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School: Berkeley
Course: Linear Algebra
MATH 110 QUIZ 5 AUGUST 7, 2014 NAME: No notes or books are allowed on the following quiz. Please justify all of your answers unless indicated otherwise. 1. Consider C3 with the standard inner product. Determine whether each of the following linear operato
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 7: DUE WEDNESDAY, AUGUST 6, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5, 6, 7.A-7.B, any of the handouts on the website, or previously solved homework pro
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 3 SOLUTIONS Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2 and 3.A-D, the Fields and Polynomials supplementary reading handouts, or previously solved homework problems wit
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 5: DUE WEDNESDAY, JULY 23, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5, any of the handouts on the website, or previously solved homework problems without
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 4: DUE WEDNESDAY, JULY 16, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5.A, any of the handouts on the website, or previously solved homework problems witho
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 2: DUE THURSDAY, JULY 3, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2 and 3.A, the Fields and Polynomials supplementary reading handouts, or previously solved homew
School: Berkeley
Course: Linear Algebra
MATH 54 FINAL EXAM PRACTICE QUESTIONS 1. Let M be an m n matrix. In terms of the pivots of M , how do we tell if the matrix equation M x = b always has at least one solution? At most one solution? 2. T/F: In some cases, it is possible for 4 vectors to spa
School: Berkeley
Course: Linear Algebra
MATH 54 MIDTERM 1 REVIEW 1. Make sure you review the denitions of the following terms. (1) augmented matrix vs. coecient matrix (2) echelon form and reduced echelon form (3) pivots, pivot rows, pivot columns (4) parametric form of solution set to Ax = b (
School: Berkeley
Course: Linear Algebra
MATH 110 MIDTERM 2 REVIEW PROBLEMS The midterm will cover Section 3.E, and Chapters 5-6. Below are given some midterm-level problems to give you an idea of what to expect. 1. Determine whether the following statements are true or false. (a) If a linear op
School: Berkeley
Course: Linear Algebra
MIDTERM 1 REVIEW SOLUTIONS 1.Make sure you know the denitions of the following terms. eld vector space over F subspace sum and direct sum linear combination span, linear dependence vs. linear independence nite vs. innite dimensional vector space b
School: Berkeley
Course: Linear Algebra
MATH 110 FINAL REVIEW PROBLEMS The nal will cover the entire class, but at least half of the questions will cover Chapters 7 and 8. Here are some sample problems to give you an idea of the type of coverage. 1. Determine whether the following statements ar
School: Berkeley
Course: Discrete Mathematics
MATH 55 MIDTERM EXTRA PRACTICE QUESTIONS The following questions have all appeared on previous midterms for Math 55. 1. Match each proposition or predicate in the rst column with one that is logically equivalent to it in the second column. (For expression
School: Berkeley
Course: Linear Algebra
MATH 110 SUPPLEMENTARY MATERIAL: POLYNOMIALS If you are interested in working over an arbitrary eld, then Denitions 2.11 and 2.12 in your book arent quite the right denitions for the vector space of polynomials over F. In this handout well give a denition
School: Berkeley
Course: Linear Algebra
MATH 110 SUPPLEMENTARY MATERIAL: CHANGE OF BASIS Let V and W be nite-dimensional vector spaces over a eld F, and T : V W a linear transformation. In this handout we will study what matrices can occur as MC (T ), for B varying bases B of V and C of W . Fir
School: Berkeley
Course: Linear Algebra
MATH 110 SUPPLEMENTARY MATERIAL: SUMMARY OF THE RELATION BETWEEN LINEAR MAPS AND MATRICES In this handout I will summarize the propositions relating linear maps between nitedimensional matrices to matrices. There will be no proofs or examples in this hand
School: Berkeley
Course: Linear Algebra
MATH 110 SUPPLEMENTARY MATERIAL: FIELDS 1. The definition of a field Before we introduce abstract vector spaces, were going to introduce the notion of a eld. Basically, a eld is a set of objects where weve dened how to add, subtract, multiply, and divide
