# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

3 Pages

### quiz1_S06 solutions

Course: MAT 202, Spring 2008
School: Princeton
Rating:

Word Count: 806

#### Document Preview

University Princeton Department of Mathematics MAT 202 Linear Algebra with Applications QUIZ 1 Spring 2006 SOLUTIONS (1) Solve the following system: x1 + 2x2 + x3 + x4 = 1 2x1 + 2x2 - x3 - x4 = 0 x1 - 2x2 - 5x3 + x4 = 0 What is the rank of the corresponding coefficient matrix and what is the rank of the corresponding augmented matrix? Solution: We apply Gauss-Jordan elimination to the system written in...

Register Now

#### Unformatted Document Excerpt

Coursehero >> New Jersey >> Princeton >> MAT 202

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
University Princeton Department of Mathematics MAT 202 Linear Algebra with Applications QUIZ 1 Spring 2006 SOLUTIONS (1) Solve the following system: x1 + 2x2 + x3 + x4 = 1 2x1 + 2x2 - x3 - x4 = 0 x1 - 2x2 - 5x3 + x4 = 0 What is the rank of the corresponding coefficient matrix and what is the rank of the corresponding augmented matrix? Solution: We apply Gauss-Jordan elimination to the system written in matrix notation: 2 3 2 3 1 2 1 1 1 1 2 1 1 1 4 2 2 -1 -1 0 5 4 0 -2 -3 -3 -2 5 1 -2 -5 1 0 0 -4 -6 0 -1 3 3 2 2 1 0 -2 0 0 1 0 -2 -2 -1 1 3 3 3 1 54 0 1 0 4 5 4 0 1 2 2 2 0 0 0 6 3 0 0 0 1 1 2 The last matrix shows that the solutions of the system are the (x1 , x2 , x3 , x4 ) satisfying 8 < x1 - 2x3 = 0 1 x2 + 3 x3 = 4 2 : 1 x4 = 2 Taking x3 as a free variable, the 2 3 2 x1 2 6 x2 7 6 - 3 6 7=6 2 4 x3 5 4 1 x4 0 solutions form the vectors 3 2 3 0 1 7 7 6 7t + 6 4 7 , tR. 5 4 0 5 1 2 The rank of a matrix is the number of leading ones in its reduced rowechelon form (rref). The leading ones are boldfaced above. As there are three leading ones in the rref of the coefficient matrix and in the rref of the augmented matrix, we see that both of these matrices have rank 3. (2) For what values of k does the following system: x1 - x2 = 1 x1 + kx2 - 4x3 = 3 -x1 + 2x2 - (1 + k)x3 = 0 have (a) infinitely many solutions? (b) no solutions? (c) exactly one solution? Justify your answers and provide a brief geometric interpretation. 2 Solution: We use Gauss-Jordan elimination: 3 3 2 2 1 1 1 -1 0 1 -1 0 2 5 3 54 0 k+1 4 1 -4 k -4 0 1 -(1 + k) 1 -1 2 -(1 + k) 0 3 3 2 2 1 1 1 -1 0 1 -1 0 1 5 1 -(1 + k) 1 5 4 0 1 -(1 + k) 4 0 2 0 k+1 -4 0 0 -4 + (1 + k)2 1 - k (a) In order to have infinitely many solutions we need the last row to be zero: -4 + (1 + k)2 = 0 and 1 - k = 0, that is k = 1. In this case the three planes intersect along one line. (b) To have no solutions we need to be short of leading ones, i.e., -4 + (1 + k)2 = 0, yet the last righthand side to be nonzero, i.e., 1 - k = 0, for the last equation to be inconsistent, that is k = -3. In this case the three planes have empty intersection. (c) To have exactly one solution we need three leading ones, therefore we must have -4 + (1 + k)2 = 0, that is k = 1 and k = -3. In this case the three planes intersect in exactly one point. (3) Find the matrix A of the orthogonal projection onto the plane x1 - x2 + x3 = 0. Describe geometrically the transformation given A2 by . Is A invertible? - Solution: The vector = (1,-1,1) has unit length and is orthogonal to n 3 the plane x1 - x2 + x3 = (1, -1, 1) (x1 , x2 , x3 ) = 0. The orthogonal projection onto this plane is the transformation proj (- ) = - - (- - )- . x x x n n V In matrix notation this is 3 3 2 2 2 3 2 2 x1 x1 1 3 1 A 4 x2 5 = 4 x2 5- (x1 -x2 +x3 ) 4 -1 5 = 4 1 3 3 x3 x3 1 -1 3 | 1 3 2 3 1 3 -1 3 1 3 2 3 {z A 3 x1 5 4 x2 5 x3 } 32 Being a projection, the square of A represents the same projection. In general, the composition of a projection with itself is the same projection: projV (projV (- )) = projV (- ) - (projV (- ) - )- x x x n n - - (- - )- - ((- - (- - )- ) - )- = x