Math 221 - Introduction to Linear Algebra

Learning Goals

• understand and remember the key ideas, concepts, definitions, and theorems of the subject. Examples in this course include linear dependence and independence, the concept of a vector space, eigenvalues and eigenvectors, and the rank theorem. --> The quizzes, homework, and exams test this foundational knowledge.
• be able to apply mathematical theorems and computational algorithms correctly to answer questions, and interpret their results correctly, including potentially non-unique solutions or breakdowns of algorithms. Examples include choosing among several theorems that help decide if a matrix is diagonalizable and the algorithm for row reduction that is non-unique in its steps and that may break down, and we need to know how to interpret these breakdowns. --> The group discussions, homework, and tests address these skills.
• be able to choose the most useful theorem or most efficient computational algorithm in a particular circumstance. Examples include the Invertible Matrix Theorem that has many parts among we can choose and the computation of determinants which can be done more efficiently if selecting different algorithms for different matrices. --> These more subtle skills are best discussed in a learning group.
• appreciate the power of mathematical abstraction and understand how mathematical theory is developed. The classical example of mathematical abstraction in this class is the axiomatic definition of a vector space which is done in abstract generality after observing that the axioms hold true concretely for vectors in the special case of Rⁿ. --> These integration goals will be supported by the lectures.
• be able to communicate orally by discussing mathematical ideas and algorithms with the instructor as well as other students. --> The formation of learning groups will contribute to this goal.
• be able to communicate in writing effectively by using the notation and terminology of the subject correctly. --> Homework, group quizzes, and tests will give you feedback.
• have gained experience and developed skills in using all available types of learning resources discussed in the learning plan. --> This will develop self-confidence for learning mathematics and other sciences.

Role of the Instructor

So, what is the role of the instructor in this learner-centered environment? The role of the instructor is to help you achieve these learning goals. Concretely, the instructor will:

• highlight the important ideas and concepts of this course and put them into the context of mathematics and applications areas outside of this course,
• explain how to interpret mathematical definitions and theorems and introduce computational algorithms with examples,
• show how rigorous mathematical theory is developed using the topic of this course as example,
• make all supporting learning aids available in a timely fashion,
• be available for help and advice,
• give you honest feedback on your performance and be available for suggestions how to improve,
• maintain a professional learning environment in class.

Detailed Learning Plan

We use the textbook as learning tool as well as to organize the activities throughout the semester. Moreover, we will use the course management software Blackboard to post additional information in the Course Documents area, organized by the numerical order of the textbook sections (not necessarily chronologically as covered in the course). Most information for the section will be contained there, including worked examples with voice-over and the on-line quizzes. The only other area of Blackboard that we will use is the Discussion Board that is available to ask questions on the material at any time.

The learning plan divides activities in three parts -- before, during, and after class --, which apply to every covered section of the textbook:

• Before class:
  • Study the section in the textbook, supported by information posted in Blackboard including the worked examples with voice-over and taking into account any announcements in class or by e-mail specifically for this section.
  • Take the on-line quiz posted in Blackboard and receive immediate feedback on your performance. These quizzes focus only on reading comprehension and basic vocabulary and are intended to prepare you for class discussion.
  • Monitor and post questions in the Blackboard Discussion Board.
  Before you arrive in class, you should have an overview of the material in the section, have read and/or seen several examples for its use, and be ready to attempt the homework problems under the guidance of the instructor.
• During class:
  • Follow the lecture which highlights the material and puts it into context.
  • Participate actively in work with all members of your learning group to solve the group quizzes and/or homework problems at the end of class. Discuss and ask questions of your group members and the instructor.
  I will circulate from group to group during the group work periods and be available for questions and guidance. By the end of class, you should have obtained answers to your questions either from group members or the instructor and have an idea of how to approach the homework.
• After class:
  • Work all assigned homework problems. It may be helpful to re-view some of the worked examples posted in Blackboard at this point.
  • If questions arise, review the textbook, notes from class, and examples in textbook or posted on-line; monitor the Discussion Board in Blackboard for questions and answers and use it to ask your own questions if necessary.
  With the shift of work towards preparing more intensively for class as opposed to seeing material for the first time in class, the activities after class should consist mainly of putting all the pieces together. Moreover, the tightly spaced and integrated work before, during, and after class should make the preparation for the tests short and effective.

A note on the worked examples with voice-over: They are flash files and no student has so far reported problems viewing them. They were prepared during Summer 2006, some taped in dedicated sessions and others taped during actual class. The underlying idea was that it would be good for you to be able to view repeatedly, to rewind, to pause, or to fast-forward examples of many crucial algorithms in this course. Additionally, despite the term "worked example", they are more than examples. Some are in fact actual lectures, but even the examples contain usually a thorough review of the theory behind the example, similar to what I would say in a lecture when working an example. Moreover, though many examples are taken from the textbook, which is done for clarity of the organization and to keep the workload of watching the examples manageable, many contain significant extensions. Many examples are also worked out in slightly different ways so as to demonstrate that there is more than one way to execute many algorithms. One student in Summer 2006 wrote to me saying that "Worked examples were different enough (insights, tricks, examples, history/background) that reviewing them was very worth while." Therefore, I urge you to take full advantage of the worked examples both in preparation for the class meetings as well as for review later on during the work on homework or during preparation for the tests. And do help everybody by letting me know if you find anything wrong or not working with the worked examples.

This learning plan relies on your active participation in all learning activities; precisely for this reason, you will learn the material better, retain more in the long run, and increase your self-confidence as a learner!