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UC Davis | MATH 67
Advanced Linear Algebra
Professors
- Schilling,
- Tracy,
- Fukuda,
- Xie,
- Dan Romik,
- Andrew Berget,
- Lankham
69 sample documents related to MATH 67
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UC Davis MATH 67
DEPARTMENT OF MATHEMATICS SYLLABUS Course # Price: Prepared by: Lec # 1 2, 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Nachtergaele, Schilling, and Lankham Linear Algebra as an Introducti -
UC Davis MATH 67
Math32a R. Kozhan Final Summary Material for the Final includes everything that we covered in the course. Namely: Chapter 12; Chapter 13 (except Section 13.4); Sections 10.110.2. Basics of the conic sections (Section 10.5 without foci and directrices) -
UC Davis MATH 67
Math32a R. Kozhan Midterm2 Summary Midterm2 will be focused on the sections listed below, and will not explicitly test the knowledge of the material included for Midterm1. However the student is assumed to know it and be able to use it when needed. The ma -
UC Davis MATH 67
M IDTERM 1 May 20, 2002 Instructions. Please show your work. You will receive little or no credit for an answer not accompanied by appropriate explanations, even if the answer is correct. If you have a question about a particular problem, please raise you -
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UC Davis MATH 67
32A Stovall Midterm 2 Name: November 9 Section: Tu/Th Duncan/Melissa I certify that the work appearing on this exam is completely my own: Signature: There are 5 problems and a total of 8 pages. Please make sure that you have all pages. Please show your -
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UC Davis MATH 67
Solutions to HW #1 Math 67 UC Davis, Fall 2011 1. (a) x + y = 22 x 2y = 7 = x = 17, y = 5 (b) 4x + 2y = 10 2x y = 10 = 5 = 2x y = 10 = no solutions (c) x + y = 10 5x + 5y = 50 = x + y = 10 = innitely many solutions (d) x + 2y + 3z = 11 2x + 6z = 14 x+y+z -
UC Davis MATH 67
Homework Assignment #2 Homework due. Math 67 UC Davis, Fall 2011 Tuesday 10/11/11 at discussion section. Reading material. Read Sections 3.24.3 in the textbook (the proof of the Fundamental Theorem of Algebra in section 3.1 is optional material). Problems -
UC Davis MATH 67
Solutions to HW #2 Math 67 UC Davis, Fall 2011 1. (a) We rewrite the second and third equations in the system as (5 z )x + 2y = 0 2x + (8 z )y = 0 We are looking for a solution other than the zero solution x = y = 0. By a previous exercise mentioned in th -
UC Davis MATH 67
Homework Assignment #3 Homework due. Math 67 UC Davis, Fall 2011 Tuesday 10/18/11 at discussion section. Reading material. Read Sections 4.45.4 and 12.3.112.3.2 in the textbook. Problems 1. Solve the following problems in the textbook: (a) Calculational e -
UC Davis MATH 67
Solutions to HW #3 Math 67 UC Davis, Fall 2011 1. Solve the following problems in the textbook: (a) Calculational exercises 1, 2, 3 in Chapter 5. (b) Proof-writing exercise 4 in Chapter 5. Solution to calculational exercise 1. If a1 , a2 , a3 are scalars -
UC Davis MATH 67
Homework Assignment #4 Math 67 UC Davis, Fall 2011 This homework will not be collected. It is meant to help you prepare for the midterm exam on October 24. Reading material. Read Sections 5.46.5. Problems 1. Solve the following problems in the textbook: ( -
UC Davis MATH 67
Solutions to HW #4 Math 67 UC Davis, Fall 2011 1. Solve the following problems in the textbook: (a) Proof-writing exercise 6 in Chapter 5. Solution. Since U + V R9 , we have dim(U + V ) 9, and therefore, by theorem 5.4.6 in the textbook, dim(U V ) = dim(U -
UC Davis MATH 67
Homework Assignment #5 Homework due. Math 67 UC Davis, Fall 2011 Tuesday 11/01/11 at discussion section. Reading material. Read Sections 6.56.6 in the textbook. Problems 1. Compute the coordinate vector [v ]B , where: (a) v = (1, 0, 1), B = cfw_(1, 0, 0), -
UC Davis MATH 67
Solutions to HW #5 Math 67 UC Davis, Fall 2011 1. Compute the coordinate vector [v ]B , where: (a) v = (1, 0, 1), B = cfw_(1, 0, 0), (1, 1, 0), (1, 1, 1). 1 Solution. [v ]B = 1 , since v = 1 (1, 0, 0) + (1) (1, 1, 0) + 1 (1, 1, 1). 1 (b) v = (1, 0, 1), B -
UC Davis MATH 67
Homework Assignment #6 Homework due. Math 67 UC Davis, Fall 2011 Tuesday 11/08/11 at discussion section. Reading material. Read Sections 6.66.7, 8.1 in the textbook. Problems 1. For each of the following pairs of matrices A, B , compute the matrix product -
