2: Linear Transformations and Matrices

The general approach to the foundations of mathematics is to study certain spaces, and then to study functions between these spaces. In this course we follow this paradigm. Up until now, we have been studying properties of vector spaces. Vector spaces have a linear structure, and so it is natural to deal with functions between vector spaces that preserve this linear structure. That is, we will concern ourselves with linear transformations between vector spaces. For finite-dimensional spaces, it will turn out that linear transformations can be represented by the action of a matrix on a vector. However, for infinite-dimensional