Linear Algebra MATH20F Midterm 1
|
|
- Erik O’Neal’
- 5 years ago
- Views:
Transcription
1 University of California San Diego NAME TA: Linear Algebra Wednesday, October st, 9 :am - :5am No aids are allowed Be sure to write all row operations used Remember that you can often check your answers Answers without full justification are worth no credit This test consists of 4 questions Each question is worth a total of 5 points Page of 6
2 Let A = 3 5 5, b = (a) Find the full solution set of the nonhomogeneous system Ax = b, written in parametric vector form [3] 4 We construct the augmented matrix associated with the system and perform the row reduction algorithm, R + R 5 R 3 + R R R ( ) R 3 + R ( ) R R Using the reduced Echelon form obtained above, we can write the following system of equations, x + x 3 = x x 3 = Hence, x x 3 x = x = + x 3 = + x 3 x 3 x 3 is the solution of the above system written in parametric vector form (b) What is the parametric vector form for the solution of the corresponding homogeneous system Ax = [] Recall that the solution set of Ax = b is the set of all vectors of the form w = p+v h, where p is a particular solution of the nonhomogeneous system Ax = b and v h is any solution of the corresponding homogeneous system Ax = It is easy to see that p = [ ] T is a solution of the nonhomogeneous system Ax = b Consequently, the solution set for the homogeneous system Ax = is x = x x x 3 = x 3 Page of 6
3 (c) Write the vector b as a linear combination of the columns of the matrix A [] Let A = [a a a 3 ], then by definition of matrix-vector multiplication, b = Ax = x a + x a + x 3 a 3 In order to write b as a linear combination of the columns of the matrix A, it is sufficient to choose some x from the solution set found in a) For example, let x 3 =, then x = + ( ) = Substituting this x into the system leads to 4 b = = Ax = ( ) 3 5 Page 3 of 6
4 Let V = {[ 4 8 ] [ 6, ]} (a) Is the set V linearly independent? Justify your answer [] The set is not linearly independent, because one of the vectors is a multiple of another In particular, [ ] 4 = [ ] In other words, there exist non-zero weights c =, c = /3, such that [ ] 4 + [ ] [ ] 6 =, 8 3 which by definition means that the vectors are linearly dependent (b) For v = (4, 8), v = ( 6, ), give a definition of Span{v, v } [] For v, v R, the set of all linear combinations of v, v is defined to be a span of v, v That is, Span{v, v }, is the collection of all vectors, with c, c scalars c v + c v, (c) Give a geometric description of Span{v, v } (ie, draw a picture and give explanation) [] Since the set of vectors {v, v } is linearly dependent, any linear combination c v + c v degenerates to a linear combination of a single vector In more detail, c v + c v = (c 3 c )v Since c, c are arbitrary scalars, we can write the span as just γv, where γ is any scalar Geometrically, this corresponds to a line through the origin Page 4 of 6
5 3 Consider T : R R 3, T (x, x ) = (x x, x + 3x, x + x ) (a) Show that T is a one-to-one linear transformation Does T map R onto R 3? Justify your answer [3] We can determine the standard matrix of T by inspection, x x [ T (x) = x + 3x = 3 x x x + x One could also use the definition, by which the matrix of T is constructed as [T (e ) T (e )], where [e e ] is the identity matrix Since T (x) = Ax, where A = 3, the transformation is linear The columns of the matrix A are not multiples of each other, so they are linearly independent By the theorem from class (Theorem, Section 9), this is equivalent to the transformation T being one-to-one By the same theorem, the transformation is onto, if the columns of A span R 3 The latter is not possible because two vectors in R cannot span the whole space Therefore, the transformation T is not onto ] (b) Consider S : R 3 R, S(x, x, x 3 ) = (x + x 3, x x 3 ) Compute the composition transformation x S(T (x)) (from R to R ) Find the standard matrix for S(T (x)) [] For convenience, let us write S(y, y, y 3 ) = (y + y 3, y y 3 ) The the composition is, S(T (x, x )) = S(x x, x + 3x, x + x ) = ((x x ) + (x + x ), (x x ) (x + x )) = (x, x ) It remains to note that [ x S(T (x, x )) = x ] = [ ] [ x x ] Page 5 of 6
6 4 Let A = (a) Find A [3] In order to obtain A, we row-reduce the augmented matrix [A Hence, R R 4 R 3 + R R + R 3 R R 3 A = I] R 3 R R + R (b) Explain why Ax = has only the trivial solution [] There are many ways to answer this question One possible argument is due to Theorem 5 in Section By this theorem, if A is invertible, then for any b R 3, the equation Ax = b has the unique solution x = A b In particular, for b =, the unique solution of the homogeneous system, Ax =, is x = A = (the latter follows easily from the definition of matrix-vector multiplication), which is the trivial solution (c) What is (A T )? [] Here we use the fact that (A T ) = (A ) T So 3 6 (A T ) = Page 6 of 6
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 informationMATH 152 Exam 1-Solutions 135 pts. Write your answers on separate paper. You do not need to copy the questions. Show your work!!!
