Underneath everything we have looked at so far, we have been concerned with finding solutions to systems of linear equations. Our principal tool for doing that has been Gaussian elimination, which we first met back in Section 1.2. When confronted with a linear system, we frequently find the reduced row echelon form of the system's augmented matrix to read off the solution.
While this is a convenient approach to learning linear algebra, in the real world, people rarely find the reduced row echelon form of a matrix. In this chapter, we will describe why this is the case and then explore some alternatives. The intent here is to demonstrate how we perform linear algebraic computations in the real world. In particular, we will improve our techniques for solving linear systems and for finding eigenvectors through Gaussian elimination.