Linear Algebra
Spring 2500, Spring 2017
This course is devoted to the basics of linear algebra: the study of systems of linear equations, vectors, matrices, and those things which arise when pursuing basic questions about these objects further.
Students who complete this course will learn to work as a mathematician does. To meet this challenge, we will run class as an Inquiry Based Learning (IBL) environment.
Successful students will learn to perform basic computations in linear algebra, but will also build mental models of the geometry underlying linear algebra. They will be able to describe and use the three pictures (row, column, and transformation) behind a system of linear equations, and the Fundamental Theorem of Linear Algebra.
Finally, student will learn the basics of the computer algebra system SageMath, including how it can be used to perform unpleasant computations efficiently and how it can make relevant visualizations.
<-this is the best way
http://theronhitchman.github.io/linear-algebra
https:/cloud.sagemath.com
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site using their UNI email address.I will be using a “standards based assessment” scheme for this course. See the following discussion for more detailed information. Grades in this course are based on the following assessments:
If you feel uncertain about your progress in the course at any time, please contact me.
Accommodations: If you have a disability and require accommodations, please contact the instructor early in the semester so that your learning needs may be appropriately met. You will need to provide documentation of your disability to the Student Disability Services (SDS) office, located on the top floor of the Student Health Center, (319) 273-2677.
Academic Learning Center Syllabus Statement
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is the University’s testing center for many standardized tests, including the
PLT, GRE, and Praxis Core. Visit the website at http://www.uni.edu/unialc/
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call (319) 273-6023 for more information or to set up an appointment.
I conduct each of my classes in some sort of Inquiry Based Learning format. Essentially, this means that I strive for an active, student-centered environment. Lectures will be infrequent (if they happen at all), and instead students will engage in activities that have them participate as working mathematicians by conducting research at an appropriate level. It is important to note two fundamental differences between how Inquiry Based classes operate and traditional lecture classes do:
If you have not experienced an inquiry based learning environment before, this can be a bit unsettling. Have no fear! It is my job as the instructor to put you in a position to have success, and to help guide you back toward success when you miss the mark. Pretty soon, you will be much more comfortable.
It is my goal to help you increase your power and ability as a mathematician. Since I want you to help you improve at doing mathematics, I have structured our class as an environment where you must do mathematics. This gives you opportunity to practice and grow, and it gives me an opportunity to see what you are capable of doing and where I can help you the most.
There are two main ways in which modern technology (computers of all kinds) can make the study of mathematics easier:
It is getting to the point where calling yourself a mathematician means you can use a computer to do those things as part of problem solving, when appropriate.
The UNI Math Department has decided that Math 2500 is the course where we will teach basic mathematical computing. The main reason is that introductory linear algebra is full of interesting things that can only be understood after routine, tedious computations, and also has a important 3d visualization component.
We do not expect that you have any significant experience with computing or programming before you enroll. This semester, you will learn some of the basics of how to use a mathematical software system. This will help you with linear algebra and, if you stick with it, it can help you with other parts of your study. To get through some of the basics, you will do an assignment before our Lab Day which is modeled on a introductory workshop.
This semester, I have chosen to use the Computer Algebra System SageMath. Sage is free and open-source, which has some big benefits. The most important one is that you will learn a system you can take with you, and will always be free to use. Another point in favor of SageMath is that it is built using Python, a powerful, high-level programming language. Eventually, you will want to write some small programs, and Python is easy to learn and easy to use.
To do our work, we will use the SageMathCloud service. SMC allows us to use Sage (and lots of other software) through a web browser. It essentially turns whatever web browser you have on you into a terminal for a powerful linux machine. (You won’t need all that power at first, but you might come to appreciate it later. It is a system designed by mathematicians and programmers for doing what they need.)
Rob Beezer wrote a guide to the basic linear algebra functionality available in SageMath. I find it useful to have a copy of this guide tucked in my textbook as a bookmark.
I have mixed together ideas from two different ways of thinking about grading. The result is a system which is likely unfamiliar to you, so I want to describe things in some detail. Let’s start with the “meta-goals” for the term.
These are the big goals I have for learning for the term. In this list, I have written them as vague ideas.
As much as possible, grades should be based on demonstrated competence on the stated learning goals of the course. If everything fits properly, you will find that striving for a higher grade and working hard for deeper and more complete understanding of the course material are one and the same.
I will set a collection of learning goals for the term, and make them explicit and public. This way you will understand what things I will try to assess in your work.
I will assign final grades for the course on the basis of demonstrated competence on those learning goals.