MATH Exam -Solutions pts Write your answers on separate paper. You do not need to copy the questions. Show your work!!!. ( pts) Find the reduced row echelon form of the matrix Solution : 4 4 6 4 4 R R
More informationSection 1.5. Solution Sets of Linear Systems
Section 1.5 Solution Sets of Linear Systems Plan For Today Today we will learn to describe and draw the solution set of an arbitrary system of linear equations Ax = b, using spans. Ax = b Recall: the solution
More informationLinear Algebra Exam 1 Spring 2007
Linear Algebra Exam 1 Spring 2007 March 15, 2007 Name: SOLUTION KEY (Total 55 points, plus 5 more for Pledged Assignment.) Honor Code Statement: Directions: Complete all problems. Justify all answers/solutions.
More informationFamily Feud Review. Linear Algebra. October 22, 2013
Review Linear Algebra October 22, 2013 Question 1 Let A and B be matrices. If AB is a 4 7 matrix, then determine the dimensions of A and B if A has 19 columns. Answer 1 Answer A is a 4 19 matrix, while
More informationMath 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 information1. 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 informationAnnouncements Monday, September 18
Announcements Monday, September 18 WeBWorK 1.4, 1.5 are due on Wednesday at 11:59pm. The first midterm is on this Friday, September 22. Midterms happen during recitation. The exam covers through 1.5. About
More informationRow 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 informationReview for Chapter 1. Selected Topics
Review for Chapter 1 Selected Topics Linear Equations We have four equivalent ways of writing linear systems: 1 As a system of equations: 2x 1 + 3x 2 = 7 x 1 x 2 = 5 2 As an augmented matrix: ( 2 3 ) 7
More informationLinear Combination. v = a 1 v 1 + a 2 v a k v k
Linear Combination Definition 1 Given a set of vectors {v 1, v 2,..., v k } in a vector space V, any vector of the form v = a 1 v 1 + a 2 v 2 +... + a k v k for some scalars a 1, a 2,..., a k, is called
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
est Review-Linear Algebra Name MULIPLE CHOICE Choose the one alternative that best completes the statement or answers the question Solve the system of equations ) 7x + 7 + x + + 9x + + 9 9 (-,, ) (, -,
More informationMATH 1553, SPRING 2018 SAMPLE MIDTERM 1: THROUGH SECTION 1.5
MATH 553, SPRING 28 SAMPLE MIDTERM : THROUGH SECTION 5 Name Section Please read all instructions carefully before beginning You have 5 minutes to complete this exam There are no aids of any kind (calculators,
More informationWorksheet for Lecture 15 (due October 23) Section 4.3 Linearly Independent Sets; Bases
Worksheet for Lecture 5 (due October 23) Name: Section 4.3 Linearly Independent Sets; Bases Definition An indexed set {v,..., v n } in a vector space V is linearly dependent if there is a linear relation
More information1. TRUE or FALSE. 2. Find the complete solution set to the system:
TRUE or FALSE (a A homogenous system with more variables than equations has a nonzero solution True (The number of pivots is going to be less than the number of columns and therefore there is a free variable
More informationMATH 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 informationMath 3A Winter 2016 Midterm
Math 3A Winter 016 Midterm Name Signature UCI ID # E-mail address There are 7 problems for a total of 115 points. Present your work as clearly as possible. Partial credit will be awarded, and you must
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.5 SOLUTION SETS OF LINEAR SYSTEMS HOMOGENEOUS LINEAR SYSTEMS A system of linear equations is said to be homogeneous if it can be written in the form A x= 0, where
More informationSolution: By inspection, the standard matrix of T is: A = Where, Ae 1 = 3. , and Ae 3 = 4. , Ae 2 =
This is a typical assignment, but you may not be familiar with the material. You should also be aware that many schools only give two exams, but also collect homework which is usually worth a small part
More informationMid-term Exam #1 MATH 205, Fall 2014
Mid-term Exam # MATH, Fall Name: Instructions: Please answer as many of the following questions as possible. Show all of your work and give complete explanations when requested. Write your final answer
More informationAdditional Problems for Midterm 1 Review
Additional Problems for Midterm Review About This Review Set As stated in the syllabus, a goal of this course is to prepare students for more advanced courses that have this course as a pre-requisite.
