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Chapter3Solution Sets and Subspaces

Primary Goals

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 first object, the solution set, will be introduced in Section 3.1. The second object will be called the column space of A .

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. In addition to the concept of the span of a set of vectors from Section 1.2, we will introduce linear independence (Section 3.2), subspaces (Section 3.3), dimension (Section 3.4), and bases (Section 3.4 and Section 3.5). Finally, we relate the dimension of the column space of A and the dimension of the solution set to Ax = b (strictly speaking, the solution set to Ax = 0 ) in Section 3.6.