Any assignment that is assessed will receive feedback in a way aligned with the stated learning goals. This way, it should be clear in which ways you should focus your efforts to improve your work.
Work done is either sufficient to receive credit, or it isn’t. The expectations for “good enough” work should be public, clear, and guide a learner in their efforts to improve.
Every assessment will have a public declaration of which learning outcomes it addresses, and a declared expectation of what constitutes passing.
Every assignment in this course will be graded on a Pass/Not-Yet binary scale.
Any assessment that is not deemed “Pass” will receive feedback on how to improve, and the student will have the opportunity to reassess, up to the natural time constraints of the semester.
If I could get away with it, I would not assign grades at all, and just focus on giving you feedback on your abilities. But the registrar will ask me for grades at the end of the term, so I need a way to come up with them. Also, some students seem to need or want the motivation of grading to keep up with coursework. So I made up this funny system.
Grade | Core Content Assessments |
Next Content Assessments |
Technology Assignments |
Survey Assignments |
---|---|---|---|---|
A | Complete all 7 | Complete 5 out of 7 |
Complete the Basic Tech Interview Assessment AND Complete the SMC radiography lab assignments |
Complete both surveys |
A- | Complete all 7 | Complete 4 out of 7 |
Complete the Basic Tech Interview Assessment AND Complete the SMC radiography lab assignments |
Complete both surveys |
B+ | Complete all 7 | Complete 3 out of 7 |
Complete the Basic Tech Interview Assessment AND Complete the SMC radiography lab assignments |
Complete both surveys |
B | Complete all 7 | Complete 2 out of 7 |
Complete the Basic Tech Interview Assessment AND Complete the SMC radiography lab assignments |
Complete both surveys |
B- | Complete all 7 | Complete 1 out of 7 |
Complete the Basic Tech Interview Assessment AND Complete the SMC radiography lab assignments |
Complete both surveys |
C+ | Complete all 7 | Complete the Basic Tech Interview Assessment |
Complete both surveys | |
C | Complete 6 out of 7 CORE content assessments |
Complete the Basic Tech Interview Assessment |
Complete both surveys | |
C- | Complete 5 out of 7 CORE content assessments |
Complete the Basic Tech Interview Assessment |
Complete both surveys | |
D+ | Complete 4 out of 7 CORE content assessments |
Complete both surveys | ||
D | Complete 3 out of 7 CORE content assessments |
Complete both surveys | ||
D- | Complete 2 out of 7 CORE content assessments |
Complete both surveys | ||
F | Fail to meet the requirements for D- |
Now, let’s discuss the different types of assignments.
Every day of class where we have new material to discuss, you have a daily assignment due. This will consist of reading, possibly watching some video, and attempting to complete a set of exercises which will guide our discussion at the meeting.
Each assignment is detailed in the online Workbook I am revising.
Your work for a daily assignment should generate a couple of pages consisting of
Each class we will split into small groups to discuss the work for the day. Each group will be assigned one of the day’s tasks to prepare for presentation. We will then rotate through presenting our solutions to the tasks and having class discussion on each.
We will spend a lot of our time in class meetings with students presenting their work on the daily tasks. Each will be assessed as a successful or not, with the following standards in mind:
The class conversation which follows your presentation will provide you with feedback on the quality of your work. Mistakes happen, of course. The point is to make mistakes early and often so you can rethink things and learn.
I am doing a study about the the way I organize this class, so I have a short survey assignment that I want each of you to do at the beginning of the semester and then again at the end of the term. These assignments are very easy to complete. To earn credit for these, you simply need to complete them and return them. I will archive them for analysis during the summer.
There will be several small projects to complete using a computer to assist your work.
These assignments will be copied into your designated class project in SageMathCloud.
Complete the assignment, and leave your answer file in the same folder where the assignment showed up. You need not do anything to turn in an assignment done on SageMathCloud. I will simply grab a copy out of your designated class project.
The math department has specified Linear Algebra as a place where we explicitly teach and assess student basic competence with a computer algebra system.
The quasi-official policy document describing expectations is here:
Students can complete the technology interview assessment any time starting with the second week of class. Successful students tend to complete it some time in the first half of the semester.
To do the technology interview assessment, schedule a 20 minute meeting with me during office hours. We will sit together at a laptop and I will ask you to complete some basic tasks.
The core of the assessment system is based on demonstrated mastery of the content.
There are several class meetings set aside to attempt content assessment papers in class.
Each paper is marked as a whole with Pass or Not Yet. Generally speaking, to earn a mark of Pass, a student must demonstrate understanding and fluency with all of the standards bundled together on that assessment. I will mark each assessment out of 10 points. A score of 8/10 or better is required to pass.
