Linear Algebra Refresher: Review, Amplification, Examples

Linear algebra is an elegant subject and remarkable tool whose influence reaches well beyond uses in pure and applied mathematics. Certainly all math majors as well as majors from a growning number of STEM fields have taken a first course in linear algebra. On the other hand, as with any tool or collection of knowledge, without frequent use, one’s facility with the material wanes.

These notes are not intended as a first or second course in linear algebra, though they assume the reader has seen the material in a basic linear algebra course, covered for example in [1], [2], or [3].

These notes will undertake a review of many basic topics from a typical first course, often taking the opportunity to interleave more advanced concepts with simpler ones when convenient. It will refresh the reader’s memory of definitions, structural results, core examples, and provide some computational tools to help the reader come to a deeper appreciation of the ideas first met perhaps a long time ago.

An additional resource of which to take advantage is Robert Beezer’s A First Course in Linear Algebra¹ which explores the use of computation in far more depth than is done here. Computations in these notes uses Sage (sagemath.org²) which is a free, open source, software system for advanced mathematics. Sage can be used either on your own computer, a local server, or on SageMathCloud (https://cocalc.com³).