This tutorial from Santa Fe Institute’s Complexity Explorer introduces students to essential ideas related to vectors and matrices. These mathematical structures form the foundations for many key topics in complex systems, such as dynamical systems, stochastic processes, and network science. The prerequisite for this tutorial is knowledge of high-school algebra. The content of the tutorial is built, in a self-contained fashion, starting with basic notions of real numbers and elementary set theory. Ideas of vectors and vector operations are developed next, in an intuitive way, by appealing, simultaneously to their algebraic and geometric underpinnings. Next, the tutorial explores matrices and vector spaces, determininants and eigenvalues with, again, an eye toward understanding the intuitive geometric and algrebraic connections that tie these notions together. Finally, the tutorial concludes with a survey of applications of matrix algebra, including diagnolization, recursion, geometric transformations, differential operators and Markov Chains.

Importantly, the content and emphasis of this material differs significantly from a standard university course in linear algebra. Instead of solving and analyzing systems of linear equations of the form Ax=b, as is conventional from the perspective of linear algebra, students will instead be exposed to the fundamental ideas of matrix algebra in a less restrictive and more conceptually-integrated way. At the conclusion of this tutorial, students will be equipped with a core understanding of the breadth and power of matrix algebra as an essential tool for complex systems research.

For more information and a complete list of courses and tutorials, visit www.complexityexplorer.org

This physics video tutorial provides a basic introduction into snell’s law which shows the relationship between the angle of incidence and the angle of refraction given the index of refraction of the two materials. This video explains how to calculate the speed of light of water given the index of refraction of water or how…

Amazing, Life Changing Tutor
Before I met Jonathan, I was struggling through most of my STEM classes because I was simply not studying properly. He taught me all kinds of new study habits that would help me to save time, raise my grades, and lower my stress. Jonathan was specifically tutoring me in vector calculus and is an amazing tutor on the subject. You will walk away from a lesson with a game plan knowing exactly what you need to do before your next session with him or before your next exam in order to do well in the course. I highly recommend him as a tutor.

Amazing, Life Changing Tutor
Before I met Jonathan, I was struggling through most of my STEM classes because I was simply not studying properly. He taught me all kinds of new study habits that would help me to save time, raise my grades, and lower my stress. Jonathan was specifically tutoring me in vector calculus and is an amazing tutor on the subject. You will walk away from a lesson with a game plan knowing exactly what you need to do before your next session with him or before your next exam in order to do well in the course. I highly recommend him as a tutor.

In this video I will show you how to convert a number from rectangular and polar form and back. This is very useful for when you are working with complex numbers.

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