Math 544, Exam 2 Information.


 Darleen Hodges
 1 years ago
 Views:
Transcription
1 Math 544, Exam 2 Information. 10/12/10, LC 115, 2:003:15. Exam 2 will be based on: Sections 1.7, 1.9, 3.2, 3.3, 3.4; The corresponding assigned homework problems (see boylan/sccourses/544fa10/544.html) At minimum, you need to understand how to do the homework problems. Lecture notes: 9/1610/7. Topic List (not necessarily comprehensive): You will need to know: theorems, results, and definitions from class. 1.7: Linear independence and nonsingular matrices. You will need to know/identify the following terms: Linear combination of vectors. Linear dependence of a set of vectors; linear independence of a set of vectors. Let S = { v 1,..., v n } R m. The set S is linearly independent the vector equation x 1 v x n v n = 0 x 1 has only the trivial solution x =. x n = 0. Conversely, S is linearly dependent the vector equation has a nontrivial solution x 0. The vector equation can be written in matrix form as A x = ( v 1 v n ) x = 0, where the vectors in S form the columns of A Mat n n (R). Only sets of vectors can be linearly dependent/linearly independent. It does not make sense to speak of matrices or systems of equations being linearly dependent/independent. Notes: 1. Any set of vectors S with 0 S is linearly dependent. 2. Suppose that v 0. Then { v} is linearly independent. 3. Two vectors are linearly dependent if and only if they are scalar multiples of each other.
2 Singular matrices; nonsingular matrices. Let A Mat n n (R). The A is nonsingular the equation A x = 0 has only the trivial solution, x = 0. Conversely, A is singular this equation has a nontrivial solution x 0. Only square matrices can be singular/nonsingular. It does not make sense to speak of a system of equations or a set of vectors as being singular/nonsingular. Note: Similarly, only systems of linear equations can consistent/inconsistent. It does not make sense to speak of matrices or sets of vectors as being consistent/inconsistent. Theorem. Suppose that S = { v 1,..., v n } R m with m < n. dependent set. Then S is a linearly Facts: Let A and B Mat n n (R). 1. AB is nonsingular if and only if A and B are nonsingular. 2. AB is singular if and only if one (or both) of A and B is singular. 1.9: Matrix inverses and their properties. Matrix inverses. A matrix A Mat n n (R) is invertible if and only if there exists B Mat n n (R) with BA = I n = AB. We say that B is the inverse of A, and we write B = A Only square matrices are allowed to be invertible. 2. Suppose that A Mat n n (R) is invertible. Then the inverse, A 1, is unique. Theorem. Let A Mat n n. Then the following conditions on A are equivalent. A is nonsingular. A x = 0 has only the trivial solution, x = 0. The columns of A are a linearly independent set of vectors. For all b R n, A x = b has a unique solution. A is invertible. A is rowequivalent to I n. Facts: Suppose that A Mat n n (R) is invertible. To compute A 1, form the augmented matrix (A I n ) Mat n 2n (R). Apply the Gauss Jordan algorithm to convert it to its reduced echelon form, (I n A 1 ). For all b R n, the system A x = b is consistent (i.e., it has a solution); the unique solution is x = A 1 b. 2
3 Theorem. Suppose that A, B Mat n n (R) are invertible. Then we have 1. (A 1 ) 1 = A. 2. (AB) 1 = B 1 A 1. (Note: (AB) 1 = A 1 B 1 AB = BA.) 3. Let k 0 in R. Then we have (ka) 1 = 1 k A (A T ) 1 = (A 1 ) T. ( ) a b Proposition. Let Mat c d 2 2 (R). Then we have ad bc = 0 = A is not invertible. ad bc 0 = A is invertible. 3.2: Vector space properties of R n. Vector space. A vector space is a collection of objects called vectors and a collection of constants called scalars together with the operations of vector addition (+) and scalar multiplication ( ) which satisfy the following axioms: Two closure axioms: closure under + and. Four addition axioms: associativity, identity, inverses, commutativity. Four scalar multiplication axioms: associativity, distributivity, identity. Subspace. Suppose that V is a vector space. A subset W V is a subspace of V if and only if W is itself a vector space (with the same scalars, addition, and multiplication as V ). 