Calculus Revisited: Multivariable Calculus (Res.18-007, MIT OCW). This consists of 26 video lectures taught by
Professor Herbert Gross, on a study of the calculus of functions of several variables (vector arithmetic and vector calculus).
Calculus Revisited is a series of videos and related resources that covers the materials normally found in
freshman- and sophomore-level introductory mathematics courses. Multivariable Calculus is the second course in the series,
consisting of 26 videos, 4 Study Guides, and a set of Supplementary Notes. The series was first released in 1971 as a way
for people to review the essentials of calculus. It is equally valuable for students who are learning calculus for
the first time. (from ocw.mit.edu)

Part IV, Lecture 1 - Linearity Revisited

Herb Gross considers linearity in spaces of dimension greater than 2, including a discussion of local linearity.