More informationLinear Algebra Quiz 4. Problem 1 (Linear Transformations): 4 POINTS Show all Work! Consider the tranformation T : R 3 R 3 given by:
Page 1 This is a 60 min Quiz. Please make sure you put your name on the top right hand corner of each sheet. Remember the Honors Code will be enforced! You may use your book. NO HELP FROM ANYONE. Problem
More informationMAT 242 CHAPTER 4: SUBSPACES OF R n
MAT 242 CHAPTER 4: SUBSPACES OF R n JOHN QUIGG 1. Subspaces Recall that R n is the set of n 1 matrices, also called vectors, and satisfies the following properties: x + y = y + x x + (y + z) = (x + y)
More informationThis is a closed book exam. No notes or calculators are permitted. We will drop your lowest scoring question for you.
Math 54 Fall 2017 Practice Exam 1 Exam date: 9/26/17 Time Limit: 80 Minutes Name: Student ID: GSI or Section: This exam contains 6 pages (including this cover page) and 7 problems. Problems are printed
More informationLinear 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 informationLinear Algebra Practice Problems
Math 7, Professor Ramras Linear Algebra Practice Problems () Consider the following system of linear equations in the variables x, y, and z, in which the constants a and b are real numbers. x y + z = a
More informationSolutions 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 informationMATH 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 informationSection 3.3. Matrix Equations
Section 3.3 Matrix Equations Matrix Vector the first number is the number of rows the second number is the number of columns Let A be an m n matrix A = v v v n with columns v, v,..., v n Definition The
More informationHomework 5. (due Wednesday 8 th Nov midnight)
Homework (due Wednesday 8 th Nov midnight) Use this definition for Column Space of a Matrix Column Space of a matrix A is the set ColA of all linear combinations of the columns of A. In other words, if
More informationMidterm 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 informationSection 2.2: The Inverse of a Matrix
Section 22: The Inverse of a Matrix Recall that a linear equation ax b, where a and b are scalars and a 0, has the unique solution x a 1 b, where a 1 is the reciprocal of a From this result, it is natural
More informationMath 2940: Prelim 1 Practice Solutions
Math 294: Prelim Practice Solutions x. Find all solutions x = x 2 x 3 to the following system of equations: x 4 2x + 4x 2 + 2x 3 + 2x 4 = 6 x + 2x 2 + x 3 + x 4 = 3 3x 6x 2 + x 3 + 5x 4 = 5 Write your
More informationMATH10212 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(c)
1. Find the reduced echelon form of the matrix 1 1 5 1 8 5. 1 1 1 (a) 3 1 3 0 1 3 1 (b) 0 0 1 (c) 3 0 0 1 0 (d) 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 (e) 1 0 5 0 0 1 3 0 0 0 0 Solution. 1 1 1 1 1 1 1 1
More informationWorksheet for Lecture 15 (due October 23) Section 4.3 Linearly Independent Sets; Bases
Worksheet for Lecture 5 (due October 23) Name: Section 4.3 Linearly Independent Sets; Bases Definition An indexed set {v,..., v n } in a vector space V is linearly dependent if there is a linear relation
More informationExercise 2) Consider the two vectors v 1. = 1, 0, 2 T, v 2
Exercise ) Consider the two vectors v,, T,, T 3. a) Sketch these two vectors as position vectors in 3, using the axes below. b) What geometric object is span v? Sketch a portion of this object onto your
More information1111: Linear Algebra I
1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 13 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 13 1 / 8 The coordinate vector space R n We already used vectors in n dimensions
More informationYORK 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 information4.9 The Rank-Nullity Theorem
For Problems 7 10, use the ideas in this section to determine a basis for the subspace of R n spanned by the given set of vectors. 7. {(1, 1, 2), (5, 4, 1), (7, 5, 4)}. 8. {(1, 3, 3), (1, 5, 1), (2, 7,
More informationChoose 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 informationMath 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 informationLecture 6: Spanning Set & Linear Independency
