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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 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 : Ax = b has a solution} We refer to Nul (T ) as the null space of T and Ran (T ) and the range of T We further say; (1) T is one to one if T (x) = T (y) implies x = y or equivalently, for all b R m there is at most one solution to T (x) = b (2) T is onto if Ran (T ) = R m or equivalently put for every b R m there is at least one solution to T (x) = b Things you should know: (1) Linear systems like 1 TEST #1 REVIEW MATERIAL A = [a 1 a n ] = a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 a m1 x 1 + a m2 x a mn x n = b m are equivalent to the matrix equation Ax = b where a 11 a 12 a 1n a 21 a 22 a 2n x = x 1 x n x n and b = a m1 a m2 a mn b 1 b ṇ b m (2) Ax is a linear combination of the columns of A Ax = x i a i = m n - coefficient matrix, and T (x) := Ax defines a linear transformation from R n R m, ie T preserves vector addition and scalar multiplication 1

2 2 MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) (3) (4) span {a 1,, a n } = {b R m : Ax = b has a solution} = {b R m : Ax = b is consistent} = Ran (A) = {b = Ax: x R n } Nul (A) := {x R n : Ax = 0} { all solutions to the homogeneous equation: = Ax = 0 } (5) Theorem: If Ap = b then the general solution to Ax = b is of the form x = p + v h where v h is a generic solution to Av h = 0, ie x = p + v h with v h Nul (A) (6) You should know the definition of linear independence (dependence), ie Γ := {a 1,, a n } R m are linearly independent iff the only solution to n x ia i = 0 is x = 0 Equivalently put, Γ := {a 1,, a n } R m are LI Nul (A) = {0} Ax = 0 has only the trivial solution (7) You should know to perform row reduction in order to put a matrix into it reduced row echelon form (8) You should be able to write down the general solution to the equation Ax = b and find equation that b must satisfy so that Ax = b is consistent (9) Theorem: Let A be a m n matrix and U :=rref(a) Then; (a) Ax = b has a solution iff rref([a b]) does not have a pivot in the last column (b) Ax = b has a solution for every b R m iff span {a 1,, a n } = Ran (A) = R m iff U :=rref(a) has a pivot in every row, ie U does not contain a row of zeros (c) If m > n (ie there are more equations than unknowns), then Ax = b will be inconsistent for some b R m This is because there can be at most n -pivots and since m > n there must be a row of zeros in rref(a) (d) Ax = b has at most one solution iff Nul (A) = 0 iff Ax = 0 has only the trivial solution iff {a 1,, a n } are linearly independent, iff rref(a) has no free variables ie there is a pivot in every column (e) If m < n (ie there are fewer equation than unknowns) then Ax = 0 will always have a non-trivial solution or equivalently put the columns of A are necessarily linearly dependent This is because there can be at most m pivots and so at least one column does not have pivot and there is at least one free variable

3 MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) 3 2 TEST #2 REVIEW MATERIAL Things to know: (1) The notion of a vector space and the fact that R m and the fact that, for any set D, V (D) = functions from D to R forms a vector space (2) The notion of a subspace, and important subspaces the function spaces, eg P n = polynomials of degree at most n (3) How to determine if H V is a subspace or not (4) The notions of linear independence and span of vectors in a vector space (5) The notions of a basis, dimension, and coordinates relative to a basis (6) Be able to check if vectors in R m are a basis or not by putting the vectors in the columns of a matrix and row reducing In particular you should know that n vectors in R m are always linearly dependent if n > m and that they do not span R m is n < m (7) Matrix operations How to compute AB, A + B, ABC = (AB) C or A (BC) You should understand when these operations make sense (8) The notion of a linear transformation, T : V W (a) If V = R n and W = R m then T (x) = Ax where A = [T (e 1 ) T (e n )] (b) The derivative and integration operations are linear (c) Other examples of linear transformations from the homework, eg T : P 2 R 3 with T (p) := p (1) p ( 1) p (2) is linear (9) Nul (T ) = {v V : T (v) = 0} all vectors in the domain which are sent to zero (a) T is one to one iff Nul (T ) = {0} (b) be able to find a basis for Nul (A) R n when A is a m n matrix (10) Ran (T ) = {T (v) W : v V } the range of T Equivalently, w Ran (T ) iff there exists a solution, v V, to T (v) = w (a) Know how to find a basis for col (A) = Ran (A) (b) Know how to find a basis for row (A) (c) Know that dim row (A) = dim col (A) = dim Ran (A) = rank (A) (11) You should understand the rank-nullity theorem see Theorem 14 on p 233 (12) Theorem If A is not a square matrix then A is not invertible! (13) Theorem If A is a n n matrix then the following are equivalent (a) A is invertible (b) col (A) = Ran (A) = R n dim Ran (A) = n = dim row (A) (c) row (A) = R n (d) Nul (A) = {0} dim Nul (A) = 0 (e) det (A) 0 (f) rref(a) = I (14) Be able to find A 1 using [A I] [A 1 I] when A is invertible (15) Now that Ax = b has solution given by x = A 1 b when A is invertible (16) Understand how to compute determinants of matrices You should also know the determinants behavior under row and column operations (17) For an n n matrix, A, the following are equivalent; (a) A 1 does not exist,

