Chapter 6 Linear Transformation
We have already been working with linear transformations without giving them a formal name. Every time we compute \(A\mathbf{x}\) for a matrix \(A\text{,}\) we are applying a linear transformation(map). In this chapter, we give this idea a proper definition and study it systematically.
The good news is: you already have all the tools you need. The most important discovery of this chapter is that every linear transformation can be represented by a matrix. So questions about a transformation — What is its kernel? Is it invertible? What are its eigenvalues? — all become questions about a matrix, which you already know how to answer.
In short: to understand a linear transformation, just study its matrix.