1. Let V be a vector space. Then { 0} and V are subspaces of V. These are the trivial subspaces of V. 2. The nontrivial subspaces of R 2 are lines through the origin; the nontrivial subspaces of R 3 are lines through the origin and planes through the origin. Theorem. Let V be a vector space, and let W V. Then W is a subspace of V if and only if W satisfies the following three axioms: 1. { 0} W. 2. For all x, y W, we have x + y W. 3. For all a R and for all x W, we have a x W. Note: One can combine axioms 2 and 3: for all a R and for all x, y W, we have a x + y W. Question: How does one determine whether a subset W of a vector space V is a subspace? To show that W is a subspace, you need to verify the three subspace axioms. To show that W is not a subspace, it suffices to provide a simple numerical example in which one of the axioms is violated. 3
4 Note: If a subset W of a vector space V is defined by a system of linear homogeneous equations satisfied by the coordinates of vectors in W, then it is a subspace. Facts: Suppose that U and V are subspaces of R n. Then we have: 1. U + V = { u + v : u U, v V } is always a subspace of R n. 2. U V is always a subspace of R n. 3. U V is not generally a subspace of R n. 3.3: Subspaces. Span. The span of a set S = { v 1,..., v r } R n is the set Span(S) = {all linear combinations of vectors in S} = { y = a 1 v a r v r : a 1,..., a r R} R n. Examples. Let v 0. Then Span{ v} is a line through the origin in the direction of v. Let u, v be nonzero vectors in R n. Then u Span{ v} if and only if u and v determine the same line. Theorem. Suppose that S = { v 1,..., v r } R n. Then Span(S) is a subspace of R n. Let A Mat m n (R). Important subspaces associated to A are: The null space of A is Null(A) = { x R n : A x = 0}. It is a subspace of R n. The range of A is Range(A) = { y R m : there exists x R n such that A x = y}. It is a subspace of R m. The column space of A is Col(A) = Span{columns of A}. It is a subspace of R m. The row space of A is Row(A) = Span{rows of A}. It is a subspace of R n. Theorem. Suppose that A Mat m n (R). Then we have Range(A) = Col(A) = Row(A T ) R m. Theorem. Suppose that A, B Mat m n (R) are rowequivalent. Then we have Row(A) = Row(B). Problem: Given a matrix A Mat m n (R), give a basis (algebraic description) of the important subspaces associated to A. 3.4: Bases. A set of vectors S spans a subspace W of a vector space V if and only if Span(S) = W. Theorem. Suppose that Span(S) = W. 1. Suppose that S is linearly dependent. Then there exists T S with T < S for which Span(T ) = W. 4
5 2. Suppose that S is linearly independent. Then no set T with T < S has Span(T ) = W. Basis. Let W { 0} be a subspace of R n. Then a subset S W is a basis if and only if 1. S is linearly independent. 2. Span(S) = W. Note: Bases are not unique. Given a matrix A Mat m n (R), compute bases for Null(A), Range(A), Col(A), and Row(A). To do this, you first compute the reduced echelon form of A. Call it B. Row(A): The nonzero rows of B form a basis for Row(A). Col(A): The columns of B with the leading 1 s correspond to the columns of A which form a basis. Range(A): Since the range and column space of A agree, you compute Range(A) just as you would Col(A). Null(A): Null(A) is the set of solutions to the homogenous system A x = 0. Therefore, begin by solving A x = 0. Convert your solution to vector form. i.e., write your solution as Null(A) = Span(S). Verify that the vectors in S are linearly independent (which is usually easy to do and requires little or no justification). Theorem. Suppose that B is a basis for a subspace W or R n. Then every vector x W is expressible as linear combination of vectors from B in a unique way. 5
MATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.
