Math 416: Abstract Linear Algebra (Spring 2022) - Sec E13/F13

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Linear systems of equations show up in all areas of scientific analysis. If something (equations or spaces or functions) is not linear, then one of our best tools for understanding it is to approximate it by a linear version. We first need to have a good understanding of the linear versions for the approximation to be helpful. Also, Linear algebra is required in nearly every area of mathematics. I hope you will enjoy studying it in this course. Math 416 is a rigorous treatment of linear algebra. We will cover vector spaces, linear transformations and matrices, canonical forms, eigenvalues and eigenvectors, and inner product spaces. The essential ideas in the course are

  1. systems of linear equations, row reduction and echelon form
  2. vectors and matrices, matrix multiplication, invertibility and inverses
  3. vector spaces and linear transformations
  4. subspaces, linear combinations, spanning sets and bases
  5. representing linear transformations as matrices, change of basis
  6. kernel and image, row and column rank, Rank-Nullity theorem
  7. determinants
  8. eigenvalues and eigenvectors
  9. finding the eigenvalues of a transformation using the characteristic polynomial
  10. finding the eigenspace associated to an eigenvalue
  11. inner product spaces and their algebra and geometry, the Cauchy–Schwarz inequality
  12. orthogonal projections, Gram–Schmidt, least squares
  13. orthogonal and unitary matrices, spectral theory
  14. bilinear forms
  15. Jordan form


Instructor Partha Dey
ContactBy email with subject line: "Math 416:" and from "@illinois.edu" account.
Class E13/E14 - MWF 1:00-1:50pm in 241 Altgeld Hall
F13/F14 - MWF 2:00-2:50pm in 445 Altgeld Hall
Websitehttps://faculty.math.illinois.edu/~psdey/Math416SP22.html
CampuswireI encourage you to post and answer questions in Campuswire. You should get an email invitation to join the class Campuswire page, please email me if you haven't.
Office 35 CAB
Office Hrs M 5-6pm, W 5-6pm (zoom link) or by appointment made via e-mail
GraderKesav Krishnan
Textbook Friedberg, Insel, and Spence, Linear Algebra, 4th edition, 601 pages, Pearson 2002.
I will post the lecture notes for each class. We will also refer to the free text: Breezer, A First Course in Linear Algebra, Version 3.5 (2016). Available online or as a downloadable PDF file.
Prerequisite Math 241 required with Math 347 strongly recommended.
Electronic devices Cell phones must be turned off during all class meetings. Calculators are not allowed in exams and/or quizzes.

Homework Policy Homework will be assigned weekly on Fridays at 4pm on this website, to be handed in at Wednesday noon via compass2g.

You are encouraged to work together on the homework, but I ask that you write up your own solutions and turn them in separately.

Late homeworks will not be graded. If you will be absent when homework is due, you must turn in your homework in advance. Note that I will check my mailbox (250 Altgeld) right after collecting homework in class.

I will drop your two lowest scores.
Exams There will be three in-class midterm exams on Wednesdays February 23, March 30 and April 20. They will be technically comprehensive, but emphasizing recent material up to the most recent graded and returned homework assignment.

The final exam is tentatively scheduled for TBA in room TBA.
Make-up examNo make-up exams will be offered after the fact. However, if the exam schedule conflicts with a university-sponsored activity (conference, competition, etc.), a make-up exam in advance may be arranged. In case of documented illness or emergency, an exam may be dropped.
Quiz Quizzes may be given on Friday in the beginning of the class period. Every quiz will be announced on the main page a week ahead. As a rule, each quiz will cover material of the previous week. Quizzes will be about 10 minutes long. No make-up quizzes will be given. I will drop the lowest quiz when calculating your final grade.
Grading Policy Check your grades at compass2g. Grades will be computed by a weighted average:

Homework 15%
Quiz 10%
Midterms 45%
Final 30%
Extra Participation score 3%

Tentative curve: A(+/-): 90-100%; B(+/-): 80-90%; C(+/-): 60-80%; D(+/-): 40-60%.
I may slightly adjust the curve later to see it fit.
Remarks The course will be challenging for most students. You will have to understand the proofs of theorems and derivations of formulas. Learn the ideas, don't memorize solutions to particular examples. Express yourself clearly. Start working early. Take good notes in class. Get prepared for every class meeting so you can participate. Your grade and satisfaction will depend on your effort.

Quiz 0 (will not be graded): Quiz 0
Homework 0 (will not be collected): Homework 0, TeX File, Solutions
Quiz 1 (on 1/28): Quiz 1 Solutions
Homework 1 (Due on 2/2): Homework 1, TeX File, Solutions
Quiz 2 (on 2/4): Quiz 2 Solutions
Homework 2 (Due on 2/9): Homework 2, TeX File, Solutions
Quiz 3 (on 2/11): Quiz 3 Solutions
Homework 3 (due on 2/16): Homework 3, TeX File, Solutions
Quiz 4 (on 2/18): Quiz 4 Solutions
Midterm the First (on 2/23): Info page, Practice 1 & Solutions,
Practice 2, Actual & Solutions.
Homework 4 (due on 3/2): Homework 4, TeX File, Solutions
Quiz 5 (on 3/4): Quiz 5 Solutions
Homework 5 (due on 3/9): Homework 5, TeX File, Solutions
Quiz 6 (on 3/11): Quiz 6 Solutions
Homework 6 (due on 3/23): Homework 6, TeX File, Solutions
Quiz 7 (on 3/25): Quiz 7 Solutions
Second Midterm (on 3/30): Info page, Summary, Practice 1 & Solutions,
Practice 2, Actual & Solutions.
Homework 7 (due on 4/6): Homework 7, TeX File, Solutions
Quiz 8 (on 4/8): Quiz 8 Solutions
Homework 8 (due on 4/13): Homework 8, TeX File, Solutions
Quiz 9 (on 4/15): Quiz 9 Solutions
Homework 8a (NOT graded): Homework 8a, TeX File, Solutions
Third Midterm (on 4/20): Info page, Summary, Practice 1 & Solutions,
Practice 2, Solutions, Actual & Solutions.
Homework 9 (due on 4/27): Homework 9, TeX File, Solutions
Quiz 10 (on 4/29): Quiz 10 Solutions
Homework 10 (due on 5/4): Homework 10, TeX File, Solutions
Final (on 5/13): Info page, Summary, Practice 1 & Solution,
Practice 2, Solution, Actual.