Lecture 6: Elif Tan Ankara University Elif Tan (Ankara University) Lecture 6 / 0 Definition (Linear Combination) Let v, v 2,..., v k be vectors in (V,, ) a vector space. A vector v V is called a linear
More informationMath 2210Q (Roby) Practice Midterm #1 Solutions Fall 2017
Math Q (Roby) Practice Midterm # Solutions Fall 7 SHOW ALL YOUR WORK! Make sure you give reasons to support your answers. If you have any questions, do not hesitate to ask! For this exam no calculators
More informationLecture 03. Math 22 Summer 2017 Section 2 June 26, 2017
Lecture 03 Math 22 Summer 2017 Section 2 June 26, 2017 Just for today (10 minutes) Review row reduction algorithm (40 minutes) 1.3 (15 minutes) Classwork Review row reduction algorithm Review row reduction
More informationSpan & Linear Independence (Pop Quiz)
Span & Linear Independence (Pop Quiz). Consider the following vectors: v = 2, v 2 = 4 5, v 3 = 3 2, v 4 = Is the set of vectors S = {v, v 2, v 3, v 4 } linearly independent? Solution: Notice that the number
More informationSolutions of Linear system, vector and matrix equation
Goals: Solutions of Linear system, vector and matrix equation Solutions of linear system. Vectors, vector equation. Matrix equation. Math 112, Week 2 Suggested Textbook Readings: Sections 1.3, 1.4, 1.5
More informationLinear Independence Reading: Lay 1.7
Linear Independence Reading: Lay 17 September 11, 213 In this section, we discuss the concept of linear dependence and independence I am going to introduce the definitions and then work some examples and
More information2. Every linear system with the same number of equations as unknowns has a unique solution.
1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations
More information5.) For each of the given sets of vectors, determine whether or not the set spans R 3. Give reasons for your answers.
Linear Algebra - Test File - Spring Test # For problems - consider the following system of equations. x + y - z = x + y + 4z = x + y + 6z =.) Solve the system without using your calculator..) Find the
More informationLinear equations in linear algebra
Linear equations in linear algebra Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra Pearson Collections Samy T. Linear
More informationMATH 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 information5) homogeneous and nonhomogeneous systems 5a) The homogeneous matrix equation, A x = 0 is always consistent.
5) homogeneous and nonhomogeneous systems 5a) The homogeneous matrix equation, A x is always consistent. 5b) If A x b is also consistent, then the solution set to A x b is a translation of the solution
More informationMATH 1553, JANKOWSKI MIDTERM 2, SPRING 2018, LECTURE A
MATH 553, JANKOWSKI MIDTERM 2, SPRING 28, LECTURE A Name GT Email @gatech.edu Write your section number here: Please read all instructions carefully before beginning. Please leave your GT ID card on your
More informationMath 221 Midterm Fall 2017 Section 104 Dijana Kreso
The University of British Columbia Midterm October 5, 017 Group B Math 1: Matrix Algebra Section 104 (Dijana Kreso) Last Name: Student Number: First Name: Section: Format: 50 min long exam. Total: 5 marks.
More informationNAME MATH 304 Examination 2 Page 1
NAME MATH 4 Examination 2 Page. [8 points (a) Find the following determinant. However, use only properties of determinants, without calculating directly (that is without expanding along a column or row
More informationChapter 1: Systems of Linear Equations
Chapter : Systems of Linear Equations February, 9 Systems of linear equations Linear systems Lecture A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where
More informationAnnouncements Wednesday, September 20
Announcements Wednesday, September 20 WeBWorK 1.4, 1.5 are due on Wednesday at 11:59pm. The first midterm is on this Friday, September 22. Midterms happen during recitation. The exam covers through 1.5.
More informationLinear Algebra. Preliminary Lecture Notes
Linear Algebra Preliminary Lecture Notes Adolfo J. Rumbos c Draft date May 9, 29 2 Contents 1 Motivation for the course 5 2 Euclidean n dimensional Space 7 2.1 Definition of n Dimensional Euclidean Space...........