MATH 2331 Linear Algebra Section 2.1 Matrix Operations Definition: A : m n, B : n p ( 1 2 p ) ( 1 2 p ) AB = A b b b = Ab Ab Ab Example: Compute AB, if possible. 1 Rowcolumn rule: ijth entry of AB:
More informationSolutions to Homework 5  Math 3410
Solutions to Homework 5  Math 34 (Page 57: # 489) Determine whether the following vectors in R 4 are linearly dependent or independent: (a) (, 2, 3, ), (3, 7,, 2), (, 3, 7, 4) Solution From x(, 2, 3,
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 informationThe definition of a vector space (V, +, )
The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element
More informationReview 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 informationReview Notes for Midterm #2
Review Notes for Midterm #2 Joris Vankerschaver This version: Nov. 2, 200 Abstract This is a summary of the basic definitions and results that we discussed during class. Whenever a proof is provided, I
More informationMATH 300, Second Exam REVIEW SOLUTIONS. NOTE: You may use a calculator for this exam You only need something that will perform basic arithmetic.
MATH 300, Second Exam REVIEW SOLUTIONS NOTE: You may use a calculator for this exam You only need something that will perform basic arithmetic. [ ] [ ] 2 2. Let u = and v =, Let S be the parallelegram
More informationMath 2174: Practice Midterm 1
Math 74: Practice Midterm Show your work and explain your reasoning as appropriate. No calculators. One page of handwritten notes is allowed for the exam, as well as one blank page of scratch paper.. Consider
More informationChapter 3. Directions: For questions 111 mark each statement True or False. Justify each answer.
Chapter 3 Directions: For questions 111 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 informationIf A is a 4 6 matrix and B is a 6 3 matrix then the dimension of AB is A. 4 6 B. 6 6 C. 4 3 D. 3 4 E. Undefined
Question 1 If A is a 4 6 matrix and B is a 6 3 matrix then the dimension of AB is A. 4 6 B. 6 6 C. 4 3 D. 3 4 E. Undefined Quang T. Bach Math 18 October 18, 2017 1 / 17 Question 2 1 2 Let A = 3 4 1 2 3
More informationMath 369 Exam #2 Practice Problem Solutions
Math 369 Exam #2 Practice Problem Solutions 2 5. Is { 2, 3, 8 } a basis for R 3? Answer: No, it is not. To show that it is not a basis, it suffices to show that this is not a linearly independent set.
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 informationLecture 22: Section 4.7
Lecture 22: Section 47 Shuanglin Shao December 2, 213 Row Space, Column Space, and Null Space Definition For an m n, a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn, the vectors r 1 = [ a 11 a 12 a 1n
More informationPractice 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 informationKevin James. MTHSC 3110 Section 4.3 Linear Independence in Vector Sp
MTHSC 3 Section 4.3 Linear Independence in Vector Spaces; Bases Definition Let V be a vector space and let { v. v 2,..., v p } V. If the only solution to the equation x v + x 2 v 2 + + x p v p = is the
More informationMidterm #2 Solutions
Naneh Apkarian Math F Winter Midterm # Solutions Here is a solution key for the second midterm. The solutions presented here are more complete and thorough than your responses needed to be  in order to
More informationMATH 152 Exam 1Solutions 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 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 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 onetoone and onto 3 For each b Y there is exactly one a
More informationMATH 2030: ASSIGNMENT 4 SOLUTIONS
MATH 23: ASSIGNMENT 4 SOLUTIONS More on the LU factorization Q.: pg 96, q 24. Find the P t LU factorization of the matrix 2 A = 3 2 2 A.. By interchanging row and row 4 we get a matrix that may be easily
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 xaxes by 6
More informationChapter 1. Vectors, Matrices, and Linear Spaces
1.6 Homogeneous Systems, Subspaces and Bases 1 Chapter 1. Vectors, Matrices, and Linear Spaces 1.6. Homogeneous Systems, Subspaces and Bases Note. In this section we explore the structure of the solution
More informationMath 123, Week 5: Linear Independence, Basis, and Matrix Spaces. Section 1: Linear Independence
Math 123, Week 5: Linear Independence, Basis, and Matrix Spaces Section 1: Linear Independence Recall that every row on the lefthand side of the coefficient matrix of a linear system A x = b which could
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW SPENCER BECKERKAHN 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 information2018 Fall 2210Q Section 013 Midterm Exam II Solution
08 Fall 0Q Section 0 Midterm Exam II Solution True or False questions points 0 0 points) ) Let A be an n n matrix. If the equation Ax b has at least one solution for each b R n, then the solution is unique
More information(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 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 informationDr. Abdulla Eid. Section 4.2 Subspaces. Dr. Abdulla Eid. MATHS 211: Linear Algebra. College of Science
Section 4.2 Subspaces College of Science MATHS 211: Linear Algebra (University of Bahrain) Subspaces 1 / 42 Goal: 1 Define subspaces. 2 Subspace test. 3 Linear Combination of elements. 4 Subspace generated
More informationMath 3191 Applied Linear Algebra
Math 9 Applied Linear Algebra Lecture : Null and Column Spaces Stephen Billups University of Colorado at Denver Math 9Applied Linear Algebra p./8 Announcements Study Guide posted HWK posted Math 9Applied
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 304 Linear Algebra Lecture 10: Linear independence. Wronskian.