School: Berkeley
Course: Linear Algebra
MATH 110 QUIZ 5 AUGUST 7, 2014 NAME: No notes or books are allowed on the following quiz. Please justify all of your answers unless indicated otherwise. 1. Consider C3 with the standard inner product. Determine whether each of the following linear operato
School: Berkeley
Course: Linear Algebra
MATH 54 4/4 QUIZ There is only one problem on this quiz. 1. Find all dierential functions y satisfying the following conditions: y 4y + 4y = 0 y (0) = 1 y (1) = 0 Solution: The general solution to the given homogeneous equation is ygen = Ae2x + Bxe2x . Pl
School: Berkeley
Course: Linear Algebra
MATH 54 4/11 QUIZ 1. Find a general solution to the following dierential equation: y y = e2t + tet 1 2 MATH 54 4/11 QUIZ 2. Suppose that yp and yq are two solutions to the dierential equation y + 2y + y = tan3 (x). Suppose further that yp (0) = yp (0) = 1
School: Berkeley
Course: Linear Algebra
MATH 54 3/7 QUIZ You have 20 minutes to complete this quiz. No notes or books are allowed. 1. Let P2 be (as usual) the vector space of polynomials of degree 2. Let D : P2 P2 be the linear transformation sending p(t) to p (t). (a) Find the matrix of D rela
School: Berkeley
Course: Linear Algebra
MATH 54 3/14 QUIZ 1. Which of the following are orthogonal matrices? Circle all that apply. 0 1/2 a. 0 1/ 2 b. 0 1 1 0 c. 1 2 2 1 d. 1 0 0 1 Solution: Only b and d are orthogonal matrices. A matrix has to have orthonormal columns to be an orthogonal matri
School: Berkeley
Course: Linear Algebra
MATH 54 2/7/2013 QUIZ You have 15 minutes to complete the following quiz. Notice that there are problems on each side. No notes are allowed. You may use a 4-function calculator if you wish, but nothing more sophisticated. It really isnt necessary, though.
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 7: DUE WEDNESDAY, AUGUST 6, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5, 6, 7.A-7.B, any of the handouts on the website, or previously solved homework pro
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 3 SOLUTIONS Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2 and 3.A-D, the Fields and Polynomials supplementary reading handouts, or previously solved homework problems wit
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 5: DUE WEDNESDAY, JULY 23, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5, any of the handouts on the website, or previously solved homework problems without
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 4: DUE WEDNESDAY, JULY 16, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5.A, any of the handouts on the website, or previously solved homework problems witho
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 2: DUE THURSDAY, JULY 3, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2 and 3.A, the Fields and Polynomials supplementary reading handouts, or previously solved homew
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 6: DUE WEDNESDAY, JULY 30, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5, 6, any of the handouts on the website, or previously solved homework problems with
School: Berkeley
Course: Linear Algebra
MATH 110 QUIZ 5 AUGUST 7, 2014 NAME: No notes or books are allowed on the following quiz. Please justify all of your answers unless indicated otherwise. 1. Consider C3 with the standard inner product. Determine whether each of the following linear operato
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 7: DUE WEDNESDAY, AUGUST 6, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5, 6, 7.A-7.B, any of the handouts on the website, or previously solved homework pro
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 3 SOLUTIONS Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2 and 3.A-D, the Fields and Polynomials supplementary reading handouts, or previously solved homework problems wit
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 5: DUE WEDNESDAY, JULY 23, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5, any of the handouts on the website, or previously solved homework problems without
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 4: DUE WEDNESDAY, JULY 16, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5.A, any of the handouts on the website, or previously solved homework problems witho
