Reform the semester course Linear Algebra,
offered to 1st year students of all IST Engineering degrees

This reform was based on the same principles adopted to reform the Mathematical Analysis III and IV courses in 1983-85 and was carried out in 1985-88, leaving the responsibility of the Mathematical Analysis III and IV courses to Manuel Ricou and Carlos Rocha, both with PhDs obtained in 1985. The course was transformed to be a modern first course on Linear Algebra, with:

• Rigorous mathematical foundations,
• Emphasizing the most relevant concepts for Mathematics and Applications,
• Considering from the outset linear spaces infinite dimensional spaces, in connection with differential and integral equations,
• Diverse applications to areas of Engineering and Science,
• Reinforcing the links with Geometry and Mathematical Analysis,
• Adopting a component of students continuous assessment who had to solve weekly short quizzes taken into consideration for the final grade in the course which was mostly based in final written exams,
• Introducing in this 1st semester of 1st year Mathematics course Mathematics topics before they were needed in courses of other areas, instead of letting them be first mentioned in Physics or Engineering courses.

The adoption of the component of students continuous assessment by weekly short quizzes was mostly because it was antecipated a conservatism of the staff involved in teaching the course in previous years, and it was found necessary to adopt a system that would guarentee that all classes would timely follow the new orientation. For this, the quizzes questions were all written by two Teaching Assistants who were teaching other courses before and had full agreement with the intended changes, namely Paulo Almeida and José Ribeiro, who diligently took this heavy job with enormous energy and generosity.

The reform was so impactful that almost all the course former teachers left IST alomg the three years because "it had become too much work" and they worked part-time at IST. Some of them said they found the effort to improve the course a loudable initiative in the general interest of students and IST, and were leaving only because they did not have the time to comply with the new welcome quality of service requirements.

This initiative resulted in the book, Luis T. Magalhães, Älgebra Linear como Introdução a Matemática Aplicada, Texto Editora, Lisboa, 1989, which had 8 editions, the last one in 1998.

The adopted approach broke with the traditional perspective of teaching Linear Algebra in Portugal and influenced curricular changes in the other Science and Engineering schools. Although highly influenced by the book Gilbert Strang, Linear Algebra and Its Applications, 2nd edition, Academic Press, New York, 1980, taught at MIT, it departed from this book in several aspects, such as:

• by not shying away from the definition of linear space introducing it immediately after an initial chapter on Gaussian elimination and dealing with the following material in that general context,
• by fully addressing linear transfomations from the beginning (after all most of the Linear Algebra strength comes from the use of intrinsic notions independent of coordinate systems),
• by addressing along the whole text infinite dimensional issues and examples with differential or integral equations (not only many of the most important applications of Linear Algebra involve infinite dimensional spaces as several general concepts also useful in finite dimension first appeared in infinite dimenstion contexts, and even the teaching of Linear Algebra to 1st year Engineering, Physics and Mathematics students only became widespread afte the publication of the influential book S.Banach, Théorie des Opérations Linéaires, Monografje Matematyczne, Warszawa, 1932),
• by including much more examples and exercises on relevant applications to Engineering, Physics and other areas of Mathematics, and not only of scholar interest.

Some students enrolled for Engineering degrees who later transferred to Mathematics said that the book and, in particular, its many exercises on relevant applications to Engineering and other Sciences was very enlightening when they studied it and influenced their career choices.

Before the book was available, it was provided to students in the form of lecture notes distributed by the IST Sudent Union (AEIST), improved every year. The first version of the lecture notes was written while the course was being lectured in 1985/86 with the new approach, and each weekly content was provided in advance to students and Teaching Assistants in separate issues, what was a very demanding task requiring exceptional committment and energy from the author.

The idea of writing the lecture notes was initially mainly to avoid reactive excuses that could come of adopting a text in English and, since there was not a Portuguese text appropriate for the new approach, it had to be written. However, in the writing process it soon became apparent that the book would be innovative even globally, in particular, by the fertile use of applications in infinite dimension relevant to Engineering and Physics, to an extent that was not observed at the then existing texbooks worldwide. Indeed, this perspective began appearing in textbooks of other authors much later, of which one of the first clear examples is the book Peter J. Olver and Chehrzad Shakiban, Applied Linear Algebra. Pearson Prentice Hall, Upper Saddle River, New Jersey, 2006.