MATHMATICS 2270-1 Linear Algebra Fall Semester 1998

The course begins by studying linear systems of equations and the Gauss-Jordan method of systematically solving them. We see how to write these problems more succinctly in matrix form, the algebra of matrix operations, about inverses of non-singular square matrices, about determinants and their usefulness in solving linear problems. These topics comprise chapters 1-2 of the text.

In chapter 3 we review the linear geometry of R²,R³, and Rⁿ and discuss the geometric meaning of matrices and determinants, as well as the dot and cross products and their geometric meanings. After chapter 3 we skip to the application called linear programming (chapter 7), which is used heavily in business.

At this point the course takes a turn towards the abstract as we study general vector spaces in chapters 4 and 6. Basically a vector space is a collection of objects (called vectors) which you can add and scalar multiply, such that certain arithmetric properties hold. From these arithmetric properties one develops notions such as linear independence, bases, dimension, subspaces, coordinates with respect to a basis, change of basis, linear transformations between vector spaces, kernel and range subspaces of linear transformations. We usually visualize vectors in the concrete example of Rⁿ, but in fact there are very natural spaces of functions and of solutions to certain (homogeneous linear) differential equations which are also vector spaces, so that these abstract concepts also apply to them. It is precisely because vector spaces appear in these different disguises that it is worthwhile to discuss them in this abstract way: as characterized by properties rather than by explicit descriptions. You will appreciate this more when you take Math 2280 and apply vector space theory to your study of linear differential equations.

We will discuss the notion of eigenvectors and eigenvalues for linear transformations from Rⁿ to Rⁿ,in chapter. These will also be used heavily in Math 2280.

The dot product in Rⁿ lets one talk about orthogonality and orthogonal projections, and we discuss several applications related to this circle of ideas: Gram-Schmidt orthogonalization, methods of least squares (8.4), diagonalizing quadratic forms (8.8), rotating space to express conic sections and quadratic surfaces optimally (8.9-8.10).

We will show there is also a natural ``dot'' product on certain function spaces, and use this to motivate Fourier series (appendix B.1) which you will return to in Math 2280.

Although we will use mostly real (scalar) vector spaces, it is often important in applications to allow complex numbers as your scalars, and we discuss this in Appendix A. If there is time at the end of the semester we will also spend a couple of lectures sketching some of the applications of our work to differential equations (8.6), as a way of forshadowing your work in Math 2280.

There will be approximately 6 computer projects during the semester, to enhance and expand upon the material in the text. They will be written in the software package MAPLE. On MAPLE days we will meet in the Math Department Computer Lab located in Building 129, between JWB and LCB. We do not assume you have had any previous experience with this software and we will make the necessary introductions during the first visit to the lab.