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 2: DUE THURSDAY, JULY 3, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2 and 3.A, the Fields and Polynomials supplementary reading handouts, or previously solved homew
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 6: DUE WEDNESDAY, JULY 30, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5, 6, any of the handouts on the website, or previously solved homework problems with
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 1 SOLUTIONS Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapter 1 and 2.A, the Fields supplementary reading handout, or previously solved homework problems without proof. 1. Let F be
School: Berkeley
Course: Discrete Mathematics
MATH 55 4/21 DISCUSSION QUESTIONS 1. (a) (b) (c) (d) (e) How many relations are there on a set A = cfw_a, b with two elements? Of these relations, how many are reexive? How many are symmetric? How many are transitive? How many are equivalence relations? S
School: Berkeley
Course: Discrete Mathematics
School: Berkeley
Course: Discrete Mathematics
MATH 55 4/14 DISCUSSION QUESTIONS 1. If G(x) is the generating function for the sequence cfw_ak , what is the generating function for each of the following sequences? (a) 0, 0, 0, a3 , a4 , a5 , . (b) a0 , 0, a1 , 0, a2 , 0, . (c) 0, 0, 0, a0 , a1 , a2 ,
School: Berkeley
Course: Discrete Mathematics
School: Berkeley
Course: Linear Algebra
MATH 54 FINAL EXAM PRACTICE QUESTIONS 1. Let M be an m n matrix. In terms of the pivots of M , how do we tell if the matrix equation M x = b always has at least one solution? At most one solution? 2. T/F: In some cases, it is possible for 4 vectors to spa
School: Berkeley
Course: Linear Algebra
MATH 54 4/4 QUIZ There is only one problem on this quiz. 1. Find all dierential functions y satisfying the following conditions: y 4y + 4y = 0 y (0) = 1 y (1) = 0 Solution: The general solution to the given homogeneous equation is ygen = Ae2x + Bxe2x . Pl
School: Berkeley
Course: Linear Algebra
MATH 54 4/11 QUIZ 1. Find a general solution to the following dierential equation: y y = e2t + tet 1 2 MATH 54 4/11 QUIZ 2. Suppose that yp and yq are two solutions to the dierential equation y + 2y + y = tan3 (x). Suppose further that yp (0) = yp (0) = 1
School: Berkeley
Course: Linear Algebra
MATH 54 3/7 QUIZ You have 20 minutes to complete this quiz. No notes or books are allowed. 1. Let P2 be (as usual) the vector space of polynomials of degree 2. Let D : P2 P2 be the linear transformation sending p(t) to p (t). (a) Find the matrix of D rela
School: Berkeley
Course: Linear Algebra
MATH 54 3/14 QUIZ 1. Which of the following are orthogonal matrices? Circle all that apply. 0 1/2 a. 0 1/ 2 b. 0 1 1 0 c. 1 2 2 1 d. 1 0 0 1 Solution: Only b and d are orthogonal matrices. A matrix has to have orthonormal columns to be an orthogonal matri
School: Berkeley
Course: Linear Algebra
MATH 54 MIDTERM 1 REVIEW 1. Make sure you review the denitions of the following terms. (1) augmented matrix vs. coecient matrix (2) echelon form and reduced echelon form (3) pivots, pivot rows, pivot columns (4) parametric form of solution set to Ax = b (
School: Berkeley
Course: Linear Algebra
MATH 54 2/7/2013 QUIZ You have 15 minutes to complete the following quiz. Notice that there are problems on each side. No notes are allowed. You may use a 4-function calculator if you wish, but nothing more sophisticated. It really isnt necessary, though.
School: Berkeley
Course: Linear Algebra
MATH 54 2/14/2013 DEFINITIONS QUIZ 1. Below is shown a matrix A, together with the reduced [A|I]. 1 4 2 1 0 0 1 [A|I] = 1 3 3 0 1 0 0 3 6 12 0 0 1 0 echelon form of the augmented matrix 0 6 3 4 0 1 1 1 1 0 0 0 3 6 1 Based on this calculation, answer the
School: Berkeley
Course: Linear Algebra
MATH 54 QUIZ 2/28 1. Let P2 be the set of polynomials of degree 2. Find the change-of-basis matrix P from the CB basis B = cfw_1, t, t2 to the basis C = cfw_1 2t + t2 , 3 5t + 4t2 , 2t + 3t2 . Solution: The matrix P has as columns the vectors [bi ]C . To
School: Berkeley
Course: Linear Algebra
MATH 54 1/31/2013 QUIZ SOLUTIONS You have 15 minutes to complete the following quiz. Notice that there are two problems, one on each side. No notes are allowed. You may use a 4-function calculator if you wish, but nothing more sophisticated. It really isn
School: Berkeley
Course: Linear Algebra
