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Importance of Linear algebra in Engineering Design Methodology

Dr. Mysore Narayanan MIAMI UNIVERSITY Hamilton, Ohio – 45011-3399 PH : 513 785 3240 FAX : 513 785 3145 narayam@muohio.edu Most mathematicians define Linear Algebra as that branch of mathematics that deals with the study of vectors, vector spaces and linear equations. Modern mathematics also relies upon linear transformations andsystems of vector matrix. Analytic geometry utilizes the techniques learned during a study of linear algebra, for analytically computing complex geometrical shapes. In addition to science, engineering and mathematics, linear algebra has extensive applications in the natural as well as the social sciences. Linear algebra today has been extended to consider n-dimnsional space. Although it is verydifficult to visualize vectors in n-space, such n-dimensional vectors are extremely useful in representing data. One can easily summarize and manipulate data efficiently in this framework, when data are ordered as a list of n components.

Since the students who take pre-calculus have very little knowledge about the subject of matrices, it has become very important to treat the subject matter indepth. In 1989, the NCTM recognized the need for greater emphasis on linear algebra and stated that "matrices and their applications" should receive "increased attention" in high school curriculum. It should be recognized that linear algebra is as important as calculus to scientists and engineers.

In linear algebra one studies sets of linear equations and their transformation properties. It ispossible to consider the analysis of rotations in space, selected curve fitting techniques, differential equation solutions, as well as many other problems in science and engineering using techniques of linear algebra. Two tools are extensively used in linear algebra. They are : The Matrix and The Determinant. Solution to a vector matrix model equation is regarded as one of the most important of‘central problems’ of linear algebra. Study of vectors in two dimensional as well as three dimensional space is extremely important for design engineers. A course was specially designed to provide the engineering and engineering technology students with all the necessary mathematical tools that are essential and necessary for a four-year program. The course begins with the coverage of a wide variety ofintroductory topics such as complex numbers, partial fractions, determinants, Taylor and Maclaurin Theorems, etc. Vectors, vector spaces, scalar products and vector products are considered extremely important. About weeklong discussion is incorporated in the course to ensure that the students obtain a very strong

foundation that pertains to vector operations. Matrices and Matrix operations arecovered extensively. Eigenvalues and Eigenvectors, Diagonalization of Matrices are considered essential foundation for subsequent engineering courses and as such several homework exercises are necessarily assigned in this area. Eigenvectors are extremely important while creating engineering models whether it be a satellite or a jet engine. Study of the dynamics of a football trajectory needs theknowledge of eigenvectors. Eigenvalues can be used to explain several aspects of musical performances. It is very well known that frequencies are vital in music performance. Tuning of an instrument actually means that their frequencies are matched. If some music is pleasing to the human ear, it is because of the frequency. Musicians may not study eigenvalues, but a study of eigenvalues explains whycertain sounds are pleasant to the human ear while certain others sounds are not. When groups of people sing in harmony, the frequency matters a lot. Engineers utilize Ordinary differential equations extensively and as such, separable, exact and homogeneous equations are discussed in great detail. Riccati Equations are stressed in particular because of their importance in as Fluid Mechanics,...