Publisher Summary This chapter reviews the key results and definitions from the theory of real linear spaces, which are relevant to what follows. This chapter explains that linear transformations from one Euclidean space to another can be represented by matrices, once the coordinate systems for the two spaces have been decided upon. Vectors can be represented as long, skinny matrices (having one column). If A is any matrix, the chapter denotes the transpose of A by AT. In this discussion, all matrices, vectors (and scalars) have real entries. Matrix transposition has the following important properties.