UC Davis MATH 67
Solutions to HW #6 Math 67 UC Davis, Fall 2011 1. For each of the following pairs of matrices A, B , compute the matrix product AB and the matrix product BA, or for each of these products indicate if it is undened. 0 1 123 (a) A = ,B = 1 1 654 22 Solutio -
UC Davis MATH 67
Homework Assignment #7 Homework due. Math 67 UC Davis, Fall 2011 Tuesday 11/15/11 at discussion section. Reading material. Read Chapter 8 in the textbook. Problems 1. (a) Compute the composition of permutations, where: i. = 123456 136245 ii. = 12345 21435 -
UC Davis MATH 67
Solutions to Midterm Exam Math 67 UC Davis, Fall 2011 Problem 1 (a) Find the general form of the x1 2x1 x1 x1 solution of the following system of linear equations: + x2 x2 + x2 + 2 x2 5x3 + 7 x3 x3 8x3 + 2 x4 24x4 2x4 + 2 x4 = = = = 0 0 0 0 Soluti -
UC Davis MATH 67
Math 67: Modern Linear Algebra (UC Davis, Fall 2011) Summary of lectures Prof. Dan Romik [Version of November 4, 2011 this document will be updated occasionally with new material] Lecture 1 (9/23/11) High-level description of course goals: 1. linear alge -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Midterm Exam February 9, 2011 Instructions. You have 50 minutes to complete the 5 problems on the exam. All of your answers should be written in complete English sentences. The symbol V will always denote a vector space -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 1 Problem 1. Write down matrix equations of the form Ax = b that come from the integrals dx dx . x2 5x + 6 x2 + 9 The second matrix will have complex (imaginary) numbers in it. Both matrices will be 2-by-2. Prob -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 1 Problem 1. Write down matrix equations of the form Ax = b that come from the integrals dx dx . 2+9 5x + 6 x The second matrix will have complex (imaginary) numbers in it. Both matrices will be 2-by-2. x2 Sol -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 2 Problems 1.10, 1.13, 1.14. from Axler. Problem A. Consider the vector (12, 1, 1, 2, 7, 0) in R6 . Write this vector as a sum u + w such that u has all of its coordinates equal and the sum of the coordinates of -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 2 Problems 1.10, 1.13, 1.14. from Axler. Problem 1.10. What is U + U ? The answer is U . The reason is that U + U consists of linear combinations of elements from U . But U is closed under taking linear combinat -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 3 Problems 2.3, 2.4, 2.8, 2.13 and 2.14 from Axler. Problem A. Recall that if we have three vectors (x1 , y1 , z1 ), (x2 , y2 , z2 ), (x3 , y3 , z3 ) in R3 then the volume of the parallelapiped they generate is -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 3 Problem 2.3 Suppose that (v1 , . . . , vn ) is linearly independent and that (v1 + w, . . . , vn + w) is linearl dependent. Then there are scalars ai , not all zero, such that n=1 ai (vi + w) = 0. Expanding th -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 4 Problems 3.3, 3.4, 3.5, 3.7 and 3.9 from Axler. Problem A. Let A and B be sets and f : A B a function. Prove that C f 1 (f (C ) and that D = f (f 1 (D). Given an example where C = f 1 (f (C ). The assertion th -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 4 Problems 3.3, 3.4, 3.5, 3.7 and 3.9 from Axler. Solutions. 3.3 Let U V be a subspace and S : U W be a linear map. Pick a basis (u1 , . . . , u ) of U and extend it to a basis of V , (u1 , . . . , u , v +1 , . -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 5 Problem A. Using the row reduction method, determine the set of solutions of the system of equation Ax = b where 1 ( a) A = 1 0 1 10 (b) A = 5 3 21 1 1 ( c) A= 0 (d) 1 2 1 , b = 0 2 2 2 20 4 , b = 8 0 6 2 1 -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 5 Problem A. Using the row reduction method, determine the set of solutions of the system of equation Ax = b where 1 1 ( a) A= 0 1 10 5 3 (b) A= 21 1 1 ( c) A= 0 (d) 1 2 , b = 0 1 2 2 2 20 , b = 8 4 0 6 2 1 -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 6 Problems 5.9, 5.14 and 5.19 from Axler. Problem A. Consider the matrices 01 00 P= Q= 00 10 Compute P Q and QP . Problem