This paper deals with teaching and learning modern Linear Algebra and it elaborates on this, our most urgent tasks in Mathematics as a whole. More specifically, how should we teach modern Linear Algebra to freshmen today. It is time to create and test first Linear Algebra course Syllabi that propagate modern Matrix Theory and modern Matrix Computations. In this survey we introduce modern examples of first year Linear Alge- bra educational ways and methods, including my own comprehensive set of Lesson Plans. We expose, assess, and discuss them. To prepare our post–Covid students properly, it has become mandatory to use Interactive Teaching Meth- ods and Inversely Taught Syllabi now. Students are well versed today to let the internet help their learnings. Teachers should begin to switch from our top-down near ubiquitous Lecturing of old, long abandoned subjects to modern ways to learn.. My set of Lesson Plans is slowly creating a grass–roots fire among students and catching instructors as well. Their elementary Linear Algebra Course follows the Teaching Principles of Jean Piaget from 80 years ago who established that we only learn deeply and retain what we need to understand from our own with-in, and not what we are taught in order to pass a test for ’Pisa’ statistics, or to graduate. My modern interactive Linear Algebra course has been designed for self study and has been class tested. It relies on only three mathematical principles. First, the Row Echelon Form Reduction introduces ma- trix and vector operations hands-on. It introduces students to software coding and the benefits of pivot searches in matrix row reduction. Then Krylov Vector Iteration allows us to look inside intrinsic matrix qualities such as matrix eigenvec- tors and eigenvalues. Both of these subjects rely in turn on Riesz’s Theorem for Matrix Representations of Linear Functions in varying vector space bases. The proposed Lesson Plan course achieves deep learning of students in inversely taught classrooms. This math education paper addresses our elementary linear algebra teach- ings and deals with the globe-wide problem of ”teaching for the test”, mostly in ’top-down’ lecturing. Unfortunately, we are generally not conveying linear algebra through interactive teaching methods, nor introducing modern matrix theory and matrix computational methods to our students. We must modernize our early college syllabi in linear algebra and enliven this area of mathematics that helps us all so extraordinarily well in our daily cell- phone and internet based lives, and in our current AI and engineering ad- vances. Linear algebra now requires alternate methods of teaching, modified subject lists and modern subject development methods that are congruent with the ubiquitous modern uses of matrix theory and matrix computations.
Frank Uhlig (2026) studied this question.