We have discussed systems of linear equations and how they are related to matrices. In this chapter we learn how to write a system of linear equations succinctly as a matrix equation, which looks like where is an matrix, is a vector in and is a variable vector in We will study two related questions:
We will use geometry to answer these questions. 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 etc. It is amazing that we can say anything substantive about objects that we cannot directly visualize. Specifically, this chapter is devoted to the geometric study of:
The second object is called the column space of The two objects are related in a beautiful way by the Rank Theorem in Section 2.9.
We develop a large vocabulary in order to describe the answers to our two questions: 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 use these concepts to give a precise geometric description of the solution set of any system of equations (Section 2.4). We also learn how to express systems of equations more simply using matrix equations (Section 2.3).