MATH 304 Linear Algebra Lecture 10: Linear independence. Wronskian. Spanning set Let S be a subset of a vector space V. Definition. The span of the set S is the smallest subspace W V that contains S. If
More informationMTH 362: Advanced Engineering Mathematics
MTH 362: Advanced Engineering Mathematics Lecture 5 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 26, 2017 1 Linear Independence and Dependence of Vectors
More informationMATH 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 informationLecture 3q Bases for Row(A), Col(A), and Null(A) (pages )
Lecture 3q Bases for Row(A), Col(A), and Null(A) (pages 576) Recall that the basis for a subspace S is a set of vectors that both spans S and is linearly independent. Moreover, we saw in section 2.3 that
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 information3.4 Elementary Matrices and Matrix Inverse
Math 220: Summer 2015 3.4 Elementary Matrices and Matrix Inverse A n n elementary matrix is a matrix which is obtained from the n n identity matrix I n n by a single elementary row operation. Elementary
More informationMath 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 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 information(b) The nonzero rows of R form a basis of the row space. Thus, a basis is [ ], [ ], [ ]
Exam will be on Monday, October 6, 27. The syllabus for Exam 2 consists of Sections Two.III., Two.III.2, Two.III.3, Three.I, and Three.II. You should know the main definitions, results and computational
More informationMATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics
MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam This study sheet will not be allowed during the test Books and notes will not be allowed during the test Calculators and cell phones
More informationInverting Matrices. 1 Properties of Transpose. 2 Matrix Algebra. P. Danziger 3.2, 3.3
3., 3.3 Inverting Matrices P. Danziger 1 Properties of Transpose Transpose has higher precedence than multiplication and addition, so AB T A ( B T and A + B T A + ( B T As opposed to the bracketed expressions
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 informationb for the linear system x 1 + x 2 + a 2 x 3 = a x 1 + x 3 = 3 x 1 + x 2 + 9x 3 = 3 ] 1 1 a 2 a
Practice Exercises for Exam Exam will be on Monday, September 8, 7. The syllabus for Exam consists of Sections One.I, One.III, Two.I, and Two.II. You should know the main definitions, results and computational
More informationMath 415 Exam I. Name: Student ID: Calculators, books and notes are not allowed!
Math 415 Exam I Calculators, books and notes are not allowed! Name: Student ID: Score: Math 415 Exam I (20pts) 1. Let A be a square matrix satisfying A 2 = 2A. Find the determinant of A. Sol. From A 2
More informationMath 3C Lecture 20. John Douglas Moore
Math 3C Lecture 20 John Douglas Moore May 18, 2009 TENTATIVE FORMULA I Midterm I: 20% Midterm II: 20% Homework: 10% Quizzes: 10% Final: 40% TENTATIVE FORMULA II Higher of two midterms: 30% Homework: 10%
More information6. The scalar multiple of u by c, denoted by c u is (also) in V. (closure under scalar multiplication)
Definition: A subspace of a vector space V is a subset H of V which is itself a vector space with respect to the addition and scalar multiplication in V. As soon as one verifies a), b), c) below for H,
More informationChapter 7. Linear Algebra: Matrices, Vectors,
Chapter 7. Linear Algebra: Matrices, Vectors, Determinants. Linear Systems Linear algebra includes the theory and application of linear systems of equations, linear transformations, and eigenvalue problems.
