SPRING 2001
This information effective for Spring 2001.
Check with instructor the first day of class for any changes.
Spring 2001
Instructor: Bruce Cooperstein
MWF, 11:00 - 12:10
Text: Linear Algebra With Applications, by George Nakos and David Joyner, 1998. Brooks/Cole Publishing Company, Pacific Grove, California
Linear algebra arose from several sources: the study of integer quadratic forms by Gauss in the nineteenth century, the theory of determinants, and the solution of systems of linear equations. Present day treatments of linear algebra primarily reflect the last of these well springs.
Linear algebra has many applications - in engineering, science, and social science. Examples include, error correcting codes used for the reliable transmission of information, quantum mechanics, computer graphics, network analysis, imput-output analysis in economics and ecology, design of control systems such as those used in space flight and dozens of others
The learning objectives for this class are both algorithmic (computational) and abstract. As an example of the former, to be successful in this class, at a minimum, students will demonstrate the ability to:
In the course of mastering these skills, methods, techniques and algorithms, students will be required to learn many other topics as well, in particular: matrix notation, vectors, row reduction, the span of a set of vectors, and the algebra and properties of complex numbers.
In addition, students will also be expected to understand the notions of an abstract vector space, subspace of a vector space, linear span of a set of vectors in vector space, linear independence and linear dependence of a set of vectors in a vector space, basis of a vector space, dimension of a vector space, as well as the concept of a linear transformation from one vector space to another vector space.
My approach to lecturing this course will be novel and attempted at UCSC for the first time in a mathematics class. Instead of writing at a black board with my back to the class, I will have prepared lectures and will project these onto a screen. Subsequent to class the lectures will be on-line and available for downloading. Hard copies will also be on reserve in the Science Library for copying. Our time in class will be spent discussing new concepts and procedures and doing multiple examples. Students will be expected to come to class prepared to discuss the material, ask and answer questions, and do problems. The goal is for the lectures to be lively.
The course will be supported by a web site where a dictionary of terms and concepts will be collected, where methods, algorithms, procedures, and routines will also be available. There will also be a description of types of problems encountered and examples. Further the course web site will have a page where students and faculty can carry on threaded discussions.
Spring 2001
Instructor: Bruce Cooperstein
MWF 9:30 -10:40
This course will be unlike any mathematics class you have yet taken and probably any that you will subsequently take at UCSC. In this course you will learn mathematics by doing rather than listening. By learning to do mathematics, I mean much more than become acquainted with some concepts and objects, though there will loads of those. And it will be more than familiarity with theorems about these new concepts, though there will be loads of these, too. In addition to these traditional aspects of learning mathematics you will also learn techniques which include strategies and tactics, including psychological strategies. We will learn how to get started and how to persist. We will learn how to monitor our thinking in order to gauge whether we are making progress and should continue along the plan already formulated, or whether what initially appeared fruitful is barren and requires backtracking and the choice of a new approach. We will learn different methods of argumentation We will learn how to investigate and formulate conjectures. We will learn fundamental tactics for solving a wide variety of mathematics problems.
By problems I mean novel problems, problems that you have not previously encountered and therefore you will be unable to fall back on some memorized template which requires little more than plugging in numbers or following a recipe. You will learn how to orient yourself to a new problem and figure out what is given and what is being asked. You will learn how to formulate a plan and implement it. It is one of the objectives of this course that your attitudes and beliefs about mathematics will change, among them: what it means to do mathematics, the role of effort and persistence, what constitutes a proof, who can do difficult mathematics.
The course will be different in other ways. Not only will I not lecture, but I will not be the final authority in class. Rather, when a problem is being "solved" by a student in class, I will lead a discussion on its merits. In particular, we will together decide such questions as: "Is the solution persuasive?" "Has the solution been communicated effectively?" "Has it been argued rigorously enough?" "Have all elements been taken into account?" "Can a more general problem be solved?" "Are there alternative approaches?" "Can the proof be simplified?" At other times I will lead the class in discussing the features of a problem that are general - the application of a new tactic or strategy that might be usefully employed in different contexts.