# Homework and Syllabus

## Linear Algebra

MATH 2135 Spring 2018

Homework is due in class and must be stapled with your name and homework number on it to receive credit.  You will be graded on the clarity of your exposition.

Please read the suggested texts before class, and then after class make sure to attempt the homework for the sections we covered that day.

You may find it useful to use a computer algebra system to check your matrix computations.  The program Mathematica, for instance, is available free to students via the University of Colorado.

You may find the Mathematics Academic Resource Center ("MARC" MATH 175) to be useful as a meeting point for discussing homework.

An asterix * indicates that a homework assignment has not been finalized.

 Date Topics Reading Homework Wednesday January 17 Introduction to the course, and review of mathematical notation Sets, subsets, unions, intersections, products, equivalence relations, and maps.  Matrix multiplication, and determinant and inverse formulas for small matrices. Introduction to LaTeX. You can review some of the standard mathematical language in: Richard Hammack, The Book of Proof, Creative Commons, 2nd Edition, 2013. Please see the references for LaTeX at the bottom of this webpage.  If possible, bring a laptop with LaTeX installed. Here are some sample files we will use: .tex, .bib, .pdf. Friday January 19 Definition of a vector space Vector spaces, examples, and elementary consequences of the axioms. Tom M. Apostol, Calculus, Volume 2 (2nd Edition), Wiley, 1969, Sections 1.1-4 Read Section 1 of the following .pdf, which gives a brief overview of vector spaces and linear maps. HW 1 Hammack Chapter 12 Section 12.1: 4,6. Section 12.2: 5,10,16. Section 12.4: 2,6,10. Section 12.5: 2,8. Monday January 22 Sub-vector spaces Definition of a sub-vector space. Apostol Section 1.6 Read Section 2 of the following .pdf Wednesday January 24 Linear maps Definition of a linear map, kernel, and image. Apostol Sections 2.1-2. Read Section 3 of the following .pdf Friday January 26 Dimension and bases Definition of dimension, linear dependence, and bases. Apostol Sections 1.7-9 Read Section 4 of the following .pdf HW 2 Apostol Section 1.5: 1-4, 23, 24, 31 (a), (b). Section 2.4: 2, 4, 6, 12, 13, 16, 18, 28, 29. Monday January 29 Inner products Definition of an inner product, Euclidean spaces, Norms, and orthogonality. Apostol Sections 1.11-12 Wednesday January 31 Inner products continued Construction of orthogonal sets, the Graeme--Schmidt process, and orthogonal complements. Apostol Sections 1.14-15 Friday February 2 Inner products continued Projections, and best approximations. Apostol Section 1.16 HW 3 Apostol Section 1.10: 1, 4, 10, 11, 17, 18, 24. Section 1.13: 1, 4, 8, 13, 15, 16. 5 points extra credit for homework typed in LaTeX Monday February 5 Linear maps continued Definition of a linear map, kernel, and image.  The Nullity--Rank Theorem. Apostol Sections 2.1-3 Wednesday February 7 Linear maps continued Linear maps as a vector space Apostol Section 2.5 Friday February 9 Linear maps continued Injectivity, and inverses. Apostol Section 2.6-7 HW 4 Apostol Section 1.17: 2, 3, 6, 8. Section 2.4: 1, 2, 6, 12, 24, 26. 2 points extra credit for homework typed in LaTeX Monday February 12 Review Wednesday February 14 Review Sample Midterm I .pdf Solutions .pdf Friday February 16 MIDTERM I MIDTERM I MIDTERM I Monday February 19 Review of exam Wednesday February 21 Linear maps continued Linear maps with prescribed values, matrix representation of a linear map. Apostol Sections 2.9-10 Friday February 23 Linear maps continued Construction of a matrix representation in diagonal form, review. Apostol Section 2.11 HW 5 Apostol Section 2.8: 1, 3, 4, 8, 22-24, 27, 30. 2 points extra credit for homework typed in LaTeX Monday February 26 Linear maps continued Isomorphism between the vector space of linear maps and the vector space of matrices.  Identification of composition with matrix multiplication. Apostol Sections 2.13-15 Wednesday February 28 Computational topics Systems of linear equations. Apostol Section 2.17 Friday March 2 Computational topics continued Computation techniques, inverses of square matrices. Apostol Sections 2.18-19 HW 6 Apostol Section 2.12: 1(c), 2, 8, 11, 17. Section 2.16: 1, 2, 3(a), 4(a), 5, 6, 7, 12. 