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Linear Algebra for Chemical Engineers LINEAR ALGEBRA FOR CHEMICAL ENGINEERS KYRIACOS ZYGOURAKIS TABLE 1 Rice University Course Materials Houston, TX 77251 COURSE TEXTBOOK FIRST-YEAR GRADUATE course (or sequence of Strang, G., Linear Algebra and Its Applications, 2nd A courses) in applied mathematics has become Edition, Academic Press, (1980). an integral part of the curriculum in a large ADDITIONAL COURSE REFERENCES number of chemical engineering departments. Among the diverse subjects taught in these 1. A mundson, N. R., Mathematical Methods in Chemical Engineering: Matrices and Their Application, Pren­ courses, linear algebra usually enjoys a prominent tice Hall, (1966). position. The reason for this popularity perhaps 2. Braun, M., Differential Equations and Their Applica­ lies in the fact that linear algebra is as central a tions, 2nd Edition, Springer-Verlag (1975). subject and as applicable as calculus. The pioneer­ 3. Dahlquist, G., A. Bjorck and N. Anderson, Numerical ing work of Neal Amundson, and of his students Methods, Prentice Hall (1974). and disciples as well as other prominent scholars, 4. Friedman, B., Principles and Techniques of Applied Mathematics, John Wiley (1956). has established beyond any doubt that many sig­ 5. Hirsch, M. W. and S. Smale, Differential Equations, nificant and complex chemical engineering prob­ Dynamical Systems and Linear Algebra, Academic lems may be solved by advanced linear algebra Press (1974). techniques [1]. 6. Noble, B. and J. W. Daniel, Applied Linear Algebra, Linear algebra can also serve as an ideal 2nd Edition, Prentice Hall (1977). 7. Steinberg, D. T., Computational Matrix Algebra, Mc­ stepping stone for introducing the first-year Graw-Hill (1974). graduate student to the formal mathematical language of functional analysis. The basic con­ The theory, however, is motivated and reinforced cepts of matrix algebra, already familiar to the by examples derived from a wide range of chemi­ student, can be formulated using the abstract cal engineering problems. Particular emphasis is framework of linear vector spaces. The same placed upon the important aspects of computa­ abstraction can also be used to unify apparently tional linear algebra. In our opinion, it is impera­ diverse problems in finite dimensional spaces tive to expose the students to some fundamental under this common framework. Thus, the ground­ computational methods and to study their efficiency work is laid out for the introduction of functional as well as their convergence problems. Student analysis in infinite dimensional spaces, which is responses to the course evaluation questionnaire necessary for the study of differential and integral indicate that they particularly enjoy the compu­ operator problems [2]. tational part of the course since it points out some Our linear algebra course strives to combine of the real problems to which linear algebra theory both elements of mathematics-abstraction and can be applied. application. Many of the fundamental theorems of linear algebra are rigorously derived in class. COURSE ORGANIZATION Eleven weeks ( out of a total of fifteen) of Student responses to the course the fall semester course, "Applied Mathematics evaluation questionnaire indicate that for Chemical Engineers I," are devoted to the they particularly enjoy the computational part of the course since it points out some of the study of linear algebra and its applications. The real problems to which linear algebra remaining time is devoted to a brief review of theory can be applied. complex analysis and complex integration, which is the final preparation step for the second course © Copyright ChE Division, ABEE. 1984 in applied mathematics taught at Rice. This 176 CHEMICAL ENGINEERING EDUCATION signed to help the students with their computer projects as well as for the discussion of home­ work assignments in an informal way. The students are urged to keep a complete set of notes, which are regularly supplemented by handouts providing lengthy theorem proofs or summarizing the results established up to that point. The assigned textbook is Linear Algebra and its Applications (2nd Edition), by Gilbert Strang. Although it is an extremely well-written book, it is not followed closely ( especially in the first part of the course). The students are strongly en­ Kyriacos Zygourakis received his diploma in chemical engineering couraged to consult additional references (see from the National Technical University of Greece in 1975 and his PhD Table 1) . from the University of Minnesota in 1980. He is presently an assistant Homework problems are assigned almost professor in the Department of Chemical Engineering at Rice Uni­ every week. In addition, the students are required versity. His main research interests are in the areas of