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Study Resources. Homework Help. Log in. Sign up. All Materials. University of Waterloo. All Semesters. Reset All. Class Notes. Untitled 2 Textbook Notes. Study Guides. Popular Study Guides Everything you need to know for your next exam. Friday, may 12 lecture 6 : dot products and norms section 1. Note: - only answers are provided here and some proofs.
On the test you must provide full and complete solutions to receive full marks: short answer. Frequently-seen exam questions from - To write the sca. Student name print legibly family name given name Instructions: fill in your waterloo id number and quest login in the box at the top of this pag.
Friday, may 26 lecture 12 : matrices and matrix algebra refers to section 3. Friday, june 30 lecture 27 : elementary matrices and invertibility. The ero pij can be. Wednesday, june 21 lecture 23 : change of coordinates matrix i. Friday, may 19 lecture 9 : systems of linear young mothers breastfeeding. Wednesday, may 24 lecture 11 : homogeneous systems. Suppose we are given a vector b in 2 and another vector a. All Materials 1, CAUW 30, MATH 3,Below are both typed version and the hand written versions of the notes.
Linear Algebra 2
The typed version are intended to be a typed version of the hand written notes but might contain typos. The handwritten notes were notes I used for my Fall classes. I've also included a typed version of all notes as well as all the handouts I used in class. A zip package of all the files is also available if you would like to reproduce these. All lectures typed from 2 to All tex and source files for typed notes.
Lecture 1 [Fall version 1A] Outline. Lecture 1 [Winter version 1B] Outline. In Lecture 11, there is a mistake in the written notes. However, the lower bound could be as bad as This is fixed in the typed notes. Solution to Practice Problem in Lecture Lecture 45 Cardinality. Lecture 46 Exam Review 1. Lecture 47 Exam Review 2. Toggle navigation Carmen's Math Resources Page.
Course Notes.Advanced Algorithms (COMPSCI 224), Lecture 1
Please be sure to check LEARN daily for up to date announcements, errata and files including an up-to-date version of the course notes, sample exams, assignments and their solutions.
All lectures from 2 to 44 All lectures typed from 2 to 44 Handouts from all classes All tex and source files for typed notes. I prefer doing EEA on midterm day.Ashwin Nayak Combinatorics and Optimization, U. Waterloo and Perimeter Institute for Theoretical Physics. This course introduces first year undergraduate mathematics students to basic elements of algebra. The idea is to gain familiarity with concepts of fundamental importance on a rigorous footing.
A weekly lecture schedule is available here. Gilbert and S. The text is not a substitute for the lectures. The lectures may present the material covered in the text in a different manner, or deviate from it entirely. You are advised to take your own notes in class. Homework problems will be generally be assigned from the exercises in the text.
The final mark in the course will be based on homework, unannounced quizzes conducted during the tutorials, one midterm, and the final exam. There will be ten graded homework assignments in all.
The best eight of these will be counted towards your final mark in the course. There will be a homework assignment essentially every week, and will be posted on the web every Monday.
The homework will be based on the material covered in class from that Monday to Friday. The homework will be due on the following Tuesday, before 1 pmin Math drop box no. Late submissions will not be accepted. Graded homework will be returned to you in the next tutorial.
You should be able to solve most of the problems in the homework on your own if you have understood the lectures. However you can expect an odd question that will require additional thought.
You may consult your TA or the instructor during their office hours or during the tutorial. You should write up the solutions on your own and mention all sources of help. Solutions will be posted on the web after the homework is collected. From May 9 until July 25 except May 23, Victoria Daythere will be a tutorial every Monday at pm running for roughly one hour. The tutorial will be open for a discussion of the homework and a review of the lectures.
You are encouraged to approach the TAs with any difficulty you are facing in the lectures and homework during this time. The instructor may also be available during the tutorials for part of the time. There will be a small number of unannounced quizzes during the tutorials, which each of you is required to complete on your own.