2 points extra credit for homework typed in LaTeX Monday March 5 Determinants Introduction, and axioms. Apostol Sections 3.1-3. Wednesday March 7 Determinants continued Computation of determinants. Apostol Section 3.4 Friday March 9 Determinants continued The uniqueness theorem. Apostol Section 3.5 HW 7 Apostol Section 2.20: 2, 4, 11, 12, 16. 2 points extra credit for homework typed in LaTeX Monday March 12 Determinants continued Product formula, determinant of an inverse. Apostol Section 3.7-8 Wednesday March 14 Determinants continued Determinants and independence of vectors.  Determinants of block-diagonal matrices. Apostol Section 3.9-10 Friday March 16 Determinants continued Expansion formulas for determinants, minors, cofactors, determinant of a transpose, the cofactor matrix, and Cramer's rule. Apostol Sections 3.12-16 HW 8 Apostol Section 3.6: 1, 3, 4(a), 5, 6. Section 3.11: 1, 2, 5. 2 points extra credit for homework typed in LaTeX Recommended, but not to be turned in: Section 3.6: 2, 3, 9. Section 3.11: 6. Monday March 19 Review Wednesday March 21 Review Sample Midterm II .pdf Solutions .pdf Friday March 23 MIDTERM II MIDTERM II MIDTERM II March 26--30 SPRING BREAK SPRING BREAK SPRING BREAK Monday April 2 Review of exam Wednesday April 4 Eigenvalues and eigenvectors Introduction, linear transformations with diagonal matrix representations. Apostol Section 4.1 Friday April 6 Eigenvalues and eigenvectors continued Eigenvalues and eigenvectors defined, linear independence of eigenvectors with distinct eigenvalues. Apostol Section 4.2-3 HW 9 Apostol Section 3.17: 1(a),(b), 2(a),(b), 3, 4, 5, 7. 2 points extra credit for homework typed in LaTeX Recommended, but not to be turned in: Section 3.17: 6, 8. Monday April 9 Eigenvalues and eigenvectors continued Characteristic polynomials, the Cayley--Hamilton Theorem. Apostol Section 4.5 Wednesday April 11 Eigenvalues and eigenvectors continued Calculation of eigenvalues and eigenvectors, trace of a matrix. Apostol Sections 4.6-7 Friday April 13 Eigenvalues and eigenvectors continued Matrices representing the same linear transformation, similar matrices, review. Apostol Section 4.9 HW 10 Apostol Section 4.4: 2, 3, 5, 7, 9. Section 4.8: 3, 4, 5, 7. 2 points extra credit for homework typed in LaTeX Recommended, but not to be turned in: Section 4.4: 1, 11, 12. Section 4.8: 1, 12, 14. Monday April 16 Eigenvalues of operators acting on Euclidean spaces Overview of Hermitian operators, and the spectral theorem. Apostol Sections 5.1-2, and Theorems 5.4, 5.7. Wednesday April 18 Applications of linear algebra: Quadratic forms, and maximizing their value subject to constraints. Friday April 20 Applications of linear algebra: Principal component analysis, with applications to statistics and image processing. HW 11 Apostol Section 4.10: 1, 2, 4, 7, 8. 2 points extra credit for homework typed in LaTeX Monday April 23 Applications of linear algebra: Singular value decomposition, and applications. Wednesday April 25 Applications of linear algebra: Linear regression, and applications. Friday April 27 Applications of linear algebra: Markov chains, and applications to the Google page rank algorithm. HW 12 Review for final exam Here are a few problems to look at for fun on Hermitian operators (not to be turned in): Apostol Section 5.5: 1, 3, 7, 9. Section 5.11: 1, 2, 3, 14. Monday April 30 Review Wednesday May 2 Review Sample Final .pdf Solutions .pdf Friday May 4 NO CLASS NO CLASS NO CLASS Monday May 7 FINAL EXAM 1:30 PM - 4:00 PM ECCR 105 (Lecture Room) FINAL EXAM FINAL EXAM

I strongly encourage everyone to use LaTeX for typing homework.  If you have a mac, one possible easy way to get started is with texshop. If you are using linux, there are a number of other possible ways to go, using emacs, ghostview, etc. If you are using windows, you're on your own, but I'm sure there's something online. Here is a sample homework file to use: (the .tex file, the .bib file, and the .pdf file).  This site can help you find LaTeX symbols by drawing: http://detexify.kirelabs.org/classify.html.  You may also want to try https://cocalc.com (formerly https://cloud.sagemath.com/) or https://www.sharelatex.com/ for a cloud version.