TABLE OF CONTENTS Most of the existing practical iterative techniques for solving large linear systems of equations utilize a projection process in one way or another. A projection process represents a canonical way of extracting an approximation to the solution of a linear system from a subspace. This chapter describes these techniques in a very general framework and presents some theory. The one-dimensional case is covered in detail at the end of the chapter, as it provides a good preview of the more complex projection processes to be seen in later chapters.
Consider the linear system
where A is an n n real matrix. In this chapter, the same symbol A is often used to denote the matrix and the linear mapping in 
that it represents. The idea of projection techniques is to extract an approximate solution to the above problem from a subspace of 
. If 
is this subspace of candidate approximants, also called the search subspace, and if m is its dimension, then, in general, m constraints must be imposed to be able to extract such an approximation. A typical way of describing these constraints is to impose m (independent) orthogonality conditions. Specifically, the residual vector b ? Ax is constrained to be orthogonal to m linearly independent vectors. This defines another subspace 
of dimension m, which will be called the subspace of constraints or left subspace
