### Linear Algebra Resources

##### Introduction

Most linear algebra texts have a wealth of information to impart, but courses are often challenged to find the time to get to the "good stuff," meaning inner product spaces and associated applications. Consequently, these crucial ideas often get short shrift.

Moreover, examples which are done in class are often too small to demonstrate effectively all facets of the theory, and time constraints limit the number of examples which can be done. However, computational software such as SageMath allow both the student and instructor to work more complex examples, hopefully providing a more nuanced understanding of the material.

Learning is an iterative process, enhanced by understanding examples and turning ideas over and over until they feel natural. It is hoped these materials begin to advance that mission. This page contains links to resources which will hopefully serve three different audiences:

• Students currently taking linear algebra; at Dartmouth, either Math 22 or 24.
• Students who have previously taken an undergraduate linear algebra course, but need a quick refresher before using those tools in another course, or are about to venture into more abstract mathematics for which finite-dimensional vector spaces and inner product spaces are intended as a familiar stepping off point.
• Instructors of linear algebra who want to generate examples with enough complexity to demonstrate salient properties.
##### The Resources

Both the Linear Algebra Companion and Refresher were written using PreTeXt which allows for multiple formats to be generated from a single XML source. Salient features of the HTML versions below are: Live Sage computations and most links are generally presented as knowls. What the latter means is that when the text says "by Theorem 3.2" or "equation (4.1)", clicking the link with that reference displays the content in place, obviating the need to scroll back through the material to find the reference. Clicking again rolls the content back up. This convenience dramatically enhances the readability of the material.

Neither resource is intended as a textbook, though each summarizes topics and give examples, offering proofs when instructive. The understanding is that these resources have been preceded by course lectures, or in the case of the refresher, an entire course. In that context, these material provide a second pass through the material hopefully adding a layer of understanding.

Caveat: This project remains under development, and these resources are simply the current draft. Comments and suggestions are welcome, well some more than others.

• The Linear Algebra Companion (html) (pdf) is written as an accompaniment to a standard course like our Math 22 or 24. It reviews most of the material in a standard syllabus, provides new examples and overviews, and provides the templates to use Sage as a computational tool. The section on inner product spaces is more thoroughly developed since this is the material that is often hard to fit into a typical course. It develops the singular value decomposition and gives an application to image compression. (Last modified: December 31, 1969.)
• The Linear Algebra Refresher (html) (pdf), is intended for those who have previously taken a linear algebra course, does a quick review of core topics up to inner product spaces and then spends significant time with inner product spaces. The section on inner product spaces is the same as in the Companion. (Last modified: December 31, 1969.)
• The Sage Jupyter notebook, Linear Algebra Tools (save the document removing the ".txt" suffix) is a starting point for instructors or students who wish to generate interesting examples using Sage such as
• A modest-sized matrix whose RREF has all integer entries and whose pivots are not in consecutive columns.
• A matrix which is diagonalizable with prescribed eigenvalues and multiplicities. What's hard about that? Well one only needs an invertible (preferably unimodular) matrix of the appropriate size.
• Given a matrix known to be diagonalizable, provide a basis of eigenvectors.
• Compute characteristic, minimal polynomials, invariant factors, rational and Jordan forms.
• Helper tools for SVD, e.g., how to complete an orthogonal set of vectors to an orthonormal basis for a space.

Here is a link for the Sage Reference Manuals.