Applied Mathematics - P. Oliver, C. Shakiban
The source of linear algebra is the solution of systems of linear algebraic equations.
Linear algebra is the foundation upon which almost all applied mathematics rests. This is
not to say that nonlinear equations are less important; rather, progress in the vastly more
complicated nonlinear realm is impossible without a firm grasp of the fundamentals of
linear systems. Furthermore, linear algebra underlies the numerical analysis of continuous
systems, both linear and nonlinear, which are typically modeled by differential equations.
Without a systematic development of the subject from the start, we will be ill equipped
to handle the resulting large systems of linear equations involving many (e.g., thousands
of) unknowns.
This first chapter is devoted to the systematic development of direct i algorithms for
solving systems of linear algegbraic equations in a finite number of variables. Our primary
focus will be the most important situation involving the same number of equations as
unknowns, although in Section 1.8 we extend our techniques to completely general linear
systems. While the former usually have a unique solution, more general systems more
typically have either no solutions, or infinitely many, and so tend to be of less direct physical
relevance. Nevertheless, the ability to confidently handle all types of linear systems is a
basic prerequisite for the subject.
The basic solution algorithm is known as Gaussian elimination, in honor of one of
the all-time mathematical greats the nineteenth century German mathematician Carl
Friedrich Gauss. As the father of linear algebra, his name will occur repeatedly throughout
this text. Gaussian elimination is quite elementary, but remains one of the most important
techniques in applied (as well as theoretical) mathematics. Section 1.? discusses some
practical issues and limitations in computer implementations of the Gaussian elimination
method for large systems arising in applications.
The systematic development of the subject relies on the fundamental concepts of
scalar, vector, and matrix, and we quickly review the basics of matrix arithmetic. Gaus-
sian elimination can be reinterpreted as matrix factorization, the (permuted) L U decom-
position, which provides additional insight into the solution algorithm. Matrix inverses
and determinants are discussed in Sections 1.5 and 1.9, respectively. However, both play a
relatively minor role in practical applied mathematics, and so will not assume their more
traditional central role in this applications-oriented text.
The source of linear algebra is the solution of systems of linear algebraic equations.
Linear algebra is the foundation upon which almost all applied mathematics rests. This is
not to say that nonlinear equations are less important; rather, progress in the vastly more
complicated nonlinear realm is impossible without a firm grasp of the fundamentals of
linear systems. Furthermore, linear algebra underlies the numerical analysis of continuous
systems, both linear and nonlinear, which are typically modeled by differential equations.
Without a systematic development of the subject from the start, we will be ill equipped
to handle the resulting large systems of linear equations involving many (e.g., thousands
of) unknowns.
This first chapter is devoted to the systematic development of direct i algorithms for
solving systems of linear algegbraic equations in a finite number of variables. Our primary
focus will be the most important situation involving the same number of equations as
unknowns, although in Section 1.8 we extend our techniques to completely general linear
systems. While the former usually have a unique solution, more general systems more
typically have either no solutions, or infinitely many, and so tend to be of less direct physical
relevance. Nevertheless, the ability to confidently handle all types of linear systems is a
basic prerequisite for the subject.
The basic solution algorithm is known as Gaussian elimination, in honor of one of
the all-time mathematical greats the nineteenth century German mathematician Carl
Friedrich Gauss. As the father of linear algebra, his name will occur repeatedly throughout
this text. Gaussian elimination is quite elementary, but remains one of the most important
techniques in applied (as well as theoretical) mathematics. Section 1.? discusses some
practical issues and limitations in computer implementations of the Gaussian elimination
method for large systems arising in applications.
The systematic development of the subject relies on the fundamental concepts of
scalar, vector, and matrix, and we quickly review the basics of matrix arithmetic. Gaus-
sian elimination can be reinterpreted as matrix factorization, the (permuted) L U decom-
position, which provides additional insight into the solution algorithm. Matrix inverses
and determinants are discussed in Sections 1.5 and 1.9, respectively. However, both play a
relatively minor role in practical applied mathematics, and so will not assume their more
traditional central role in this applications-oriented text.
