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Chapter3Systems of Linear Equations: Geometry

Primary Goals

  1. Understand what the solution set of Ax = b looks like.
  2. Understand the set of b such that Ax = b is consistent.

This chapter is devoted to the geometric study of two objects:

  1. the solution set of a system of linear equations, and
  2. the set of all constants that makes a particular system consistent.

These objects are related in a beautiful way by the rank theorem in Section 3.9.

We will develop a large amount of vocabulary that we will use to describe the above objects: vectors (Section 3.1), spans (Section 3.2), linear independence (Section 3.5), subspaces (Section 3.6), dimension (Section 3.7), coordinate systems (Section 3.8), etc. We will use these concepts to give a precise geometric description of the solution set of any system of equations (Section 3.4). We will also learn how to express systems of equations more simply using matrix equations (Section 3.3).