B. Consider the matrix P= 1 1 11 Very that P 2 2P +2I is the zero matrix. Use this to gue -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 6 Problems 5.9, 5.14 and 5.19 from Axler. Solution. (5.9) Suppose that T : V V is a linear map with rank k . If v is an eigenvector with a non-zero eigenvector then T (v/) = v . This implies that v is in the ra -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 7 Problem A. (this is Artin 1.3.1) Compute the following determinants: (1) 1 i 2i 3 (2) 11 1 1 (3) 201 010 102 (4) 1 5 8 0 0 2 6 9 0 0 3 7 0 0 0 4 (5) 1 2 4 2 4 3 1 0 1 5 0 0 3 0 0 0 Problem B. (this is Artin 1. -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 7 Problem A. (this is Artin 1.3.1) Compute the following determinants: 1 i 2i 3 (1) = 2 2i 11 1 1 (2) = 2 201 0 1 0 =3 102 (3) (4) 1 5 8 0 0 2 6 9 0 0 3 7 0 0 = 24 0 4 (5) 1 2 4 2 4 3 1 0 1 5 0 0 3 0 = 30 0 0 Pr -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Homework 8 Axler Problems 1.3, 1.9. (Hint for 1.3 aj bj = jaj bj / j .) Problem. Write [1, 2, 3, 4i] as av + w where v = [1, 0, 1, 2i] and v , w = 0. Problem. Consider the matrix 12i Q = 2 8 i i i 2 For u, v C3 , dene u -
UC Davis MATH 67
Winter 2011 Math 67 Syllabus Linear Algebra Course description. The purpose of this course is two-fold. The rst is to introduce you to the powerful language and theory of linear algebra. The second is to give you your rst experience formulating rigorous m -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Writting Assignment 1 Instructions. The purpose of this assignment is less mathematical and more to familiarize you with writing mathematics well. As such, you are to type a solution to the following problem, pretending -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Writting Assignment 2 Instructions. Same as last time. Exercise. Let Mat(n) be the complex vector space of n-by-n matrices with entries in C. Dene a linear map T : Mat(n) Mat(n) by T (X ) = X + X tr , where X tr is the t -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Writting Assignment 3 Exercise 1. Consider the 4-by-4 matrix with complex entries x1 x4 X= x3 x2 x2 x1 x4 x3 x3 x2 x1 x4 x4 x3 . x2 x1 This is called a circulant matrix. Let be a 4th root of unity (i.e., 4 = 1). Prove -
UC Davis MATH 67
Winter 2011 Math 67 Linear Algebra Writing Assignment 3 Exercise 1. The circulant matrix x1 x4 X= x3 x2 x2 x1 x4 x3 x3 x2 x1 x4 x4 x3 C44 x2 x1 has an eigenvector (1, , 2 , 3 ), where is a 4th root of unity. The associated eigenvalue is x1 + x2 + 2 x3 -
UC Davis MATH 67
Linear Algebra As an Introduction to Abstract Mathematics Lecture Notes for MAT67 University of California, Davis written Fall 2007, last updated November 19, 2009 Contents 1 What is Linear Algebra? 1.1 1.2 1.3 Introduction to MAT 67 . . . . . . . . . . . -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Homework Set 1: Exercises on Complex Numbers Directions: You are assigned the Calculational Problems 1(a, b, c), 2(b), 3(a, b), 4(b, c), 5(a, b), and the Proof-Writing Problems 8 and 11. Please su -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Solutions to Homework Set 1 1. Express the following complex numbers in the form x + yi for x, y R: (a) (2 + 3i) + (4 + i) Solution: By direct computation, (2 + 3i) + (4 + i) = (2 + 4) + (3 + 1)i -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Homework Set 2: Exercises on Linear Equations and Vector Spaces Directions: Submit your solutions to Problems 1, 2 and 4. Separately, please also submit the Proof-Writing-Problems 3 and 5. This ho -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Solutions to Homework Set 2 As usual, we are using F to denote either R or C. 1. Solve the following systems of linear equations and characterize their solution set (unique solution, no solution, -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Homework Set 3: Exercises on Linear Spans and Bases Directions: Please work on all exercises! Hand in Problems 1 and 2 as your \"Calculational Homework\" and Problems 5 and 7 as your \"Proof-Writing -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Solutions to Homework Set 3 1. Show that the vectors v1 = (1, 1, 1), v2 = (1, 2, 3), and v3 = (2, -1, 1) are linearly independent in R3 . Write the vector v = (1, -2, 5) as a linear combination of -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Homework Set 4: Exercises on Linear Maps Directions: Please work on all of the following problems! Hand in the Calculational Problems 1 and 2, and the Proof-Writing Problems 6 and 7 at the beginni -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Solutions to Homework Set 4 1. Define the map T : R2 R2 by T (x, y) = (x + y, x). (a) Show that T is linear. (b) Show that T is surjective. (c) Find dim nullT . (d) Find the matrix for T with res -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Homework Set 5: Exercises on Matrices and Linear Maps Directions: Please work on all problems! Hand in solutions to the Calculational Problems 1, 2(i,m,r), 5(a), 6(a) and the Proof-Writing Problem -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Solutions to Homework Set 5 1. Suppose that A, B, C, D, and E are matrices over F having the following sizes: A is 4 5, B is 4 5, C is 5 2, D is 4 2, E is 5 4. Determine whether the followin -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Homework Set 6: Exercises on Eigenvalues Directions: Please work on all exercises and hand in your solutions to Problems 6 and 7 at the beginning of lecture on February 16, 2006. (Because of the m -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Solutions to Homework Set 6 As usual, we are using F to denote either R or C, and F[z] denotes the set of polynomials with coefficients over F. 1. Let V be a finite-dimensional vector space over F -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Homework Set 7: More Exercises on Eigenvalues Directions: Please work on all of the following exercises. Submit your solutions to Problems 3(d) and 5(b) as your Calculational Problems and Problems -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Solutions to Homework Set 7 As usual, we are using F to denote either R or C. 1. Let a, b, c, d F and consider the system of equations given by ax1 + bx2 = 0 cx1 + dx2 = 0. (1) (2) Note that x1 -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Homework Set 8: Exercises on Inner Product Spaces Directions: Please work on all of the following exercises and then submit your solutions to the Calculational Problems 1 and 8, and the Proof-Writ -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Solutions to Homework Set 8 As usual, we are using F to denote either R or C. We also use , to denote an arbitrary inner product and to denote its associated norm. 1. Let (e1 , e2 , e3 ) be the -
UC Davis MATH 67
MAT067 University of California, Davis Winter 2007 Homework Set 9: Exercises on Orthogonality and Diagonalization Directions: Please work on all of the problems and submit your solutions to the Calculational Problems 1 and 2, and Proof-Writing Pro -
UC Davis MATH 67
MAT 67, Homework 7, due 2/25/08 1. Solve the following systems of linear equations by using Gaussian elimination. (a) x1 + 2x2 2x3 + 3x4 = 2 2x1 + 4x2 3x3 + 4x4 = 5 5x1 + 10x2 8x3 + 11x4 = 12 (b) x1 + 2x2 3x3 = 4 x1 + 3x2 + x3 = 11 2x1 + 5x2 4x -
UC Davis MATH 67
MAT 67, Homework 6, due 2/20/08 1. Read section 6.4 2. Exercises 1, 2, 3, 4, 6 on page 79-80 3. Proof writing exercises 1, 2, 3, 5 on page 80-81 1 -
UC Davis MATH 67
MAT 67, Homework 3, due 1/28/08 1. Exercises 1, 3, 5 on page 46-47 2. Proof writing exercise 2, 3, 4 on page 47 3. Find the span of the vectors (1, 1), (1, 2), (1, 0) in R2 . Are the vectors linearly independent? 4. Find the span of the vectors (1, -
UC Davis MATH 67
UC Davis Summer Sessions Special Program 2008 Program/Course(s): Program Dates: Application Deadline: Fee Payment Deadline: Office Use Only MATH 67 (4 units) TBA TBA June 23 August 15, 2008 Term: Rate: Course: CRN: Units: 200806 None MAT 67 612 -
UC Davis MATH 67
Winter 2009 Math 67 - UC Davis TA: Efrem Rensi erensi@math.ucdavis.edu www.math.ucdavis.edu/~erensi office: 2131 MSB my office hours are wed 2-3pm, thurs 4:10-5:10 (after section) or you can stop by and ask me a question if I\'m around and not bus -
UC Davis MATH 67
3>;) 7v* 77r xt *7v -uq Xv -X q -?/r = f -2 X f >O >h / l t - *z: :Y| -L->l r\\F\"R;Gz)+Rz / | | l -z -Ll r\\ j,:,:;li.\\#rr Y.(! ftl:-ijlt ) \" / o c2 -t -l o \\R3=R,\'t-u+Rt -J -Z I z -r f \\ : \"-te ).-<l o ):) l e e r-rL 3 l -t -t J \\or \'lo / ( -
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