4 4 MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) (b) Nul (A) {0}, (c) Ran (A) R n, (d) det A = 0

5 Things you should know: MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) 5 3 AFTER TEST #2 REVIEW MATERIAL (1) You should understand the definition of Eigenvalues and Eigenvectors for linear transformations and especially for matrices This includes being able to test if a vector is an eigenvector or not for a given linear transformation (2) If T : V V is a linear transformation and {v 1,, v n } are eigenvectors of T such that T (v i ) = λ i v i with all the eigenvalues {λ 1,, λ n } being distinct, then {v 1,, v n } is a linearly independent set (3) The characteristic polynomial of a n n matrix, A, is p A (λ) := det (A λi) You should know; (a) how to compute p A (λ) for a given matrix A using the standard methods of evaluating determinants (b) The roots {λ 1,, λ n } (with possible repetitions) are all of the eigenvalues of A (c) The matrix A is diagonalizable if all of the roots of p A (λ) is distinct, ie there will be n -distinct eigenvectors in this case If there are repeated roots A may or may not be diagonalizable (4) After finding an eigenvalue, λ i, of A you should be able to find a basis of the corresponding eigenvectors to this eigenvalue, ie find a basis for Nul (A λ i I) (5) An n n matrix A is diagonalizable (ie may be written as A = P DP 1 where D is a diagonal matrix and P is an invertible matrix) An n n matrix A is diagonalizable iff A has a basis (B) of eigenvectors Moreover if B = {v 1,, v n }, with Av i = λ i v i, and λ λ P = [v 1 v n ] and D = 2 0, 0 0 λ n then A = P DP 1 (6) If A = P DP 1 as above then A k = P D k P 1 where λ k D k 0 λ = k λ k n (7) The basic definition of the dot product on R n ; u v = u i v i = u T v = v T u

6 6 MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) and its associated norm or length function, u := u u and know that the distance between u and v is dist (u, v) := u v = n u i v i 2 (8) The meaning of orthogonal sets, orthonormal sets, orthogonal bases, and orthonormal bases Moreover you should know; (a) If B = {u 1,, u n } is an orthogonal basis for a subspace, W R m, then proj W v = v u i u i 2 u i is the orthogonal projection of v onto W The vector proj span(b) v is the unique vector in W closest to v and also is the unique vector w W such that v w is perpendicular to W, ie (v w) w = 0 for all w W (b) If B = {u 1,, u n } is an orthogonal basis for R n then ie v = v u i u i 2 u i for all v R n, [v] B = v u 1 u 1 2 v u 2 u 2 2 v u n u n 2 (c) If U = [u 1 u n ] is a n n matrix, then the following are equivalent; (i) U is an orthogonal matrix, ie U T U = I = UU T or equivalently put U 1 = U T (ii) The columns {u 1,, u n } of U form an orthonormal basis for R n (d) You should be able to carry out the Gram Schmidt process in order to make an orthonormal basis for a subspace W out of an arbitrary basis for W (9) If A is a m n matrix you should know that A T is the unique n m matrix such that Ax y = x A T y for all x R n and y R m (10) You should be able to find all least squares solutions to an inconsistent matrix equation, Ax = b (ie b / Ran (A) by solving the normal equations, A T Ax = A T b Recall the Least squares theorem states that x R n is a minimizer of Ax b iff x satisfies A T Ax = A T b (11) You should know and be able to show that if A is a symmetric matrix and u and v are eigenvectors of A with distinct eigenvalues, then u v = 0 (12) The spectral theorem; every symmetric n n matrix A has an orthonormal basis of eigenvectors Equivalently put A may be written as A = UDU T where U is a orthogonal matrix and D is a diagonal matrix In particular every symmetric matrix may be diagonalized

7 MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) 7 (13) Given a symmetric matrix A you should be able to find U = [u 1 u n ] as described in the spectral theorem The method is to find (using the Gram Schmidt process if necessary) an orthonormal basis of eigenvectors {u 1,, u n } of A and then we take U = [u 1 u n ]

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

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

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

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

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

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

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

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

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

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

Assignment 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. 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 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

