In Section 2.7, we encountered the concepts of span and linear independence, and saw that these lead naturally to certain systems of equations. %, the column space of a matrix, and the null space of a matrix.
In each case we were able to explain the concept, but unable to compute any examples, since we lacked the machinery for solving the systems of equations that arose.
You have probably encountered simple systems of linear equations in high school; you can might be able to remember solving systems of equations where you had two or three equations in two or three unknowns, and you tried to find the value of the unknowns. In this chapter we will uncover some of the fundamental principles guiding the solution to such problems.
Solving such systems was a bit time consuming, but not terribly difficult. So why bother? We bother because, in addition to the theoretical applications mentioned above, there are many, many, many practical applications where systems of linear equations arise, from business and finance to engineering to computer graphics to understanding more mathematics. And not only are there many applications of systems of linear equations, on most occasions where these systems arise we are using far more than three variables. (Engineering applications, for instance, often require thousands of variables.) So getting a good understanding of how to solve these systems effectively is important.