x n n x x n n n n | {z } 0 - - (- - )- = x x n n = proj (- ) . x V The matrix A is not invertible as all the vectors proportional to - (vectors n orthogonal to the plane) project to zero, so its kernel is more than {0}. (4) Find the inverse of the matrix 0 0 A= 4 1 0 0 1 0 0 1 2 0 1 0 3 0 3 Solution: Changing the order of rows in the first step and then using the leading ones in the first, third and fourth rows to also bring the second row to the reduced row-echelon form, we obtain 3 3 2 2 1 1 1 1 7 7 6 6 1 1 1 7 76 4 1 2 3 6 5 5 4 4 4 1 2 3 1 1 1 1 1 1 1 2 1 1 6 -3 -2 1 -4 1 6 4 1 1 1 1 where the omitted entries are zero. We conclude that 2 3 0 0 0 1 6-3 -2 1 -47 7 A-1 = 6 40 1 0 05 1 0 0 0 3 7 7 5 (5) Determine whether the following statements are true or false. No explanations, no credit. (a) Given two square matrices A and B, if AB is invertible, then so is A. Solution: TRUE Let C be the inverse of AB, i.e., (AB)C = C(AB) = I. By associativity A(BC) = I, which implies that A is invertible with inverse BC. (b) The composition of a shear, a reflection and a rotation in the plane is invertible. Solution: TRUE A shear S, a reflection R and a rotation T are invertible transformations, hence their composition (in any order) is invertible. In particular, (SRT )-1 = T -1 R-1 S -1 . (c) There exist matrices A of size 3 2 and B of size 2 3 such that AB = I3 is the identity matrix. Solution: FALSE Since B is of size 2 3, we can find a nonzero vector - R3 such v - = - (the corresponding system has 2 equations for 3 variables, that B v 0 hence cannot have a unique solution, and zero is always a solution for a - homogeneous system). Therefore (AB)- = A(B - ) = 0 which shows v v that AB cannot be the identity matrix.
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Princeton - MAT - 202
MAT 202 Spring 2004, Quiz 1 on Chapters 1 and 2 1. (8 pts) Find all solutions to the given system by using elementary row operations to bring the matrix of the system into reduced row echelon form. Be sure to indicate which variables are free variabl
Princeton - MAT - 202
MATH 202, MIDTERM Wednesday, October 20, 7.30-9.00 PM Covers Chapters 15.1 of textbook Time: 1 hour 30 minutesYour name (print): . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Your instructor: . . . . . . . . . . . . . . . . . . .
Princeton - MAT - 202
Princeton University Department of MathematicsName: Instructor: A. Cannas J. Holland N. Pavlovic A. Yang P. Raphael Class time:A. Salehi GolsefidyJ. SzeftelMAT 202 Linear Algebra with Applications QUIZ 1 Spring 2007 This quiz is due
Princeton - MAT - 202
Princeton University Department of MathematicsName:Instructor:Class time:MAT 202 Linear Algebra with Applications QUIZ 1 Spring 2006 This quiz is due Friday, March 3, at 2 pm at your instructor's office. You have 60 minutes to compl
Princeton - MAT - 202
NameClass Time MATH 202 - QUIZ # 1 Wednesday, October 10, 2007 Covers Chapter 1 and Chapter 2 of the textPlease show all work. Books, notes, calculators, are not permitted on this quiz. As part of your obligations under the Honor Code, do not dis
Princeton - MAT - 202
MATH 202, QUIZ #1 Due Friday, October 6, before 3 PM strict Covers Chapters 12 of textbook Time: one hourYour name (print): . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Your instructor: . . . . . . . . . . . . . . . . . . . Time
Ball State - PEP - 291
Mike Copley PEP 291 Journal 1 2-27-08 Reflect on your first contact with the children Q. How did they react to you? Did they listen to you? Did they accomplish the tasks you stated? A. At first they were shy and they looked scared. But when we got t
Ball State - PEP - 291
Mike Copley 3-24-08 Journal 2 PEP 291 Talk about the child that you selected for your child case study: 1. Identify two skills (one object control, one locomotor) taught to the children. a. Report if the child that you have been observing for your C
