ABSTRACT

Just about anybody can solve two equations in two unknowns by somehow manipulating the equations. In this chapter, we will develop a systematic way for finding the solution, simply by checking the underlying geometry. This approach will later enable us to solve much larger systems of equations in future chapters. We begin with Cramer's rule and discuss the concepts of consistent and inconsistent linear system. We move on to Gauss elimination, including pivoting. We give examples of unsolvable and homogeneous linear systems. Techniques for finding the space of solutions to homogeneous systems is outlined and a section on the kernel describes the issues of the existence and uniqueness of a solution. The inverse of a matrix is defined and illustrated. Applications include intersecting two lines and defining a linear map given source and target vectors. A dual view of 2 x 2 linear systems is discussed: viewing the system in terms of the columns of the system versus the rows of the system. An extensive introduction to change of basis with example applications is provided. Sketches and figures illustrate the concepts.