More informationDEPARTMENT OF MATHEMATICS
DEPARTMENT OF MATHEMATICS. Points: 4+7+4 Ma 322 Solved First Exam February 7, 207 With supplements You are given an augmented matrix of a linear system of equations. Here t is a parameter: 0 4 4 t 0 3
More informationChapter 3: Theory Review: Solutions Math 308 F Spring 2015
Chapter : Theory Review: Solutions Math 08 F Spring 05. What two properties must a function T : R m R n satisfy to be a linear transformation? (a) For all vectors u and v in R m, T (u + v) T (u) + T (v)
More information(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 informationAnnouncements Monday, September 17
Announcements Monday, September 17 WeBWorK 3.3, 3.4 are due on Wednesday at 11:59pm. The first midterm is on this Friday, September 21. Midterms happen during recitation. The exam covers through 3.4. About
More informationICS 6N Computational Linear Algebra Vector Equations
ICS 6N Computational Linear Algebra Vector Equations Xiaohui Xie University of California, Irvine xhx@uci.edu January 17, 2017 Xiaohui Xie (UCI) ICS 6N January 17, 2017 1 / 18 Vectors in R 2 An example
More informationMath 3C Lecture 25. John Douglas Moore
Math 3C Lecture 25 John Douglas Moore June 1, 2009 Let V be a vector space. A basis for V is a collection of vectors {v 1,..., v k } such that 1. V = Span{v 1,..., v k }, and 2. {v 1,..., v k } are linearly
More informationLinear Equations in Linear Algebra
1 Linear Equations in Linear Algebra 1.9 THE MATRIX OF A LINEAR TRANSFORMATION THE MATRIX OF A LINEAR TRANSFORMATION Theorem 10: Let T: R n R m be a linear transformation. Then there exists a unique matrix
More informationMath 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 informationLinear Algebra. Preliminary Lecture Notes
Linear Algebra Preliminary Lecture Notes Adolfo J. Rumbos c Draft date April 29, 23 2 Contents Motivation for the course 5 2 Euclidean n dimensional Space 7 2. Definition of n Dimensional Euclidean Space...........
More informationQuizzes for Math 304
Quizzes for Math 304 QUIZ. A system of linear equations has augmented matrix 2 4 4 A = 2 0 2 4 3 5 2 a) Write down this system of equations; b) Find the reduced row-echelon form of A; c) What are the pivot
More informationExam 2 MAS 3105 Applied Linear Algebra, Spring 2018
Exam 2 MAS 3105 Applied Linear Algebra, Spring 2018 (Clearly!) Print Name: Mar 8, 2018 Read all of what follows carefully before starting! 1. This test has 6 problems and is worth 100 points. Please be
More information2018 Fall 2210Q Section 013 Midterm Exam I Solution
8 Fall Q Section 3 Midterm Exam I Solution True or False questions ( points = points) () An example of a linear combination of vectors v, v is the vector v. True. We can write v as v + v. () If two matrices
More informationMath 301 Test I. M. Randall Holmes. September 8, 2008
Math 0 Test I M. Randall Holmes September 8, 008 This exam will begin at 9:40 am and end at 0:5 am. You may use your writing instrument, a calculator, and your test paper; books, notes and neighbors to
More informationMATH 2360 REVIEW PROBLEMS
MATH 2360 REVIEW PROBLEMS Problem 1: In (a) (d) below, either compute the matrix product or indicate why it does not exist: ( )( ) 1 2 2 1 (a) 0 1 1 2 ( ) 0 1 2 (b) 0 3 1 4 3 4 5 2 5 (c) 0 3 ) 1 4 ( 1
More informationLinear Algebra Math 221
Linear Algebra Math 221 Open Book Exam 1 Open Notes 3 Sept, 24 Calculators Permitted Show all work (except #4) 1 2 3 4 2 1. (25 pts) Given A 1 2 1, b 2 and c 4. 1 a) (7 pts) Bring matrix A to echelon form.
More informationSolutions to Math 51 First Exam October 13, 2015
Solutions to Math First Exam October 3, 2. (8 points) (a) Find an equation for the plane in R 3 that contains both the x-axis and the point (,, 2). The equation should be of the form ax + by + cz = d.
