Better Standards for Linear Algebra: Organize by the Big Questions

I want to write a big post with lots of pictures summarizing my experience at the last Iowa Section of the MAA meeting. But that will have to wait as I have lots of real work to do. One piece of that work I want to share makes up this post.

In response to a bunch of whining on Google+, many internet friends who have tried standards based grading gave me advice on how to handle my linear algebra class better. Basically, by trying to be complete, I was making an enormous, unwieldy list of tiny little standards to track. This was clearly a disaster in the making, so I begged for help.

The main piece of advice I got was this: group the standards. Josh Bowman told me this quite clearly, and remarked that he had a standard for every day or two of a calculus course. Then either Bret Benesh or Kate Owens reminded me of Kate’s organizational scheme: use the big questions from your course to decide your standards.

With this pair of wonderful ideas, I ran back to my office to rework things. This took a lot of thought to sort out. But I am so grateful to my internet friends for sharing their expertise with me. I am slowly becoming a better teacher in part because they are willing to help me.

Anyway, I worked hard on this, but I don’t feel like it is totally mine because I needed so much help to get rolling. So, I am sharing it with you, my faithful readers. (uh, hi… Doug? I am pretty sure there is a nonzero chance someone named Doug will read this.) This is version 0.12, or something like it. It could use some criticism, which I welcome. Still, I feel like this is a public beta product. Maybe an official release could happen the next time I am assigned linear algebra.

Without further ado, here is the current status of my Linear Algebra Content Standards, Organized Thematically by The Big Questions of a first matrix algebra course. In this course, I like to use Gil Strang’s Introduction to Linear Algebra (4th Ed), so items are keyed to the chapters in that text where the relevant material is developed. I have done this for my student’s sake, but I bet it can’t hurt that someone tells you that this is text worth reading.

Standards for Linear Algebra Content Learning

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, and are definitely at a “second level.”

Foundational Goals

Question One: What are the basic objects of linear algebra?

• Vector Algebra (Chapter 1)

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)

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)

compute the dot product of two vectors, compute angles between vectors, compute length of a vector, normalize a vector, use connection between dot product and linear equations to work with normal vectors

Question Two: How can we solve a square system of linear equations?

• Gauss-Jordan Elimination (Chapter 2)

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

Question: How can we solve a general system of linear equations?

How can we tell if there is a solution? What shape will the solution set have? When will the solution be unique? Is there a computationally effective way to find the solution set?

• Solving Systems of Equations (Chapter 3)

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

• The Four Subspaces (Chapter 3)

Compute the nullspace, column space, row space, and left nullspace of a matrix. describe these subspaces by giving bases

Question: What are the good ways to understand subspaces?

• Implicit and Explicit Descriptions (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

Question: Can we find approximation solutions to systems that do not have an actual solution?

• 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

Question: Is there a good way to test if a square matrix is invertible?

• Determinants and the Invertible Matrix Theorem (Chapter 5)

Question: How can we understand the geometry of square matrices as transformations?

• Eigenvalues, Eigenvectors, and the Spectral Theorem (Chapter 6)

Question: Are there any good geometric interpretations of a system of linear equations?

• 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.

Question: How do we understand matrices as transformations?

• Four Subspaces and the Fundamental Theorem of Linear Algebra (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

Question: Is there a way to choose a geometrically good basis for a subspace?

• 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

Question: Is there a good geometric way to understand the behavior of a general matrix as a function?

• Singular Value Decomposition (Chapter 6)

2 thoughts on “Better Standards for Linear Algebra: Organize by the Big Questions”

1. Hi TJ,

These look good. You have everything that was in my most recent set of Learning Goals.

How are you assessing these? Quizzes? In-class? If they “add vectors,” does that count just as much for “Vector Algebra” as if they “plot vectors?” How do you turn these things into a grade?

Also, how are your technology “gateway assessments” working?
Bret

2. TJ says:

I assess these through exams, for a start. (Mistake here, will change in the future.) Basically, We cover a little less than six chapters of Strang. So after chapters 1 & 2, I gave an exam on the standards that came up. There were five. Most people only passed one. Since then, I used two other class days to do reassessments with quizzes. I will give an exam for Ch 3 & 4 on Monday (six new standards). the Exam on Ch 5 & 6 will happen in the last week. (only three new standards). In the mean time, I have given lots of other one standard quizzes, and I have done a few “oral exams” for people who were missing only one final piece of a standard after a paper reassessment.

This will need better planning next time. More frequent, and smaller, assessments. And starting earlier in the term!

Right now, I have a few students who are at four out of five, most at two, a few at one, and even one still at zero. But it is up to them to seek reassessment, and the “LAW” is clear.

The tech thing is going fine. I have done four of them, and they seem to work out okay. Takes about twenty minutes to do one.