USC - BISC - 330L
Midterm 2 Statistics: 11:00 Section 139 26 170 32 Noon Section 143 18 168 72Class Average (out of 175 pts) Standard Deviation High Score Low ScorePlease remember re-grade is not your opportunity to haggle for points. If you feel a factual or arit
Ball State - PEP - 291
Mike Copley 4-14-08 PEP 291 - Journal 3 Reflect on how your teaching has improved over the semester, including specific areas of improvement: What mistakes did you make in your teaching at the beginning? What did you learn from your group of children
USC - BISC - 330L
Midterm 1 Statistics: 11:00 Section 130 21.9 164.5 36 Noon Section 138 20.2 165.75 56.5Class Average (out of 175 pts) Standard Deviation High Score Low ScorePlease remember re-grade is not your opportunity to haggle for points. If you feel a fact
Ball State - PEP - 291
PEP 291 LESSON PLANS Striking, Leaping, and Sliding Name: Mike Copley Date: 4-11-08 Objectives: Psychomotor: During the activities, the students will be able to perform correctly at least two of the cues taught consistently. Cognitive: During the ac
USC - BISC - 330L
Midterm 1 Statistics: 11:00 Section 130 21.9 164.5 36 Noon Section 138 20.2 165.75 56.5Class Average (out of 175 pts) Standard Deviation High Score Low ScorePlease remember re-grade is not your opportunity to haggle for points. If you feel a fact
Ball State - PEP - 291
PEP 291 LESSON PLAN Dribbling, Running, Jumping, and LeapingName: Mike Copley Date: 4-25-08 Objectives: Psychomotor: During the activities, the students will be able to perform correctly all the cues taught consistently. Cognitive: During the activ
USC - BISC - 330L
Midterm 3 Statistics:11:00 Section Average Score (out of 175 pts) Standard Deviation High Score Low Score 145.6 13.7 164.6 90.8 Noon Section 147.0 11.7 164.6 76.6Please remember re-grade is not your opportunity to haggle for points. If you feel a
Ball State - PEP - 291
PEP 291 LESSON PLANS Kicking, Running, and Jumping Name: Mike Copley Date: 3-7-08 Objectives: Psychomotor: During the activities, the students will be able to perform correctly at least one of the cues taught consistently. Cognitive: During the acti
USC - BISC - 330L
Midterm 2 Statistics: 11:00 Section 139 26 170 32 Noon Section 143 18 168 72Class Average (out of 175 pts) Standard Deviation High Score Low ScorePlease remember re-grade is not your opportunity to haggle for points. If you feel a factual or arit
Ball State - PEP - 222
CuesPivoting -Plant the foot -Maintain point of control of the floor -Protect the ball Passing -Crisp, but not too hard to catch -Use appropriate pass -Step in the direction of the ball -Stay balanced -Don't telegraph the pass -Aim the ball between
Ball State - PEP - 222
SoftballBy Mike Copley Skills/Ques Catching -Watch the ball into the hands or gloves -Position fingers up and thumbs together for balls chest high or higher. Position fingers down and little fingers together for balls waist high and lower -Relax the
Texas San Antonio - POL - 1013.009
POL 1013-009Introduction to American PoliticsLecture #26 April 29thExam ReviewThe University of Texas at San AntonioToday We Will.Review all of the key material we have covered since Exam #3 Combined with:readings Prior review lecture
Christopher Newport University - COMM - 250
Ruth Fuller Comm 211 - Manning 3/28/08&quot;What do the Experts Say&quot; Paper TopicHow do children's facial expressions and childrens reactions to facial expressions affect a situation?People often speak of the maternal bond between a mother and child
Tennessee - MATH - 119
Do not open this cover sheet until instructed to do so.Math 119 Unit 1 Test Instructor: Jeneva Moseley Thursday, February 1st, 2007Read and Sign Below. The Honor Statement An essential feature of the University of Tennessee is a commitment to main