More informationMath 313 Chapter 1 Review
Math 313 Chapter 1 Review Howard Anton, 9th Edition May 2010 Do NOT write on me! Contents 1 1.1 Introduction to Systems of Linear Equations 2 2 1.2 Gaussian Elimination 3 3 1.3 Matrices and Matrix Operations
More informationSUMMARY 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 informationPRACTICE 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 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 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 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 informationMATH 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 informationMath 22 Fall 2018 Midterm 2
Math 22 Fall 218 Midterm 2 October 23, 218 NAME: SECTION (check one box): Section 1 (S. Allen 12:5) Section 2 (A. Babei 2:1) Instructions: 1. Write your name legibly on this page, and indicate your section
More informationSECTION 3.3. PROBLEM 22. The null space of a matrix A is: N(A) = {X : AX = 0}. Here are the calculations of AX for X = a,b,c,d, and e. =
SECTION 3.3. PROBLEM. The null space of a matrix A is: N(A) {X : AX }. Here are the calculations of AX for X a,b,c,d, and e. Aa [ ][ ] 3 3 [ ][ ] Ac 3 3 [ ] 3 3 [ ] 4+4 6+6 Ae [ ], Ab [ ][ ] 3 3 3 [ ]
More informationMath 308 Discussion Problems #4 Chapter 4 (after 4.3)
Math 38 Discussion Problems #4 Chapter 4 (after 4.3) () (after 4.) Let S be a plane in R 3 passing through the origin, so that S is a twodimensional subspace of R 3. Say that a linear transformation T
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 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 informationFind the solution set of 2x 3y = 5. Answer: We solve for x = (5 + 3y)/2. Hence the solution space consists of all vectors of the form
Math 2 Homework #7 March 4, 2 7.3.3. Find the solution set of 2x 3y = 5. Answer: We solve for x = (5 + 3y/2. Hence the solution space consists of all vectors of the form ( ( ( ( x (5 + 3y/2 5/2 3/2 x =
More informationMath Linear algebra, Spring Semester Dan Abramovich
Math 52 0  Linear algebra, Spring Semester 20122013 Dan Abramovich Fields. We learned to work with fields of numbers in school: Q = fractions of integers R = all real numbers, represented by infinite
More informationMath 2030 Assignment 5 Solutions
Math 030 Assignment 5 Solutions Question 1: Which of the following sets of vectors are linearly independent? If the set is linear dependent, find a linear dependence relation for the vectors (a) {(1, 0,
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationMATH 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 informationMath 4377/6308 Advanced Linear Algebra
2. Linear Transformations Math 4377/638 Advanced Linear Algebra 2. Linear Transformations, Null Spaces and Ranges Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/
More informationSept. 26, 2013 Math 3312 sec 003 Fall 2013
Sept. 26, 2013 Math 3312 sec 003 Fall 2013 Section 4.1: Vector Spaces and Subspaces Definition A vector space is a nonempty set V of objects called vectors together with two operations called vector addition
More informationLecture 13: Row and column spaces
Spring 2018 UWMadison Lecture 13: Row and column spaces 1 The column space of a matrix 1.1 Definition The column space of matrix A denoted as Col(A) is the space consisting of all linear combinations
More information1 Systems of equations
Highlights from linear algebra David Milovich, Math 2 TA for sections 6 November, 28 Systems of equations A leading entry in a matrix is the first (leftmost) nonzero entry of a row. For example, the leading
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 information7.6 The Inverse of a Square Matrix
7.6 The Inverse of a Square Matrix Copyright Cengage Learning. All rights reserved. What You Should Learn Verify that two matrices are inverses of each other. Use GaussJordan elimination to find inverses
More informationFall 2016 MATH*1160 Final Exam
Fall 2016 MATH*1160 Final Exam Last name: (PRINT) First name: Student #: Instructor: M. R. Garvie Dec 16, 2016 INSTRUCTIONS: 1. The exam is 2 hours long. Do NOT start until instructed. You may use blank
More informationStudy 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 informationVector space and subspace
Vector space and subspace Math 112, week 8 Goals: Vector space, subspace, span. Null space, column space. Linearly independent, bases. Suggested Textbook Readings: Sections 4.1, 4.2, 4.3 Week 8: Vector
More informationORIE 6300 Mathematical Programming I August 25, Recitation 1
ORIE 6300 Mathematical Programming I August 25, 2016 Lecturer: Calvin Wylie Recitation 1 Scribe: Mateo Díaz 1 Linear Algebra Review 1 1.1 Independence, Spanning, and Dimension Definition 1 A (usually infinite)
More informationHomework 11/Solutions. (Section 6.8 Exercise 3). Which pairs of the following vector spaces are isomorphic?
