A central achievement of classical linear algebra is a thorough examination of the problem of solving systems of linear equations. The results – definite, timeless, and profound – give the subject a completely settled appearance. As linear systems of equations are the core engine in many engineering developments and solutions, much of this knowledge is practically and successfully deployed in applications. Surprisingly, within this well-understood arena there is an elementary problem that has to do with sparse solutions of linear systems, which only recently has been explored in depth; we will see that this problem has surprising answers, and it inspires numerous practical developments. In this chapter we shall concentrate on defining this problem carefully, and set the stage for its answers in later chapters.
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