It is not uncommon for students to earn a “not yet” on a first attempt! This is how learning works, the feedback from a faltering attempt is the important thing. Students can schedule further attempts on content standards by making an office hour appointment with me, or some other mutually agreeable arrangement.
Multiple attempts are expected and encouraged. But please plan ahead and leave time to get through all of these. It is difficult (for you and for me) to complete many standards in the last two weeks of the term. I advise you to get to work right away on a content standard that you have not mastered, so they don’t pile up.
Each student is allowed to do at most one reassessment on any given day. Since we will discuss a lot of material, it will be important to keep up with things. It will be impossible to go from F to A (or even a B) in the last week.
These are organized by the “big questions” that we address throughout the course. At first, the questions are pretty straightforward and focus on solving systems of equations. Eventually, the questions become more internal to linear algebra, and address things that come up in our study of systems, but are definitely at a “second level.”
I have labelled seven of these as “CORE” competencies. Those are the standards required to earn grades up to C+. The other standards help you move to deeper understanding, and grades of B- or better.
Vector Algebra (Chapter 1, CORE)
add vectors, plot vectors, compute scalar multiplication of number and vector, compute linear combinations, geometric interpretations of these operations
Matrix Algebra (Chapter 1 and 2, CORE)
add matrices, take transpose, multiply matrix times vector (two ways), multiply two matrices (three ways), identify troubles with matrix multiplication: commutativity, inverses
The Dot Product (Chapter 1, CORE)
compute the dot product of two vectors, compute angles between vectors, interpret sign of dot product in terms of angle between the vectors, compute length of a vector, normalize a vector
The Geometry of Hyperplanes (Chapter 1)
connect the geometry of the dot product to the linear equation representing a hyperplane, understand the effect of changes to the form of a linear equation on the location of a hyperplane, understand the hyperplane as a level set of the dot product, sketch a line or plane in \(\mathbb{R}^2\) or \(\mathbb{R}^3\).
The Three Viewpoints (chapters 1 and 2)
The row picture, the column picture, and the transformational picture. pass back and forth cleanly pass between the representations, and describe what a solution means in each case.
When will the solution be unique? Is there a computationally effective way to find the solution set?
Gauss-Jordan Elimination (Chapter 2, CORE)
Use Gauss-Jordan and back-solving to solve a system, find LU decomposition, identify when Gauss-Jordan breaks, identify when matrix does not have an LU decomposition and discuss workaround, compute determinant of a square matrix, compute the inverse of a square matrix
Solving Systems of Equations (Chapter 3, CORE)
Solve a general (rectangular) system of linear equations using the reduced row-echelon form, special solutions, a particular solution. Give the general solution to a system of linear equations. Compute the rank of a matrix. Use pivots and free variables to reason about the solution set to a system of equations
Implicit and Explicit Descriptions of Subspaces (Chapter 3)
determine when a set of vectors is linearly dependent or linearly independent, determine the span of a set of vectors. determine if a collection of vectors is a basis for a subspace Find a basis for a subspace described using equations, find equations to describe a subspace described using a basis use the row space algorithm and the column space algorithm to find a basis
Approximate Solutions and Least Squares (Chapter 4)
Find the “best” available approximate solution to an unsolvable system of equations, draw pictures explaining how orthogonal projection is relevant, use approximate solutions to fit curves to data
The Four Subspaces (Chapter 3)
Compute the null space, column space, row space, and left null space of a matrix. describe these subspaces by giving bases
Matrices as Functions (Chapters 3 and 4)
Use the four subspaces to describe the action of a matrix as a transformation (function) Draw reasonably accurate schematic of the transformational picture using information about the four subspaces, make conclusions about the nature of a matrix using the four subspaces
Orthonormal Bases and the QR Decomposition (Chapter 4)
Use Gram-Schmidt to compute an orthonormal basis for a subspace, decide if a matrix is orthogonal or not, compute the QR decomposition of a matrix
Determinants and the Invertible Matrix Theorem (Chapter 5, CORE)
Use properties of determinants, compute determinants in a variety of ways, Use the invertible matrix theorem
Eigenvalues, Eigenvectors, and the Spectral Theorem (Chapter 6, CORE)
Compute eigenvalues and eigenvalues, diagonalize matrices when possible (and recognize limitations), know and use spectral theorem
Singular Value Decomposition (Chapter 6)
Compute the SVD of a matrix, reason about the structure of a matrix using the SVD, build matrices with given properties using the SVD.