Chapter 5: Using Matrices to Transform Space

Linear transformations are one of the most central topics in linear algebra. Now that we have defined vectors and vector spaces, we need to be able to do things with them. In Chapter 3, Foundations, we manipulated mathematical objects with functions. When we manipulate vectors in vector spaces, mathematicians use the term linear transformations. Why the change of terminology? As with most things in linear algebra, the wording is inspired by Euclidean geometry. We will see that, geometrically, these "functions" actually "transform" vectors from one direction and length to another. But this visual transformation has been generalized algebraically to all types of vectors (n-tuples of numbers, functions, and so on).

We also go through the crucial link between linear transformations and matrices. The most important point of this chapter is that linear transformations can always be represented by matrices when the vector...