# Major Results of Undergraduate Linear Algebra

I have grown steadily unhappier with my linear algebra course. So this term, I am going to rewrite it in a radical way. I would like to make the experience a bit closer to my (successful) IBL Euclidean Geometry course. In EG, we use Euclid’s Elements as our “extant literature,” and then I have an IBL task sequence designed to go with it.

To make this project work, I need to write something analogous for Linear Algebra. For lack of a better title, I am calling this Elements of Linear Algebra. I would like to emulate the structure of Euclid’s work, too, where each book is just enough material to get to an important goal result. Then I can design tasks & homework for the students to do so that they work around the text and extend it. For context, here are the big results at the end of the first four books of Euclid’s Elements:

1. The Pythagorean Theorem and its converse
2. The equi-decomposability of rectilineal planar figures with squares
3. If x lies outside a circle O, and a line through x meets O at points b and c, then the line xa is tangent to O if and only if (xa)^2 = (xb)(xc).
4. the construction of a regular pentagon

Those are four beautiful theorems.

How should I choose the Beautiful Theorems for an undergraduate linear algebra course?

For context: (1) my course is is pre-rigorous, we don’t write “proofs.” Instead, we argue from examples to generality. (2) this course has been entirely stuck in Euclidean space, but it is not quite a matrix algebra course. (3) I have taught out of Strang for several years.

I want your feedback! Here is my current list of potential big results for the to explore. Each of these would be a chapter of my text.

1. Descriptions of lines and planes in the plane and in 3-space using both standard forms and parametric forms.
2. How to solve a system of equations, including how to decide if it is solvable, how big the solution set is, and a reasonably efficient algorithm for writing down the solution set.
3. the invertible matrix theorem
4. the fundamental theorem of linear algebra (borrowed from Strang — the four subspaces and how they fit together)
5. How to find a ‘best approximate solution’ when there is not a given solution
6. The spectral theorem for diagonalization of symmetric matrices

I would love to include the singular value decomposition, but I can barely get to #6 right now.

Note that these are just the big results! Each chapter would have other material in it to help us understand and contextualize the big result. For example, the invertible matrix theorem will necessitate determinants coming up in that chapter.

Anyway… What do you think?