reaction engineering, applied mathematics and numerical methods. to complete one or two computational projects. They also have to take a mid-semester and a final exam, which consist of both open- and closed-book second course covers the theory of differential and parts. integral operators, again using the functional analysis approach. COURSE CONTENTS The course meets twice a week for two hours and runs largely as a lecture, although active The linear algebra part of the course (see student participation is encouraged by frequent Table 2) consists of four parts: questions from the instructor. The lectures are • Vector spaces and linear transformations accompanied by tutoring sessions which are de- • The solution of systems of linear equations TABLE 2 Topical Outline of the Linear Algebra Course 1. VECTOR SPACES AND LINEAR • Overview of iterative methods for solving linear TRANSFORMATIONS equations. • Comparison of the various numerical algorithms. • Overview of the problem of solving systems of linear equations. Which applications give rise 3. THE EIGENVALUE PROBLEM A x = AX to such systems? Which are the theoretical porblems that must be answered? • Determinants. • Vector spaces and subspaces. • Inner products, norms, orthogonality. • Linear dependence, basis and dimension. • Eigenvalues and eigenvectors of matrices. • Linear transformations between finite-dimension­ • Diagonalization and similarity transformations. al spaces and their matrix representation. • Systems of difference equations. • Rank and nullity of linear transformations. • Functions of matrices. • Elementary matrices and the computation of the • Solution of systems of ordinary differential rank of a matrix. equations. Stability. • The theory of simultaneous linear equations. • Unitary transformations. Normal matrices. Homogeneous and nonhomogeneous systems. • Spectral decomposition of operators. The Fredholm alternative. 4. QUADRATIC FORMS AND VARIATIONAL 2. SOLUTION OF SYSTEMS OF LINEAR PRINCIPLES EQUATIONS A x = b • Positive definite quadratic forms. • Gaussian elimination. LU---decomposition, pivot­ • Minimization problems. Least squares. ing, operation count. • Rayleigh quotient. Maximum and minimax • Error analysis. Ill-conditioned matrices. principles. • Band matrices and how they arise in practice. • Numerical computation of eigenvalues and Finite differences solution of partial differ­ eigenvectors. ential equations. • Overview of the finite elements method. FALL 1984 177 The students are thus presented with our objectives for the first part of the course. A brief review of the algebra of matrices follows, reminding the student of the familiar concepts of multiplying a matrix by a scalar to obtain another matrix and of summing two matrices to obtain a third one. • The Eigenvalue problem problem of linear algebra A x = b. This is ac­ • Quadratic forms and variational principles complished in one lecture using the previously The Linear Equation Problem A x = b derived theorems. Throughout this part of the course, emphasis The course starts with an introduction to the is placed on the generality of this approach, and problem of solving systems of linear equations of the students have the opportunity to see how the the form A x = b. Several applications that give results apply to linear differential and integral rise to such large systems are discussed and the operators, as well as to chemical engineering three fundamental questions are introduced: problems. Such examples include first-order re­ • Do these problems have a solution? action systems and the determination of the • If they do, is the solution unique? number of independent chemical reactions in a • How can the solution be computed? closed system using experimental measurements. The students are thus presented with our ob­ The practical problem of efficiently computing jectives for the first part of the course. A brief the solution of systems of linear equations can now review of the algebra of matrices follows, remind­ be considered. The Gauss elimination procedure ing the student of the familiar concepts of multi­ and the LU decomposition are introduced, which plying a matrix by a scalar to obtain another lead naturally to the idea of the operation count matrix and of summing two matrices to obtain a as a measure of the computational effort required. third one. It is also pointed out that these opera­ An important application which gives rise to large tions satisfy certain properties such as associativi­ systems of linear equations is then studied by ty, commutativity, distributivity, etc. This dis­ introducing the finite-difference method for solv­ cussion serves as the motivation to introduce the ing ordinary and partial differential equations notion of abstract linear vector spaces. Several subject
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