MATH 110 MIDTERM 2 JULY 31, 2014 NAME: You are allowed one 2-sided sheet of notes on this midterm. Please clear your desk of all materials except for this sheet of notes, writing utensils, and scratch paper. You have 65 minutes to complete the test. Answe
School: Berkeley
Course: Linear Algebra
MATH 110 SUPPLEMENTARY MATERIAL: POLYNOMIALS If you are interested in working over an arbitrary eld, then Denitions 2.11 and 2.12 in your book arent quite the right denitions for the vector space of polynomials over F. In this handout well give a denition
School: Berkeley
Course: Linear Algebra
MATH 110 MIDTERM 2 REVIEW PROBLEMS The midterm will cover Section 3.E, and Chapters 5-6. Below are given some midterm-level problems to give you an idea of what to expect. 1. Determine whether the following statements are true or false. (a) If a linear op
School: Berkeley
Course: Linear Algebra
MIDTERM 1 REVIEW SOLUTIONS 1.Make sure you know the denitions of the following terms. eld vector space over F subspace sum and direct sum linear combination span, linear dependence vs. linear independence nite vs. innite dimensional vector space b
School: Berkeley
Course: Linear Algebra
MATH 110 MIDTERM 1 JULY 10, 2014 NAME: You are allowed one 2-sided sheet of notes on this midterm. Please clear your desk of all materials except for this sheet of notes, writing utensils, and scratch paper. You have 65 minutes to complete the test. Answe
School: Berkeley
Course: Linear Algebra
MATH 110 SUPPLEMENTARY MATERIAL: CHANGE OF BASIS Let V and W be nite-dimensional vector spaces over a eld F, and T : V W a linear transformation. In this handout we will study what matrices can occur as MC (T ), for B varying bases B of V and C of W . Fir
School: Berkeley
Course: Linear Algebra
MATH 110 SUPPLEMENTARY MATERIAL: SUMMARY OF THE RELATION BETWEEN LINEAR MAPS AND MATRICES In this handout I will summarize the propositions relating linear maps between nitedimensional matrices to matrices. There will be no proofs or examples in this hand
School: Berkeley
Course: Linear Algebra
MATH 110 CHAPTER 5 REVIEW PROBLEMS 1. Let M be the matrix 4 0 M = 0 0 2 1 0 0 2 7 6 4 3 0 . 0 0 Find 6 distinct M -invariant subspaces of R4 . Solution: In addition to cfw_0 and R4 , the spans of the following lists of vectors are M invariant: (v1 ),
School: Berkeley
Course: Linear Algebra
MATH 110 SUPPLEMENTARY MATERIAL: FIELDS 1. The definition of a field Before we introduce abstract vector spaces, were going to introduce the notion of a eld. Basically, a eld is a set of objects where weve dened how to add, subtract, multiply, and divide
School: Berkeley
Course: Linear Algebra
MATH 110 FINAL REVIEW PROBLEMS The nal will cover the entire class, but at least half of the questions will cover Chapters 7 and 8. Here are some sample problems to give you an idea of the type of coverage. 1. Determine whether the following statements ar
School: Berkeley
Course: Linear Algebra
MATH 110 CHAPTER 1 AND 2 PRACTICE PROBLEMS: SOLUTIONS Below are some brief answers to the practice problems for Chapters 1 and 2. These solutions are only meant for you to check if you are on the right track. Several of the answers would not receive full
School: Berkeley
Course: Discrete Mathematics
MATH 55 MIDTERM EXTRA PRACTICE QUESTIONS The following questions have all appeared on previous midterms for Math 55. 1. Match each proposition or predicate in the rst column with one that is logically equivalent to it in the second column. (For expression
School: Berkeley
Course: Linear Algebra
Math 110 Midterm Exam Professor K. A. Ribet October 31, 2002 Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players, PDAs, and other electronic devices. You may refer to a single 2-sided sheet of notes. Please write yo
School: Berkeley
Course: Linear Algebra
Math 110 PROFESSOR KENNETH A. RIBET C Final Exam December 12, 2002 12:303:30 PM The scalar field F will be the field of real numbers unless otherwise specified. Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players, P
School: Berkeley
Course: Linear Algebra
MATH 54 FINAL EXAM PRACTICE QUESTIONS 1. Let M be an m n matrix. In terms of the pivots of M , how do we tell if the matrix equation M x = b always has at least one solution? At most one solution? 2. T/F: In some cases, it is possible for 4 vectors to spa
School: Berkeley
Course: Linear Algebra