DIAGONALIZATION. In order to see the implications of this definition, let us consider the following example Example 1. Consider the matrix

DIAGONALIZATION. 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 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

Chapter 6: Orthogonality

Chapter 6: Orthogonality Chapter 6: Orthogonality (Last Updated: November 7, 7) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). A few theorems have been moved around.. Inner products

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

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

Solutions to Review Problems for Chapter 6 ( ), 7.1

Solutions to Review Problems for Chapter 6 ( ), 7.1 Solutions to Review Problems for Chapter (-, 7 The Final Exam is on Thursday, June,, : AM : AM at NESBITT Final Exam Breakdown Sections % -,7-9,- - % -9,-,7,-,-7 - % -, 7 - % Let u u and v Let x x x x,

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

Elementary Linear Algebra Review for Exam 2 Exam is Monday, November 16th.

Elementary Linear Algebra Review for Exam 2 Exam is Monday, November 16th. Elementary Linear Algebra Review for Exam Exam is Monday, November 6th. The exam will cover sections:.4,..4, 5. 5., 7., the class notes on Markov Models. You must be able to do each of the following. Section.4

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

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

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

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

22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices

22m:033 Notes: 7.1 Diagonalization of Symmetric Matrices m:33 Notes: 7. Diagonalization of Symmetric Matrices Dennis Roseman University of Iowa Iowa City, IA http://www.math.uiowa.edu/ roseman May 3, Symmetric matrices Definition. A symmetric matrix is a matrix

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

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

1. Select the unique answer (choice) for each problem. Write only the answer.

1. Select the unique answer (choice) for each problem. Write only the answer. MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +

More 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

Solutions to practice questions for the final

Solutions to practice questions for the final Math A UC Davis, Winter Prof. Dan Romik Solutions to practice questions for the final. You are given the linear system of equations x + 4x + x 3 + x 4 = 8 x + x + x 3 = 5 x x + x 3 x 4 = x + x + x 4 =

More information

235 Final exam review questions

235 Final exam review questions 5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an

More 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 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

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

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

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

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

1 Last time: least-squares problems

1 Last time: least-squares problems MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that

More information

Math 323 Exam 2 Sample Problems Solution Guide October 31, 2013

Math 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 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

SUMMARY OF MATH 1600

SUMMARY OF MATH 1600 SUMMARY OF MATH 1600 Note: The following list is intended as a study guide for the final exam. It is a continuation of the study guide for the midterm. It does not claim to be a comprehensive list. You

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

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

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

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

Definitions for Quizzes

Definitions 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 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

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

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

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

Sample Final Exam: Solutions

Sample Final Exam: Solutions Sample Final Exam: Solutions Problem. A linear transformation T : R R 4 is given by () x x T = x 4. x + (a) Find the standard matrix A of this transformation; (b) Find a basis and the dimension for Range(T

More information

Solutions to Final Practice Problems Written by Victoria Kala Last updated 12/5/2015

Solutions to Final Practice Problems Written by Victoria Kala Last updated 12/5/2015 Solutions to Final Practice Problems Written by Victoria Kala vtkala@math.ucsb.edu Last updated /5/05 Answers This page contains answers only. See the following pages for detailed solutions. (. (a x. See

More information

MATH 221, Spring Homework 10 Solutions

MATH 221, Spring Homework 10 Solutions MATH 22, Spring 28 - Homework Solutions Due Tuesday, May Section 52 Page 279, Problem 2: 4 λ A λi = and the characteristic polynomial is det(a λi) = ( 4 λ)( λ) ( )(6) = λ 6 λ 2 +λ+2 The solutions to the

More information

Linear Algebra Highlights

Linear 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 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 3191 Applied Linear Algebra

Math 3191 Applied Linear Algebra Math 9 Applied Linear Algebra Lecture 9: Diagonalization Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./9 Section. Diagonalization The goal here is to develop a useful

More information

Math Linear Algebra II. 1. Inner Products and Norms

Math Linear Algebra II. 1. Inner Products and Norms Math 342 - Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,

More information

LINEAR ALGEBRA REVIEW

LINEAR 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 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

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

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 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2.

MATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2. MATH 304 Linear Algebra Lecture 23: Diagonalization. Review for Test 2. Diagonalization Let L be a linear operator on a finite-dimensional vector space V. Then the following conditions are equivalent:

More 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

Solutions to Math 51 Midterm 1 July 6, 2016

Solutions to Math 51 Midterm 1 July 6, 2016 Solutions to Math 5 Midterm July 6, 26. (a) (6 points) Find an equation (of the form ax + by + cz = d) for the plane P in R 3 passing through the points (, 2, ), (2,, ), and (,, ). We first compute two

More information

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v

More information

Check that your exam contains 30 multiple-choice questions, numbered sequentially.