The quizzes will be of 10 to 15 minute duration, and will be conducted at the end of the tutorial. It is your responsibility to be present for all the tutorials.
The problem s assigned in the quiz will be of similar difficulty to the ones in the homework.Following toggle tip provides clarification. Topics include orthogonal and unitary matrices and transformations; orthogonal projections; the Gram-Schmidt procedure; and best approximations and the method of least squares. Inner products; angles and orthogonality; orthogonal diagonalization; singular value decomposition; and other applications will also be explored.
In this module, we will look at the fundamental subspaces of a matrix and of a linear mapping, and prove some useful results. Part of the purpose of this module is to help review and recall many of the concepts from Linear Algebra I that are needed for this course.
In this module, we will extend the concept of a linear mapping from R n to R m to linear mappings from a vector space V to a vector space W. It may be helpful to briefly review linear mappings from R n to R mwhich includes the matrix of a linear mapping and diagonalization.
Previously, we studied the important concepts of length, orthogonality, and projections. Since these concepts are so amazingly useful in R nin this module, we will generalize all of these concepts to general vector spaces.
To do this, we will need to first generalize the idea of the dot product in R n to the concept of an inner product on a general vector space. This will lead us to an extremely important theorem in linear algebra. Finally, we will use the theory we developed in this module to actually look at a real-world application, called the method of least squares. In this module, we will use the theory we have previously developed to extend the idea of diagonalization to something even better: orthogonal diagonalization.
This will lead us to the extremely useful and important topic of quadratic forms. Finally, using all of the theory we have developed, we will look at how to mimic diagonalization for non-square matrices.
In this module, we are going to revisit many concepts covered previous, now allowing the use of complex numbers. We will see that much of the theory remains the same, but there will be some differences. Moreover, we will see that quite a lot of the computations will now be a little more complex. Toggle navigation System Homepage. Linear Algebra 2 Class Homepage. Fundamental Subspaces In this module, we will look at the fundamental subspaces of a matrix and of a linear mapping, and prove some useful results.
Lesson: Fundamental Subspaces of a Matrix. Lesson: Bases for Fundamental Subspaces. Quiz: Fundamental Subspaces of Matrix. Linear Mappings In this module, we will extend the concept of a linear mapping from R n to R m to linear mappings from a vector space V to a vector space W. Lesson: Linear Mappings. Quiz: General Linear Mappings. Lesson: The Rank-Nullity Theorem. Quiz: Rank-Nullity Theorem.
Lesson: Matrix Mappings. Lesson: Matrix of a Linear Mapping Continued. Quiz: Matrix of a Linear Mapping. Lesson: Isomorphisms. Lesson: Isomorphisms Continued.Vector integral calculus-line integrals, surface integrals and vector fields, Green's theorem, the Divergence theorem, and Stokes' theorem. Applications include conservation laws, fluid flow and electromagnetic fields. An introduction to Fourier analysis.
Fourier series and the Fourier transform. Parseval's formula. Frequency analysis of signals. Discrete and continuous spectra.
A rigorous introduction to the field of computational mathematics. The focus is on the interplay between continuous models and their solution via discrete processes. Topics include pitfalls in computation, solution of linear systems, interpolation, discrete Fourier transforms, and numerical integration.
Applications are used as motivation. Offered: W,S]. Physical systems which lead to differential equations examples include mechanical vibrations, population dynamics, and mixing processes. Dimensional analysis and dimensionless variables. Solving linear differential equations: first- and second-order scalar equations and first-order vector equations.
Laplace transform methods of solving differential equations. This course offers a more theoretical treatment of differential equations and solution methods. In addition, emphasis will be placed on computational analysis of differential equations and on applications in science and engineering.
