## Introduction to Linear and Matrix Algebra

Abstract:

Linear algebra, moreso than any other mathematical subject, can be approached in numerous ways. Many textbooks present the subject in a very concrete and numerical manner, spending much of their time solving systems of linear equations and having students perform laborious row reductions on matrices. Many other books instead focus very heavily on linear transformations and other basis-independent properties, almost to the point that their connection to matrices is considered an inconvenient after-thought that students should avoid using at all costs.

This book is written from the perspective that both linear transformations and matrices are useful objects in their own right, but it is the connection between the two that really unlocks the magic of linear algebra. Sometimes when we want to know something about a linear transformation, the easiest way to get an answer is to grab onto a basis and look at the corresponding matrix. Conversely, there are many interesting families of matrices and matrix operations that seemingly have nothing to do with linear transformations, yet can nonetheless illuminate how some basis-independent objects behave.

For this reason, we introduce both matrices and linear transformations early in Chapter 1, and frequently switch back and forth between these two perspectives. For example, we motivate matrix multiplication in the standard way via composition of linear transformations, but are also careful to say that this is not the only useful way of looking at matrix multiplication—for example, multiplying the adjacency matrix of a graph with itself gives useful information about walks on that graph (see Section 1.B), despite there not being a linear transformation in sight.

Because we spend much of the first chapter discussing the geometry of vectors and linear transformations, we emphasize the geometric nature of matrix properties repeatedly throughout the book. For example, invertibility of matrices (see Section 2.2) is not just presented as an abstract algebraic concept that we determine via Gaussian elimination, but its geometric interpretation as a linear transformation that does not “squish” space is also emphasized. Even more dramatically, the determinant, which is notoriously difficult to motivate algebraically, is first introduced geometrically as the factor by which a linear transformation stretches space (see Section 3.2).

We believe that repeatedly emphasizing this interplay between algebra and geometry (i.e., between matrices and linear transformations) leads to a deeper understanding of the topics presented in this book. It also better prepares students for future studies in linear algebra, where linear transformations take center stage.

Authors:

• Nathaniel Johnston