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Linear Algebra

Linear Algebra

When I started teaching this subject I found three kinds of texts. There were applications books that avoid proofs and cover the linear algebra only as needed for their applications. There were advanced books that assume that students can understand their elegant proofs and know how to answer the homework questions having seen only one or two examples. And, there were books that spend a good part of the semester multiplying matrices and computing determinants and then suddenly change level to working with definitions and proofs. In my classroom each of these types was a problem. The applications were interesting but I wanted to focus on the linear algebra. The advanced books were beautiful but my students were not ready for them. And the level-switching books resulted in a lot of grief: students estimated that these were like calculus books, where there is material labeled `proof' that can skipped in favor of computations, and when the level switched no amount of prompting by me could convince them otherwise. That is, my students cannot now perform at the level assumed by the advanced books. But my goal is to work steadily to have them come up to that level over the undergraduate program. This course is a great place to make progress on this goal. This goal leads straight to a number of tasks. It means first that we must prove things. It means also that we must step away from the rote computations of the applications books in favor of understanding the concepts (for instance, students must understand matrix-vector multiplication as representing the application of a linear function). But, it means also being sure that the approach is not too advanced for the current level of the students: the presentation must emphasize motivation and naturalness, have many examples, and have many exercises, particularly the medium-difficult questions that challenge a learner without overwhelming them.

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