This free, online text is designed to be used as a supplement in multivariate calculus courses.

The central concept of differential calculus is that of the *local
linearization*. Most introductory multivariable calculus texts define
and utilize vectors and vector operations, but stop short of defining
*linear transformations*. The purpose of these notes is to fill that
gap. By developing the basic properties and vocabulary of linear
transformations and the corresponding matrix algebra, the aim is to
provide a natural, unifying, and useful language for understanding the
definition of the derivative and facts including the chain rule.

The material in these notes is not meant to be a short version of a full semester linear algebra course. It is designed to provide enough basic linear algebra and matrix algebra to work effectively with the material in a standard introductory multivariable calculus course.

A secondary purpose of these notes is to address the common practice in undergraduate textbooks (even in courses on linear algebra!) of introducing matrix multiplication as a "voodoo" practice, that is, a complicated operation on boxes of numbers given without motivation. Here, we develop matrix multiplication as the natural consequence of the fact that matrices represent linear transformations: the product of matrices that represent two linear transformations is the matrix that represents their composition. For this development alone, the first three sections of these notes could be of use in a linear algebra course.

The text is designed for active engagement, with carefully structured exercises throughout.

Here is an example of a possible schedule for integrating these notes into a multivariate calculus course in a total of 3 to 5 50-minute class meetings.

- Sections 1,2,3: Add two or three days to the end of the multivariate calculus text's chapter on vectors in two-dimensional and three-dimensional real space (material on dot products and cross products, etc)
- Section 4: Add one or two days at the end of the multivariate calculus text's chapter on partial derivatives (material on tangent planes, chain rule, etc)

David W. Lyons is a professor of mathematics at Lebanon Valley College in Annville, Pennsylvania, USA, where he has taught and conducted research since 2000. Lyons works in mathematical physics, leading a student-faculty research program in quantum information science since 2002. For more information, visit his academic website at the URL below.