Check that your exam contains 30 multiple-choice questions, numbered sequentially. MATH EXAM SPRING VERSION A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER On your scantron, write and bubble your PSU ID, Section Number, and Test Version. Failure to correctly code these items may result

More information

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination

Math 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column

More information

MATH 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. 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 information

BASIC ALGORITHMS IN LINEAR ALGEBRA. Matrices and Applications of Gaussian Elimination. A 2 x. A T m x. A 1 x A T 1. A m x

BASIC ALGORITHMS IN LINEAR ALGEBRA. Matrices and Applications of Gaussian Elimination. A 2 x. A T m x. A 1 x A T 1. A m x BASIC ALGORITHMS IN LINEAR ALGEBRA STEVEN DALE CUTKOSKY Matrices and Applications of Gaussian Elimination Systems of Equations Suppose that A is an n n matrix with coefficents in a field F, and x = (x,,

More information

Linear Algebra- Final Exam Review

Linear 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 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

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

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

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

Problem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show

Problem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show MTH 0: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology - Kanpur Problem Set Problems marked (T) are for discussions in Tutorial sessions (T) If A is an m n matrix,

More 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 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

Name: Final Exam MATH 3320

Name: Final Exam MATH 3320 Name: Final Exam MATH 3320 Directions: Make sure to show all necessary work to receive full credit. If you need extra space please use the back of the sheet with appropriate labeling. (1) State the following

More information

Math 3191 Applied Linear Algebra

Math 3191 Applied Linear Algebra Math 9 Applied Linear Algebra Lecture : Orthogonal Projections, Gram-Schmidt Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./ Orthonormal Sets A set of vectors {u, u,...,

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

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

W2 ) = dim(w 1 )+ dim(w 2 ) for any two finite dimensional subspaces W 1, W 2 of V.

W2 ) = dim(w 1 )+ dim(w 2 ) for any two finite dimensional subspaces W 1, W 2 of V. MA322 Sathaye Final Preparations Spring 2017 The final MA 322 exams will be given as described in the course web site (following the Registrar s listing. You should check and verify that you do not have

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 220 FINAL EXAMINATION December 13, Name ID # Section #

MATH 220 FINAL EXAMINATION December 13, Name ID # Section # MATH 22 FINAL EXAMINATION December 3, 2 Name ID # Section # There are??multiple choice questions. Each problem is worth 5 points. Four possible answers are given for each problem, only one of which is

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

Lecture 12: Diagonalization

Lecture 12: Diagonalization Lecture : Diagonalization A square matrix D is called diagonal if all but diagonal entries are zero: a a D a n 5 n n. () Diagonal matrices are the simplest matrices that are basically equivalent to vectors

More 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

1. In this problem, if the statement is always true, circle T; otherwise, circle F.

1. 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 information

Chapter 5 Eigenvalues and Eigenvectors

Chapter 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 information

Third Midterm Exam Name: Practice Problems November 11, Find a basis for the subspace spanned by the following vectors.

Third Midterm Exam Name: Practice Problems November 11, Find a basis for the subspace spanned by the following vectors. Math 7 Treibergs Third Midterm Exam Name: Practice Problems November, Find a basis for the subspace spanned by the following vectors,,, We put the vectors in as columns Then row reduce and choose the pivot

More information

1 9/5 Matrices, vectors, and their applications

1 9/5 Matrices, vectors, and their applications 1 9/5 Matrices, vectors, and their applications Algebra: study of objects and operations on them. Linear algebra: object: matrices and vectors. operations: addition, multiplication etc. Algorithms/Geometric

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

Eigenvalues and Eigenvectors A =

Eigenvalues and Eigenvectors A = Eigenvalues and Eigenvectors Definition 0 Let A R n n be an n n real matrix A number λ R is a real eigenvalue of A if there exists a nonzero vector v R n such that A v = λ v The vector v is called an eigenvector

More information

1. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal

1. Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal . Diagonalize the matrix A if possible, that is, find an invertible matrix P and a diagonal 3 9 matrix D such that A = P DP, for A =. 3 4 3 (a) P = 4, D =. 3 (b) P = 4, D =. (c) P = 4 8 4, D =. 3 (d) P

More information

Question 7. Consider a linear system A x = b with 4 unknown. x = [x 1, x 2, x 3, x 4 ] T. The augmented

Question 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

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

(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

Chapter 7: Symmetric Matrices and Quadratic Forms

Chapter 7: Symmetric Matrices and Quadratic Forms Chapter 7: Symmetric Matrices and Quadratic Forms (Last Updated: December, 06) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). A few theorems have been moved

More 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