Oregon State - ENGLISH - 101
Angie Peterson November 14, 2007 Odyssey ALS Class Success InterviewI interviewed a family friend, Juleigh Cozzetto, for this success project. She is a successful Interior Designer who has pretty much done everything. Juleigh began her life with a
Kettering - CS - 101L
file:/X|/cs101/Exercise2_1.txtimport javax.swing.JOptionPane; public class Exercise2_1 { / Main method public static void main(String[] args) { / Enter a temperature in Fahrenheit String fahrenheitString = JOptionPane.showInputDialog(null, &quot;Enter a
Kettering - CS - 101L
file:/X|/cs101/Homework3_0.txt/* = Solve 4.20 from page 125 = Write a program that will print all the prime numbers between 2 and 100, inclusively. Display eight prime numbers per line. */ public class Homework3_0 { public static void main(String[]
Kettering - CS - 101L
file:/X|/cs101/Homework2_0.txt/*= Solve 3.5 from page 94 = Write a program that generates three single-digit integers and prompt the user to enter the addition of these three integers.*/ import java.util.Scanner; public class Homework2_0 { publ
Kettering - CS - 101L
CS101 Lab3 Objectives Loop control and selection control statements Java Input and output Numerical computation Documentation and formatingLab descriptionIn this lab, you are going to develop a program that can compares loans with various i
Kettering - CS - 101L
file:/X|/cs101/Homework7.txt/* 3 functions: open new account, perform account transaction, quit. */ import java.util.Scanner; public class Homework7 { public static void main(String[] args) { Scanner input = new Scanner(System.in); String buffer; d
Kettering - CS - 101L
file:/X|/cs101/Lincoln.txtpublic class Lincoln { /-/ Prints a presidential quote. /-public static void main (String[] args) { System.out.println (&quot;A quote by Abraham Lincoln:&quot;); System.out.println (&quot;Whatever you are, be a good one.&quot;); } }file:/X|
Kettering - CS - 101L
Lab 5 for CS101 Objective: Arrays This Java lab project addresses the ability to work with single and multidimensional arrays in Java. In this lab project, we consider how we can create arrays, index elements of arrays, set elements of arrays, pass
Kettering - CS - 101L
Objective: Lab 7 class definition and object creation UML diagram Write multiclass Java programLab descriptionIn this lab, you are going to design a class named Stock that contains A string data field named symbol for the stock's symbol
Kettering - CS - 101L
Lab 7Objective File I/O, Scanner, PrintWriter Class and Objects Array object array sorting Lab description In this lab, you are going to write a program that read in from a text file about the information of a bunch of students. For each stu
Kettering - MECH - 210
Chapter One: Basic Concepts2Irwin, Basic Engineering Circuit Analysis, 8/EChapter One: Basic Concepts3SOLUTION:4Irwin, Basic Engineering Circuit Analysis, 8/ESOLUTION:Chapter One: Basic Concepts5SOLUTION:6Irwin, Basic Engine
Kettering - PHYS - 114
Page 2 of 6PHYS-114Test #1Summer 2006[1] (10 pts) A car driving South slows down slightly as it approaches a curve in the road ahead. The car rounds the curve at constant speed and then speeds up slightly as it heads off East. Draw a complete
Kettering - CS - 101L
file:/X|/cs101/Homework6_0.txt/*[6.5 from p204] Write a program that reads in ten numbers and displays distinct numbers (i.e., if a number appears multiple times, it is displayed only once).*/ import java.util.Scanner; public class Homework6_0
Kettering - CS - 101L
Lab 8 CS101 Fall 2007 Objective File I/O Class definition and object creation Object array Inheritance and polymorphism Lab description In this lab, you are going to write a program that read a set of geometry objects from a text file. The ge
Kettering - PHYS - 114