MATH 54 MIDTERM 1 REVIEW 1. Make sure you review the denitions of the following terms. (1) augmented matrix vs. coecient matrix (2) echelon form and reduced echelon form (3) pivots, pivot rows, pivot columns (4) parametric form of solution set to Ax = b (
School: Berkeley
Course: Linear Algebra
MATH 110 MIDTERM 2 REVIEW PROBLEMS The midterm will cover Section 3.E, and Chapters 5-6. Below are given some midterm-level problems to give you an idea of what to expect. 1. Determine whether the following statements are true or false. (a) If a linear op
School: Berkeley
Course: Linear Algebra
MIDTERM 1 REVIEW SOLUTIONS 1.Make sure you know the denitions of the following terms. eld vector space over F subspace sum and direct sum linear combination span, linear dependence vs. linear independence nite vs. innite dimensional vector space b
School: Berkeley
Course: Linear Algebra
MATH 110 FINAL REVIEW PROBLEMS The nal will cover the entire class, but at least half of the questions will cover Chapters 7 and 8. Here are some sample problems to give you an idea of the type of coverage. 1. Determine whether the following statements ar
School: Berkeley
Course: Discrete Mathematics
MATH 55 MIDTERM EXTRA PRACTICE QUESTIONS The following questions have all appeared on previous midterms for Math 55. 1. Match each proposition or predicate in the rst column with one that is logically equivalent to it in the second column. (For expression
School: Berkeley
Course: Linear Algebra
MATH 110 SUPPLEMENTARY MATERIAL: POLYNOMIALS If you are interested in working over an arbitrary eld, then Denitions 2.11 and 2.12 in your book arent quite the right denitions for the vector space of polynomials over F. In this handout well give a denition
School: Berkeley
Course: Linear Algebra
MATH 110 SUPPLEMENTARY MATERIAL: CHANGE OF BASIS Let V and W be nite-dimensional vector spaces over a eld F, and T : V W a linear transformation. In this handout we will study what matrices can occur as MC (T ), for B varying bases B of V and C of W . Fir
School: Berkeley
Course: Linear Algebra
MATH 110 SUPPLEMENTARY MATERIAL: SUMMARY OF THE RELATION BETWEEN LINEAR MAPS AND MATRICES In this handout I will summarize the propositions relating linear maps between nitedimensional matrices to matrices. There will be no proofs or examples in this hand
School: Berkeley
Course: Linear Algebra
MATH 110 SUPPLEMENTARY MATERIAL: FIELDS 1. The definition of a field Before we introduce abstract vector spaces, were going to introduce the notion of a eld. Basically, a eld is a set of objects where weve dened how to add, subtract, multiply, and divide
School: Berkeley
Course: Linear Algebra
MATH 110 QUIZ 5 AUGUST 7, 2014 NAME: No notes or books are allowed on the following quiz. Please justify all of your answers unless indicated otherwise. 1. Consider C3 with the standard inner product. Determine whether each of the following linear operato
School: Berkeley
Course: Linear Algebra
MATH 54 4/4 QUIZ There is only one problem on this quiz. 1. Find all dierential functions y satisfying the following conditions: y 4y + 4y = 0 y (0) = 1 y (1) = 0 Solution: The general solution to the given homogeneous equation is ygen = Ae2x + Bxe2x . Pl
School: Berkeley
Course: Linear Algebra
MATH 54 4/11 QUIZ 1. Find a general solution to the following dierential equation: y y = e2t + tet 1 2 MATH 54 4/11 QUIZ 2. Suppose that yp and yq are two solutions to the dierential equation y + 2y + y = tan3 (x). Suppose further that yp (0) = yp (0) = 1
School: Berkeley
Course: Linear Algebra
MATH 54 3/7 QUIZ You have 20 minutes to complete this quiz. No notes or books are allowed. 1. Let P2 be (as usual) the vector space of polynomials of degree 2. Let D : P2 P2 be the linear transformation sending p(t) to p (t). (a) Find the matrix of D rela
School: Berkeley
Course: Linear Algebra
MATH 54 3/14 QUIZ 1. Which of the following are orthogonal matrices? Circle all that apply. 0 1/2 a. 0 1/ 2 b. 0 1 1 0 c. 1 2 2 1 d. 1 0 0 1 Solution: Only b and d are orthogonal matrices. A matrix has to have orthonormal columns to be an orthogonal matri
School: Berkeley
Course: Linear Algebra
MATH 54 2/7/2013 QUIZ You have 15 minutes to complete the following quiz. Notice that there are problems on each side. No notes are allowed. You may use a 4-function calculator if you wish, but nothing more sophisticated. It really isnt necessary, though.
