University of Memphis Mathematical Sciences Numerical Analysis

1. University of MemphisMathematical SciencesNumerical Analysis MATH 4721/6721Spring 2012Dwiggins Chapter Seven Linear Algebra “Algebra” (from the Arabic al-jabr, (the mending of) “broken bones”) literally refers to the process of taking pieces of given information concerning the value of a certain numerical quantity and deducing its value through a sequence of logical deductions. For example, finding a value of x which is the positive root of some cubic polynomial is an algebraic problem. The Fundamental Theorem of Algebra states that every polynomial has a root, but does not tell how one should go about actually finding the root. That process requires many steps in its sequence of logical deductions, a sequence that may never end, as is the case when x is an irrational number. Thus, as a first step in our understanding of how algebraic equations are to be solved in general, we consider only those equations which are linear in x, which means the variable x only occurs in the first power, i.e. no quadratic, cubic, fractional, or otherwise nonlinear terms. Thus, the basic linear equation in one variable is of the form mx + n = p, where m, n, p are given numbers. This equation can also be written in the form ax = b, and it is easy to tell from this form whether or not there is a solution, in this case No if a = 0 but not b, and also what the solution is in terms of arithmetical operations, x = b/a except when a = 0. We studied methods of solving nonlinear equations in the first part of this course, but those were all equations involving a single variable. We now wish to consider systems of equations in which there are many unknowns, perhaps hundreds or even thousands, for which we want to determine values, at least approximately. While a computer solution to a single-variable problem ax = b involves negligible round-off error, you may well imagine that solving a thousand such equations simultaneously can involve enormous amounts of accumulated error, and so we need a way to determine if any answers we obtain from a computer-based process even make sense, let alone how close they are to the actual values. The general problem to be considered in linear algebra is the existence and computation of solutions x, which will be elements of an n-dimensional vector space, to equations of the form Ax = b, where b is also an n-dimensional vector, and A is an element of the space of square matrices of order n. Thus, before we can begin the numerical analysis of such a problem, we all have to first agree on what is meant by terms such as vector, matrix, n-dimensional vector spaces, and so on. We will investigate the numerical analysis of vector-matrix equations in Chapters 8 and 9, but in Chapter 7 we will just be giving a review of topics covered in a Linear Algebra course (MATH 3242) plus some topics covered in a first-year graduate Functional Analysis course (normed spaces, linear spaces).