Winter 2008PHYS-114Test #1Page 1 of 6PHYS-114, Newtonian Mechanics Test #1 (Chapter 1,2,3,4)Rules:[1] This exam has 6 pages. There are five shorter questions and one long problem. [2] You must show ALL work in order to receive complete cred
Kettering - PHYS - 114
PHYS-114 Test #3Chapters 9,10,11(Conservation of Momentum and Energy) Name (please print): _ Raw Score: _ Test Grade: _[1] (10 pts) A 3.7kg box is sliding across an icy pond with a velocity of 8.0 m/s. A gust of wind exerts a force as shown in
Kettering - PHYS - 114
PYYS-114Practice Test #2Page 1 of 6[1] A 1.0-kg physics book is given an initial velocity of 3.0 m/s up a ramp. The ramp makes an angle of 20 above the horizontal. If the coefficient of kinetic friction between the book and the surface is 0.2,
Kettering - PHYS - 114
PHYS-114 Test #3Chapters 9,10,11Winter 2008(Conservation of Momentum and Energy) Name (please print): _ Raw Score: _ Test Grade: _[1] (8 pts) In an episode of Futurama Professor Farnsworth uses his Smell-O-Scope to detect a gigantic 5 ball of
Kettering - CHEM - 136
Kettering - PHYS - 114
Winter 2008PHYS-114Test #2Page 1 of 7PHYS-114, Newtonian Mechanics Test #2 (Chapter 5,6,7,8)Rules:[1] This test has 4 worked out problems with plenty of room for your work. [2] You must show ALL work in order to receive complete credit for
Hofstra - ENGG - 10
THRUA2537 7 R2156x6x27B128APROPRIETARY AND CONFIDENTIAL THE INFORMATION CONTAINED IN THIS DRAWING IS THE SOLE PROPERTY OF &lt;INSERT COMPANY NAME HERE&gt;. ANY REPRODUCTION IN PART OR AS A WHOLE WITHOUT THE WRITTEN PERMISSION OF &lt;INSER
Cornell - MPS/ILR/NY - OB 525
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46Organizational Behavior (OB): The Action Perspective ILROB 525; Spring 2008 Sam Bacharach Notes by Christine Early Class
University of Florida - PHY - 2048
Ch 4 notes3d-motion: general equations of motion Projectile motion (special case)^ ^ ^ 3d-position: r = xi + yj + zkyrr1r2x^ ^ r1 = x1i + y1 ^ + z1k j ^ ^ r2 = x2i + y2 ^ + z2 k j ^ ^ ^ r = xi + yj + zk ^ ^ r (t ) = x(t )i + y (t ) ^ + z (
University of Florida - PHY - 2048
Ch3: VectorsScalar: just a number mass (m) time (t) pressure (P) temperature (T) energy (E) . Vector: magnitude + direction displacement r velocity v accelerationa force F momentum p .Adding Vectorsc = a +bbac acbTo add vec
University of Florida - PHY - 2048
1-d Motion: Position &amp; Displacement The x-axis:We locate objects by specifying their position along an axis (in this case x-axis). The positive direction of an axis is in the direction of increasing numbers. The opposite is the negative direction.
University of Florida - PHY - 2048
Force &amp; Motion: Friction Coefficient of static friction s:Case 1: v = 0 &amp; a = 0FN Block M FextIf on object does not move (i.e. v = 0 a = 0) then the static frictional force, fs, and the component of the external force Fext that is parallel to th
University of Florida - PHY - 2048
Force &amp; Motion: Newton's Laws (1st Law) If no net force acts on a body then the body's velocity cannot change. Inertial mass!Zero net force implies zero acceleration. (2nd Law) The net force on a body is equal to its mass r all r times its accele
Texas A&M - PHYS - 222
PHYSICS 222 EXAM C April 17, 2008N AME (PLEASE PRINT)w jJoffJeeqjjatojtsheets nd 1 site your nameor, the hack of the. exam.)wave function (30) 1 consider an electron trapped in an infinite 14e11 with width a 1 he will be 0 for x 0 and x a and to
Texas A&M - PHYS - 222
_____PHYSICS 222 EXAM B a February 28, 2008NAME (PLEASE PRINT)(Tear off the equation sheets and write your name on the back of the exam.) (10) 1. At what peak wavelength does an object at 37 C radiate ?c. 9.4x I0 m 6 d. 7.8 x 10 mT
Texas A&M - MATH - 311
Math 311 midterm exam I answers1. We need to solve the system 1 -1 1 1 t t 1 - s 1 = 1 -1 s 1 0 1 0 3 1 . = 2Using row reduction, 1 1 4 1 1 4 1 0 3 1 1 4 1 -1 2 0 -2 -2 0 1 1 0 1 1 . 1 0 3 0 -1 -1 0 0 0 0 0 0 Thus t =