2. Basic Concepts
Society for Industrial and Applied Mathematics eBooks · 2000
The purpose of this and the succeeding three chapters is to set the stage for what follows. We use two basic tools, linear algebra and real analysis, throughout the book. Of course, both disciplines are vast, and we have chosen to review only those topics that are directly relevant to the rest of our book. However, even with this goal in mind, the level of exposition we offer will vary from topic to topic. This is deliberate. Some of the material we review is so basic that we feel it inappropriate that we should more than summarize what is known. If the reader feels we are going too fast, we provide references to the books and papers where this material is covered in considerably more depth. The remainder of the material is more directly relevant, and in this case we try to motivate and develop it so that we hope the reader need look no further. 2.1 Basic Notation We shall denote the ith component of a vector by , or simply if no confusion can arise from other suffixes attached to x. Similarly, the (i, j)th component of the matrix will be written as or simply (note that matrices are traditionally written in upper case, while their components are in lower case). if is a subset of {1,…,n}, we shall write or simply for the vector whose components are , . There are a number of scalar functions that we shall extend quite naturally for vectors. The vector |x| is simply that whose ith component is . Similarly, we shall write for the vector whose ith component is , and for that whose ith component is . The vectors max[x, y] and min[x, y] are simply those whose ith components are and , respectively, while mid(x, y, z) is that whose ith component is the median of , , and .