School: Berkeley
Course: Linear Algebra
MATH 54 2/14/2013 DEFINITIONS QUIZ 1. Below is shown a matrix A, together with the reduced [A|I]. 1 4 2 1 0 0 1 [A|I] = 1 3 3 0 1 0 0 3 6 12 0 0 1 0 echelon form of the augmented matrix 0 6 3 4 0 1 1 1 1 0 0 0 3 6 1 Based on this calculation, answer the
School: Berkeley
Course: Linear Algebra
MATH 54 QUIZ 2/28 1. Let P2 be the set of polynomials of degree 2. Find the change-of-basis matrix P from the CB basis B = cfw_1, t, t2 to the basis C = cfw_1 2t + t2 , 3 5t + 4t2 , 2t + 3t2 . Solution: The matrix P has as columns the vectors [bi ]C . To
School: Berkeley
Course: Linear Algebra
MATH 54 1/31/2013 QUIZ SOLUTIONS You have 15 minutes to complete the following quiz. Notice that there are two problems, one on each side. No notes are allowed. You may use a 4-function calculator if you wish, but nothing more sophisticated. It really isn
School: Berkeley
Course: Linear Algebra
MATH 110 MIDTERM 2 JULY 31, 2014 NAME: You are allowed one 2-sided sheet of notes on this midterm. Please clear your desk of all materials except for this sheet of notes, writing utensils, and scratch paper. You have 65 minutes to complete the test. Answe
School: Berkeley
Course: Linear Algebra
MATH 110 MIDTERM 1 JULY 10, 2014 NAME: You are allowed one 2-sided sheet of notes on this midterm. Please clear your desk of all materials except for this sheet of notes, writing utensils, and scratch paper. You have 65 minutes to complete the test. Answe
School: Berkeley
Course: Linear Algebra
Math 110 PROFESSOR KENNETH A. RIBET C Final Exam December 12, 2002 12:303:30 PM The scalar field F will be the field of real numbers unless otherwise specified. Please put away all books, calculators, electronic games, cell phones, pagers, .mp3 players, P
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 7: DUE WEDNESDAY, AUGUST 6, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5, 6, 7.A-7.B, any of the handouts on the website, or previously solved homework pro
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 3 SOLUTIONS Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2 and 3.A-D, the Fields and Polynomials supplementary reading handouts, or previously solved homework problems wit
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 5: DUE WEDNESDAY, JULY 23, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5, any of the handouts on the website, or previously solved homework problems without
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 4: DUE WEDNESDAY, JULY 16, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5.A, any of the handouts on the website, or previously solved homework problems witho
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 2: DUE THURSDAY, JULY 3, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2 and 3.A, the Fields and Polynomials supplementary reading handouts, or previously solved homew
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 6: DUE WEDNESDAY, JULY 30, 2014 Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapters 1, 2, 3, 4, 5, 6, any of the handouts on the website, or previously solved homework problems with
School: Berkeley
Course: Linear Algebra
MATH 110 HWK 1 SOLUTIONS Solve the following problems. Prove all assertions. For each problem, you may use any of the results in Chapter 1 and 2.A, the Fields supplementary reading handout, or previously solved homework problems without proof. 1. Let F be
School: Berkeley
Course: Discrete Mathematics
MATH 55 4/21 DISCUSSION QUESTIONS 1. (a) (b) (c) (d) (e) How many relations are there on a set A = cfw_a, b with two elements? Of these relations, how many are reexive? How many are symmetric? How many are transitive? How many are equivalence relations? S
School: Berkeley
Course: Discrete Mathematics
School: Berkeley
Course: Discrete Mathematics
MATH 55 4/14 DISCUSSION QUESTIONS 1. If G(x) is the generating function for the sequence cfw_ak , what is the generating function for each of the following sequences? (a) 0, 0, 0, a3 , a4 , a5 , . (b) a0 , 0, a1 , 0, a2 , 0, . (c) 0, 0, 0, a0 , a1 , a2 ,
School: Berkeley
Course: Discrete Mathematics
School: Berkeley
Course: Linear Algebra
MATH 110 CHAPTER 5 REVIEW PROBLEMS 1. Let M be the matrix 4 0 M = 0 0 2 1 0 0 2 7 6 4 3 0 . 0 0 Find 6 distinct M -invariant subspaces of R4 . Solution: In addition to cfw_0 and R4 , the spans of the following lists of vectors are M invariant: (v1 ),
School: Berkeley
Course: Linear Algebra
MATH 110 CHAPTER 1 AND 2 PRACTICE PROBLEMS: SOLUTIONS Below are some brief answers to the practice problems for Chapters 1 and 2. These solutions are only meant for you to check if you are on the right track. Several of the answers would not receive full