ABSTRACT
In this chapter we continue introducing computational skills relevant to lin- ear algebra. In Section 2.1 system of equations are introduced. In Section 2.2 the main algorithm for solving systems of equations is presented called Gaussian Elimina- tion. Section 2.3 is an application section on Markov Chains. A method is presented for solving in the case that the transition matrix is regular. Section 2.4 is another application section introducing the Simplex Method on a sub-collection of linear pro- gramming problems. In Section 2.5 the discussion is less computational and more theoretical where matrix equivalence is discussed, a notion necessary for further the- oretical developments. In Section 2.6 the inverse of a matrix is presented, when it exists and how to find it. In Section 2.7 the Simplex Method is revisited and redone using matrix multiplication in place of elementary row operations. In Section 2.8 lin- ear systems of equations are divided into homogeneous and non-homogeneous. The rank of a matrix is also presented including its theoretical significance. In Section 2.9 the determinant of a matrix is computed in several ways and a connection to the existence of an inverse for a matrix is made. In Section 2.10 certain linear systems are solved entirely in terms of determinant and the inverse of a matrix is computed entirely in terms of determinant, when it exists. One final numerical methods application is presented in Section 2.11 called the LU factorization.