More informationMath 313 Chapter 5 Review
Math 313 Chapter 5 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 5.1 Real Vector Spaces 2 2 5.2 Subspaces 3 3 5.3 Linear Independence 4 4 5.4 Basis and Dimension 5 5 5.5 Row
More information1 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 informationMath 308 Spring Midterm Answers May 6, 2013
Math 38 Spring Midterm Answers May 6, 23 Instructions. Part A consists of questions that require a short answer. There is no partial credit and no need to show your work. In Part A you get 2 points per
More informationis Use at most six elementary row operations. (Partial
MATH 235 SPRING 2 EXAM SOLUTIONS () (6 points) a) Show that the reduced row echelon form of the augmented matrix of the system x + + 2x 4 + x 5 = 3 x x 3 + x 4 + x 5 = 2 2x + 2x 3 2x 4 x 5 = 3 is. Use
More informationSolutions to Exam I MATH 304, section 6
Solutions to Exam I MATH 304, section 6 YOU MUST SHOW ALL WORK TO GET CREDIT. Problem 1. Let A = 1 2 5 6 1 2 5 6 3 2 0 0 1 3 1 1 2 0 1 3, B =, C =, I = I 0 0 0 1 1 3 4 = 4 4 identity matrix. 3 1 2 6 0
More informationChapter 2 Notes, Linear Algebra 5e Lay
Contents.1 Operations with Matrices..................................1.1 Addition and Subtraction.............................1. Multiplication by a scalar............................ 3.1.3 Multiplication
More informationChapter 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 informationSections 1.5, 1.7. Ma 322 Fall Ma 322. Sept
Sections 1.5, 1.7 Ma 322 Fall 213 Ma 322 Sept. 9-13 Summary ˆ Solutions of homogeneous equations AX =. ˆ Using the rank. ˆ Parametric solution of AX = B. ˆ Linear dependence and independence of vectors
More informationLecture 9: Vector Algebra
Lecture 9: Vector Algebra Linear combination of vectors Geometric interpretation Interpreting as Matrix-Vector Multiplication Span of a set of vectors Vector Spaces and Subspaces Linearly Independent/Dependent
More informationSpan and Linear Independence
Span and Linear Independence It is common to confuse span and linear independence, because although they are different concepts, they are related. To see their relationship, let s revisit the previous
More informationMATH 2050 Assignment 6 Fall 2018 Due: Thursday, November 1. x + y + 2z = 2 x + y + z = c 4x + 2z = 2
MATH 5 Assignment 6 Fall 8 Due: Thursday, November [5]. For what value of c does have a solution? Is it unique? x + y + z = x + y + z = c 4x + z = Writing the system as an augmented matrix, we have c R
More informationSections 1.5, 1.7. Ma 322 Spring Ma 322. Jan 24-28
Sections 1.5, 1.7 Ma 322 Spring 217 Ma 322 Jan 24-28 Summary ˆ Text: Solution Sets of Linear Systems (1.5),Linear Independence (1.7) ˆ Solutions of homogeneous equations AX =. ˆ Using the rank. ˆ Parametric
More informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More informationMath 1553, Introduction to Linear Algebra
Learning goals articulate what students are expected to be able to do in a course that can be measured. This course has course-level learning goals that pertain to the entire course, and section-level
More informationMATH240: Linear Algebra Exam #1 solutions 6/12/2015 Page 1
MATH4: Linear Algebra Exam # solutions 6//5 Page Write legibly and show all work. No partial credit can be given for an unjustified, incorrect answer. Put your name in the top right corner and sign the
More informationProblem Point Value Points
Math 70 TUFTS UNIVERSITY October 12, 2015 Linear Algebra Department of Mathematics Sections 1 and 2 Exam I Instructions: No notes or books are allowed. All calculators, cell phones, or other electronic
More informationMath 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 informationProperties 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 informationMath 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 informationFinish section 3.6 on Determinants and connections to matrix inverses. Use last week's notes. Then if we have time on Tuesday, begin:
Math 225-4 Week 7 notes Sections 4-43 vector space concepts Tues Feb 2 Finish section 36 on Determinants and connections to matrix inverses Use last week's notes Then if we have time on Tuesday, begin
More informationAPPM 2360 Exam 2 Solutions Wednesday, March 9, 2016, 7:00pm 8:30pm
APPM 2360 Exam 2 Solutions Wednesday, March 9, 206, 7:00pm 8:30pm ON THE FRONT OF YOUR BLUEBOOK write: () your name, (2) your student ID number, (3) recitation section (4) your instructor s name, and (5)
More informationLINEAR 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 informationEK102 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 informationThe scope of the midterm exam is up to and includes Section 2.1 in the textbook (homework sets 1-4). Below we highlight some of the important items.
AMS 10: Review for the Midterm Exam The scope of the midterm exam is up to and includes Section 2.1 in the textbook (homework sets 1-4). Below we highlight some of the important items. Complex numbers
More informationSSEA Math 51 Track Final Exam August 30, Problem Total Points Score
Name: This is the final exam for the Math 5 track at SSEA. Answer as many problems as possible to the best of your ability; do not worry if you are not able to answer all of the problems. Partial credit
More information