Offered: F]. Newtonian dynamics, gravity and the two-body problem, introduction to Lagrangian mechanics, introduction to Hamiltonian mechanics, non-conservative forces, oscillations, introduction to special relativity [Offered: F]. Topology of Euclidean spaces, continuity, norms, completeness. Contraction mapping principle. Fourier series. Various applications, for example, to ordinary differential equations, optimization and numerical approximation.
Offered: F,W]. Complex numbers, Cauchy-Riemann equations, analytic functions, conformal maps and applications to the solution of Laplace's equation, contour integrals, Cauchy integral formula, Taylor and Laurent expansions, residue calculus and applications. An introduction to numerical methods for ordinary and partial differential equations.
Ordinary differential equations: multistep and Runge-Kutta methods; stability and convergence; systems and stiffness; boundary value problems. Partial differential equations: finite difference methods for elliptic, hyperbolic and parabolic equations; stability and convergence. The course focuses on introducing widely used methods and highlights applications in the natural sciences, the health sciences, engineering, and finance.
Difference equations and discrete dynamical systems. Mathematical models are taken from ecology, biology, economics, and other fields. First order ordinary differential equations. Applications to continuous compounding and the dynamics of supply and demand. Higher order linear ordinary differential equations. Systems of linear ordinary differential equations. Introduction to linear partial differential equations.MATH is a foundational course for students in the Faculty, yet too many were struggling to learn the material.
Modifications to high school curricula required a different strategy for the delivery of this course. The time had come to try something new to improve student outcomes and their intellectual stamina in learning mathematics. In the fall term, 15 instructors led by Professor J. Pretti implemented a different version of the course for first-year students. All of this was accomplished while maintaining the academic rigour students expect at Waterloo. MATH is centered on proof, the defining property of mathematics.
Proof defines the transition from high school to university mathematics. It must be thoroughly understood for students to be successful in their upper year math courses.
Classes were smaller. Instead of students, MATH was offered in sections of just 60 students. Instructors got to know their students by name, and provide more one-on-one support during office hours.
Smaller classes also helped the students to better connect and help each other. In-class participation increased, along with attendance. Instructors found that being able to call on the students by name made them more comfortable offering answers. The classes met four times a week with their primary instructor, instead of the previous model of three times a week plus an hour tutorial with a teaching assistant. Lectures were designed so that there was time devoted to working through and discussing in-class exercises.
This transition meant more preparation time for the instructors, but they were happy to see that the new approach worked. The students were pushed further, in terms of the difficulty of assignments and exams, yet showed a more solid understanding of the material. Instead of just listening to an instructor present the course material while they took notes, the students worked through problems in class.
Spending this in-class time practicing proofs helps instill confidence. When students believe they can do it, they enjoy the material more and are ready to tackle even the most challenging assignment questions and exams. There were over 10, posts over the term for this course alone. They were also chiming in to support classmates with questions, and offering insightful comments on the course material.
Increasing student engagement and active learning with these changes to MATH was a very successful experiment. Not only were the grades better overall, but instructors noted that students seemed to show more interest in the material and to enjoy the course more. MATH instructors were pleased with the outcome of the course delivery transition, and they credit Professor J.
His collegiality and organization were noted by his colleagues, who also appreciated his trouble-shooting and clear direction. Support Mathematics. Department of Combinatorics and Optimization. Department of Statistics and Actuarial Science.A forum for news and discussion relevant to the university. Admissions Megathread. No personally identifiable information: real names, Facebook profiles, screenshots of private forums e.
No explicit sharing or insinuation of illegal information and activities, including drug dealing e. Does anybody have a copy of the most recent edition of the MATH course notes? I'm taking it online this term and you would normally get a link to the PDF on learn once you enroll in the course, but it appears they aren't giving a link for the online course.
Does anybody have the course notes from last term or the term before? I had to buy the course notes last spring term while taking it online. They did not give us the course notes like math Yeah, they didn't provide the pdf this term either. Luckily I was able to find the first edition online, but it's from Was just hoping I'd be able to get a more recent edition.
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