Week Date Due Content






1 W Jan 19 Introduction. Sec 1.1 of [FIS]. E13 Notes and Video, F13 Notes and Video.
F Jan 21 Quiz 0 Vector Spaces. Sec 1.2 of [FIS]. E13 Notes and Video, F13 Notes and Video.






2 M Jan 24 Subspaces. Sec 1.3 of [FIS]. Notes
W Jan 26 Linear combinations, systems of equations. Sec 1.4 of [FIS] and SSLE of [B]. Notes
F Jan 28 Quiz 1 Using matrices to encode and solve linear systems. Sec RREF of [B]. Notes






3 M Jan 31 Row echelon form and Gaussian elimination. Sec RREF of [B].Notes
W Feb 2 HW 1 Solution spaces to linear systems. Sec TSS of [B]. Notes, E13 and F13.
F Feb 4 Quiz 2 Linear dependence and independence. Sec 1.5 of [FIS]. Notes, E13 and F13.






4 M Feb 7 Basis and dimension. Sec 1.6 of [FIS]. Notes
W Feb 9 HW 2 Basis and dimension. Sec 1.6 of [FIS]. Notes
F Feb 11 Quiz 3 Basis, dimension, and linear systems. Notes






5 M Feb 14 Intro to linear transformations. Sec 2.1 of [FIS]. Notes
W Feb 16 HW 3 The Dimension Theorem. Sec 2.1 of [FIS]. Notes
F Feb 18 Quiz 4 Encoding linear transformations as matrices. Sec 2.2 of [FIS]. Notes






6 M Feb 21 Composing linear transformations, matrix multiplication. Sec 2.3 of [FIS]. Notes
W Feb 23 MT 1 Information page, Review 1, Review 2
F Feb 25 More on matrix multiplication. Sec 2.3 of [FIS]. Notes






7 M Feb 28 Isomorphisms and invertibility. Sec 2.4 of [FIS]. Notes
W Mar 2 HW 4 Invertibility and rank. Sec 2.4 of [FIS] and Secs MINM and CRS of [B]. Notes
F Mar 4 Quiz 5 Changing coordinates. Sec 2.5 of [FIS]. Notes






8 M Mar 7 Introduction to determinants. Sec 4.1 of [FIS]. Notes
W Mar 9 HW 5 Definition of the determinant. Sec 4.2 of [FIS]. Notes
F Mar 11 Quiz 6 The determinant and row operations. Sec 4.2 of [FIS]. Notes






10 M Mar 14 No class. Spring Break.
W Mar 16 No class. Spring Break.
F Mar 18 No class. Spring Break.






9 M Mar 21 Elementary matrices and the determinant. Secs 3.1 and 4.3 of [FIS]. Notes
W Mar 23 HW 6 Determinants and volumes. Sec 4.3 of [FIS]. Notes
F Mar 25 Quiz 7 Diagonalization and eigenvectors. Sec 5.1 of [FIS]. Notes






11 M Mar 28 Finding eigenvectors. Secs 5.1 and 5.2 of [FIS]. Notes
W Mar 30 MT 2 Information page, Summary, Review 1, Review 2
F Apr 1 Diagonalization Criteria. Sec 5.2 of [FIS]. Notes






12 M Apr 4 Proof of the Diagonalization Criteria. Sec 5.2 of [FIS]. Notes
W Apr 6 HW 7 Matrix powers and Markov Chains. Sec 5.3 of [FIS]. Notes
F Apr 8 Quiz 8 Convergence of Markov Chains. Sec 5.3 of [FIS]. Notes






13 M Apr 11 Inner products. Sec 6.1 of [FIS]. Notes
W Apr 13 HW 8 Inner products and orthogonality. Secs 6.1 and 6.2 of [FIS]. Notes
F Apr 15 Quiz 9 Gram-Schmidt and friends. Sec 6.2 of [FIS]. Notes






14 M Apr 18 Orthogonal complements and projections. Secs 6.2 and 6.3 of [FIS]. Notes
W Apr 20 MT 3 Information page, Summary, Review 1, Review 2
F Apr 22 Projections and adjoints. Sec 6.3 of [FIS]. Notes






15 M Apr 25 Normal and self-adjoint operators. Sec 6.4 of [FIS]. Notes
W Apr 27 HW 9 Diagonalizing self-adjoint operators. Sec 6.4 of [FIS]. Notes
F Apr 29 Quiz 10 Orthogonal and unitary operators. Sec 6.5 of [FIS]. Notes






16 M May 1 Dealing with nondiagonalizable matrices. Sec 6.7 and 7.1 of [FIS]. Notes
W May 4 HW 10 Final Review. Notes






17 F May 13 Final Information page, Summary, Review 1.








Adjacency Matrices Block Inverses Choleski Decomposition
Dual Spaces Incidence Matrices Jordan Canonical Forms
LU Decomposition Markov Chains Modules
Minimal Polynomial Permanent Pfaffian
Principle Component Analysis QR Decomposition Quadratic Forms
Quotient Spaces Theory of Special Relativity