MTH 94 Linear Algebra I F Section Exercises 6.8,4,5 7.,b 7.,, Homework /Solutions (Section 6.8 Exercise ). Which pairs of the following vector spaces are isomorphic? R 7, R, M(, ), M(, 4), M(4, ), P 6,
More 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 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 information1. 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 informationProblem 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 informationLECTURES 14/15: LINEAR INDEPENDENCE AND BASES
LECTURES 14/15: LINEAR INDEPENDENCE AND BASES MA1111: LINEAR ALGEBRA I, MICHAELMAS 2016 1. Linear Independence We have seen in examples of span sets of vectors that sometimes adding additional vectors
More informationMath 290, Midterm IIkey
Math 290, Midterm IIkey Name (Print): (first) Signature: (last) The following rules apply: There are a total of 20 points on this 50 minutes exam. This contains 7 pages (including this cover page) and
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 informationMath Linear Algebra Final Exam Review Sheet
Math 151 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a componentwise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationChapter 3. Vector spaces
Chapter 3. Vector spaces Lecture notes for MA1111 P. Karageorgis pete@maths.tcd.ie 1/22 Linear combinations Suppose that v 1,v 2,...,v n and v are vectors in R m. Definition 3.1 Linear combination We say
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 informationLinear Independence. Linear Algebra MATH Linear Algebra LI or LD Chapter 1, Section 7 1 / 1
Linear Independence Linear Algebra MATH 76 Linear Algebra LI or LD Chapter, Section 7 / Linear Combinations and Span Suppose s, s,..., s p are scalars and v, v,..., v p are vectors (all in the same space
More informationGENERAL VECTOR SPACES AND SUBSPACES [4.1]
GENERAL VECTOR SPACES AND SUBSPACES [4.1] General vector spaces So far we have seen special spaces of vectors of n dimensions denoted by R n. It is possible to define more general vector spaces A vector
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 informationLecture 14: Orthogonality and general vector spaces. 2 Orthogonal vectors, spaces and matrices
Lecture 14: Orthogonality and general vector spaces 1 Symmetric matrices Recall the definition of transpose A T in Lecture note 9. Definition 1.1. If a square matrix S satisfies then we say S is a symmetric
More informationMath 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 informationICS 6N Computational Linear Algebra Vector Space
ICS 6N Computational Linear Algebra Vector Space Xiaohui Xie University of California, Irvine xhx@uci.edu Xiaohui Xie (UCI) ICS 6N 1 / 24 Vector Space Definition: A vector space is a non empty set V of
More informationThe scope of the midterm exam is up to and includes Section 2.1 in the textbook (homework sets 14). 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 14). Below we highlight some of the important items. Complex numbers
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 informationMath 308 Practice Final Exam Page and vector y =
Math 308 Practice Final Exam Page Problem : Solving a linear equation 2 0 2 5 Given matrix A = 3 7 0 0 and vector y = 8. 4 0 0 9 (a) Solve Ax = y (if the equation is consistent) and write the general solution
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
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 informationMTH501 Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~
MTH501 Linear Algebra MCQS MIDTERM EXAMINATION ~ LIBRIANSMINE ~ Question No: 1 (Marks: 1) If for a linear transformation the equation T(x) =0 has only the trivial solution then T is Onetoone Onto Question
More informationLinear Algebra I Lecture 8
Linear Algebra I Lecture 8 Xi Chen 1 1 University of Alberta January 25, 2019 Outline 1 2 GaussJordan Elimination Given a system of linear equations f 1 (x 1, x 2,..., x n ) = 0 f 2 (x 1, x 2,..., x n
More informationWhat is on this week. 1 Vector spaces (continued) 1.1 Null space and Column Space of a matrix
Professor Joana Amorim, jamorim@bu.edu What is on this week Vector spaces (continued). Null space and Column Space of a matrix............................. Null Space...........................................2
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 5228734. Normally, homework
More information