ABSTRACT

In this chapter, we will provide a framework for linear spaces that are not only of dimension 2 or 3, but of possibly much higher dimension. These spaces tend to be somewhat abstract, but they are a powerful concept in dealing with many real-life problems, such as car crash simulations, weather forecasts, or computer games. Hence the term “general” in the chapter title refers to the dimension and abstraction that we will study. After covering the basics of linear spaces, we present a more general Gram-Schmidt method, and use this to introduce the QR decomposition, which will bring us back to the least squares solution to an overdetermined linear system. We visit a gallery of spaces, one of which is the space of cubic polynomials over [0,1], which is important for many applications. A music analysis application applies concepts from the chapter. Sketches and figures illustrate the concepts.