This lesson on linear transformations and their matrices explores the idea of a linear transformation and how it can make sense without a matrix. However, for most people, coordinates are necessary in order to compute anything, leading to a matrix. The lecture provides examples of linear transformations, including a projection and rotation, and explains the rules for a linear transformation to be considered linear. The goal is to understand linear transformations and find the matrix behind them, which requires bringing in coordinates and choosing a basis.
Linear Transformations and Their Matrices -- Lecture 30. The overarching idea of linear transformations.
Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed November 23, 2008). License: Creative Commons BY-NC-SA.
More info at: http://ocw.mit.edu/terms