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Chapter2Systems of Linear Equations: Geometry

Primary Goals

We have already discussed systems of linear equations and how this is related to matrices. In this chapter we will learn how to write a system of linear equations succinctly as a matrix equation, which looks like Ax = b , where A is an m × n matrix, b is a vector in R m and x is a variable vector in R n . As we will see, this is a powerful perspective. We will study two related questions:

  1. What is the set of solutions to Ax = b ?
  2. What is the set of b so that Ax = b is consistent?

The first question is the kind you are used to from your first algebra class: what is the set of solutions to x 2 − 1 = 0. The second is also something you could have studied in your previous algebra classes: for which b does x 2 = b have a solution? This question is more subtle at first glance, but you can solve it in the same way as the first question, with the quadratic formula.

In order to answer the two questions listed above, we will use geometry. This will be analogous to how you used parabolas in order to understand the solutions to a quadratic equation in one variable. Specifically, this chapter is devoted to the geometric study of two objects:

  1. the solution set of a matrix equation Ax = b , and
  2. the set of all b that makes a particular system consistent.

The second object will be called the column space of A . The two objects are related in a beautiful way by the rank theorem in Section 2.9.

Instead of parabolas and hyperbolas, our geometric objects are subspaces, such as lines and planes. Our geometric objects will be something like 13-dimensional planes in R 27 , etc. It is amazing that we can say anything substantive about objects that we cannot directly visualize.

We will develop a large amount of vocabulary that we will use to describe the above objects: vectors (Section 2.1), spans (Section 2.2), linear independence (Section 2.5), subspaces (Section 2.6), dimension (Section 2.7), coordinate systems (Section 2.8), etc. We will use these concepts to give a precise geometric description of the solution set of any system of equations (Section 2.4). We will also learn how to express systems of equations more simply using matrix equations (Section 2.3).