Problem Solutions

Linear Algebra (MATH ) Spring 2011 Final Exam Practice Problem Solutions Linear Algebra (MATH 4) Spring 2 Final Exam Practice Problem Solutions Instructions: Try the following on your own, then use the book and notes where you need help. Afterwards, check your solutions with

More information

1 Determinants. 1.1 Determinant

1 Determinants. 1.1 Determinant 1 Determinants [SB], Chapter 9, p.188-196. [SB], Chapter 26, p.719-739. Bellow w ll study the central question: which additional conditions must satisfy a quadratic matrix A to be invertible, that is to

More information

Linear Algebra 1 Exam 1 Solutions 6/12/3

Linear Algebra 1 Exam 1 Solutions 6/12/3 Linear Algebra 1 Exam 1 Solutions 6/12/3 Question 1 Consider the linear system in the variables (x, y, z, t, u), given by the following matrix, in echelon form: 1 2 1 3 1 2 0 1 1 3 1 4 0 0 0 1 2 3 Reduce

More information

Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Remark By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero. Sec 6 Eigenvalues and Eigenvectors Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called an eigenvalue of A if there is a nontrivial

More information

Math 3C Lecture 20. John Douglas Moore

Math 3C Lecture 20. John Douglas Moore Math 3C Lecture 20 John Douglas Moore May 18, 2009 TENTATIVE FORMULA I Midterm I: 20% Midterm II: 20% Homework: 10% Quizzes: 10% Final: 40% TENTATIVE FORMULA II Higher of two midterms: 30% Homework: 10%

More information

Review for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions.

Review for Exam Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. Review for Exam. Find all a for which the following linear system has no solutions, one solution, and infinitely many solutions. x + y z = 2 x + 2y + z = 3 x + y + (a 2 5)z = a 2 The augmented matrix for

More information

Linear Algebra: Lecture notes from Kolman and Hill 9th edition.

Linear Algebra: Lecture notes from Kolman and Hill 9th edition. Linear Algebra: Lecture notes from Kolman and Hill 9th edition Taylan Şengül March 20, 2019 Please let me know of any mistakes in these notes Contents Week 1 1 11 Systems of Linear Equations 1 12 Matrices

More information

MATH 2331 Linear Algebra. Section 1.1 Systems of Linear Equations. Finding the solution to a set of two equations in two variables: Example 1: Solve:

MATH 2331 Linear Algebra. Section 1.1 Systems of Linear Equations. Finding the solution to a set of two equations in two variables: Example 1: Solve: MATH 2331 Linear Algebra Section 1.1 Systems of Linear Equations Finding the solution to a set of two equations in two variables: Example 1: Solve: x x = 3 1 2 2x + 4x = 12 1 2 Geometric meaning: Do these

More information

Online Exercises for Linear Algebra XM511

Online 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 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

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

A 2. =... = c c N. 's arise from the three types of elementary row operations. If rref A = I its determinant is 1, and A = c 1

A 2. =... = c c N. 's arise from the three types of elementary row operations. If rref A = I its determinant is 1, and A = c 1 Theorem: Let A n n Then A 1 exists if and only if det A 0 proof: We already know that A 1 exists if and only if the reduced row echelon form of A is the identity matrix Now, consider reducing A to its

More information

MATH 310, REVIEW SHEET 2

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

More information

Math 320, spring 2011 before the first midterm

Math 320, spring 2011 before the first midterm Math 320, spring 2011 before the first midterm Typical Exam Problems 1 Consider the linear system of equations 2x 1 + 3x 2 2x 3 + x 4 = y 1 x 1 + 3x 2 2x 3 + 2x 4 = y 2 x 1 + 2x 3 x 4 = y 3 where x 1,,

More information

1111: Linear Algebra I

1111: Linear Algebra I 1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Michaelmas Term 2015 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Michaelmas Term 2015 1 / 10 Row expansion of the determinant Our next goal is

More information

MAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to:

MAC Module 3 Determinants. Learning Objectives. Upon completing this module, you should be able to: MAC 2 Module Determinants Learning Objectives Upon completing this module, you should be able to:. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor

More information

Fall 2016 MATH*1160 Final Exam

Fall 2016 MATH*1160 Final Exam Fall 2016 MATH*1160 Final Exam Last name: (PRINT) First name: Student #: Instructor: M. R. Garvie Dec 16, 2016 INSTRUCTIONS: 1. The exam is 2 hours long. Do NOT start until instructed. You may use blank

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

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

Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in

Chapter 4 - MATRIX ALGEBRA. ... a 2j... a 2n. a i1 a i2... a ij... a in Chapter 4 - MATRIX ALGEBRA 4.1. Matrix Operations A a 11 a 12... a 1j... a 1n a 21. a 22.... a 2j... a 2n. a i1 a i2... a ij... a in... a m1 a m2... a mj... a mn The entry in the ith row and the jth column

More information

Properties of Linear Transformations from R n to R m

Properties of Linear Transformations from R n to R m Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation

More information

Math 54 First Midterm Exam, Prof. Srivastava September 23, 2016, 4:10pm 5:00pm, 155 Dwinelle Hall.

