This chapter is dedicated to the first major problem of linear algebra: solving systems of linear equations. Restated as a set of questions, we will consider these.

• What is the set of solutions to a given system of linear equations?
• When does a given system have solution, and when does it not?
• If there is a solution, how many solutions are there?
• What ways do we have of describing the collection of solutions?
• Is there a computationally effective way to find those solutions?

Though we begin with the first question, we answer the last one first. As we explore the process of finding solutions, we will start to build the tools we need to finish answering the other questions in a later chapter. In this chapter, we aim to get as complete understanding as we can for at least a special case: square systems.

We will begin with a study of the two ways that vectors let us make pictures of systems of linear equations. Then we take up a basic process for finding solutions, where matrices will appear as a convenient notational device. But as we dig a little further, matrices will become interesting in their own way, so we will study those. But what we study will relate back to the fundamental issue of solving systems of linear equations.

This chapter has two short review sections in it. One just after we learn about elimination, and another at the end of the chapter.