Vector and Matrix Algebra

Complexity Explorer


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This tutorial 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.

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vector algebra
Vector Spaces
Mathematical structures


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. 

  1. Sets and Vectors
  2. Matrices and Vector Spaces
  3. Matrix Applications