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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