Math 54 First Midterm Exam, Prof. Srivastava September 23, 2016, 4:10pm 5:00pm, 155 Dwinelle Hall. Math 54 First Midterm Exam, Prof Srivastava September 23, 26, 4:pm 5:pm, 55 Dwinelle Hall Name: SID: Instructions: Write all answers in the provided space This exam includes two pages of scratch paper,

More information

MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics

MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam This study sheet will not be allowed during the test Books and notes will not be allowed during the test Calculators and cell phones

More information

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

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

More information

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

Algebra & Trig. I. For example, the system. x y 2 z. may be represented by the augmented matrix

Algebra & Trig. I. For example, the system. x y 2 z. may be represented by the augmented matrix Algebra & Trig. I 8.1 Matrix Solutions to Linear Systems A matrix is a rectangular array of elements. o An array is a systematic arrangement of numbers or symbols in rows and columns. Matrices (the plural

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

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

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

Equality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.

Equality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same. Introduction Matrix Operations Matrix: An m n matrix A is an m-by-n array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors 5 Eigenvalues and Eigenvectors 5.2 THE CHARACTERISTIC EQUATION DETERMINANATS n n Let A be an matrix, let U be any echelon form obtained from A by row replacements and row interchanges (without scaling),

More information

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

Calculating determinants for larger matrices

Calculating determinants for larger matrices Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det

More information

MH1200 Final 2014/2015

MH1200 Final 2014/2015 MH200 Final 204/205 November 22, 204 QUESTION. (20 marks) Let where a R. A = 2 3 4, B = 2 3 4, 3 6 a 3 6 0. For what values of a is A singular? 2. What is the minimum value of the rank of A over all a

More information

Warm-up. True or false? Baby proof. 2. The system of normal equations for A x = y has solutions iff A x = y has solutions

Warm-up. True or false? Baby proof. 2. The system of normal equations for A x = y has solutions iff A x = y has solutions Warm-up True or false? 1. proj u proj v u = u 2. The system of normal equations for A x = y has solutions iff A x = y has solutions 3. The normal equations are always consistent Baby proof 1. Let A be

More information

MAT Linear Algebra Collection of sample exams

MAT 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 information

Row Reduction and Echelon Forms

Row Reduction and Echelon Forms Row Reduction and Echelon Forms 1 / 29 Key Concepts row echelon form, reduced row echelon form pivot position, pivot, pivot column basic variable, free variable general solution, parametric solution existence

More information

University of Ottawa

University of Ottawa University of Ottawa Department of Mathematics and Statistics MAT 1302A: Mathematical Methods II Instructor: Hadi Salmasian Final Exam April 2016 Surname First Name Seat # Instructions: (a) You have 3

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 1553, C. JANKOWSKI MIDTERM 3

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

More information

MATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns.

MATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. MATRICES After studying this chapter you will acquire the skills in knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. List of

More information

3 Matrix Algebra. 3.1 Operations on matrices

3 Matrix Algebra. 3.1 Operations on matrices 3 Matrix Algebra A matrix is a rectangular array of numbers; it is of size m n if it has m rows and n columns. A 1 n matrix is a row vector; an m 1 matrix is a column vector. For example: 1 5 3 5 3 5 8

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

MATH 1553 SAMPLE FINAL EXAM, SPRING 2018

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

More information

Remark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero.

Remark 1 By definition, an eigenvector must be a nonzero vector, but eigenvalue could be zero. Sec 5 Eigenvectors and Eigenvalues In this chapter, vector means column vector Definition An eigenvector of an n n matrix A is a nonzero vector x such that A x λ x for some scalar λ A scalar λ is called

More information

Lecture 22: Section 4.7

Lecture 22: Section 4.7 Lecture 22: Section 47 Shuanglin Shao December 2, 213 Row Space, Column Space, and Null Space Definition For an m n, a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn, the vectors r 1 = [ a 11 a 12 a 1n

More information

EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016

EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016 EK102 Linear Algebra PRACTICE PROBLEMS for Final Exam Spring 2016 Answer the questions in the spaces provided on the question sheets. You must show your work to get credit for your answers. There will

More information

Math 102, Winter 2009, Homework 7

Math 102, Winter 2009, Homework 7 Math 2, Winter 29, Homework 7 () Find the standard matrix of the linear transformation T : R 3 R 3 obtained by reflection through the plane x + z = followed by a rotation about the positive x-axes by 6

More information