The Journal of Online Mathematics and Its Applications, Volume 7 (2007)
Experiments with Matrices and Vectors in Calculus and Linear Algebra, Sulewski, Kobylski, and Wattenberg
The central idea in the USMA program is cadet leadership development. This idea transforms the academic program from a collection of individual courses taught by individual departments to a 47 month program taught by a team with disciplinary frontiers that are sites of intense activity rather than boundaries. A large part of cadet development is the continual process of moving beyond and enlarging every cadet's comfort zone. In mathematics courses we try to use problems that stretch our very best students and that move all our students forward. The imageAnimations OSSLET supports a wide range of challenges from "confidence-building" exercises that virtually all of our cadets solve to problems that challenge even the most well-prepared and motivated students.
We want our students to develop the ability to see beyond technology as a "black box." This is especially important as military leaders and public policy decision-makers rely increasingly on compelling simulations -- simulations that are often judged by superficial sound and imagery but should be judged on the basis of the underlying mathematical models. This unit reveals some of the mathematics behind computer-based simulations. We hope that our students will think about vectors and matrices as they watch Superman, Star Wars and Harry Potter and as they read the specs for computers and graphics cards.
Modern technology from graphics processing units that perform trillions of floating point operations per second to digital movie cameras that cost only a few hundred dollars have opened up new means of communication and placed those means in every person's hands. Virtual reality is a means of realizing imagination. Problems like the Opening Scene Challenge and the Closing Scene Challenge (Open the OSSLET if it is not still open) provide an opportunity to talk about communicating ideas visually. Don't forget to click the Play button for the Opening Scene Challenge. What movie scene does this challenge recall?
Fundamentally this OSSLET is about using the formal language of mathematics to describe, communicate and realize visual effects.
Last and certainly not least we use this OSSLET to develop knowledge and skills related to matrices, vectors, and their use in computer graphics.
One of the key steps in problem-solving is verifying a solution. For many of our entering students this means checking the answer in the back of the book. The students check their work on the problems we've discussed here by seeing whether their animations match the challenges.