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In this module, we will discuss Gaussian elimination and elementary row operations and how they can be used to solve systems of linear equations as well as, REF, RREF and the Moore-Penrose inverse.

Matrices and vectors initially arose, and perhaps remain primarily so, as a way of representing a set of equations with unknowns that we wish to solve for. We should point out that in our representation of the system of equations, the columns will contain the coefficients for each variable for each of the equations and each row will contain an entire equation. This mimics our data table which had variables in the columns and observations in the rows. There is a certain equivalence among matrices and that is that the solutions to a set of equations may have many different matrix representations. A process that transforms a matrix to row-echelon form demonstrates the “plasticity” of the matrix representation of a system of equations. We will also introduce you to solving a system of linear equations using Gaussian elimination. In the process of attempting to solve a set of equations, we will show how to determine the number of solutions a system of linear equations has. We will also introduce the concept of a vector space, a vector space’s basis, and spanning vectors.

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