Linear Algebra 1: Linear Equations and Systems





This is the first addition to what will become an ongoing series on the basics of Linear Algebra, the foundational math behind machine learning. This article would best serve readers if read in accompaniment with Linear Algebra and Its Applications by David C. Lay, Steven R. Lay, and Judi J. McDonald. Consider this series as an external companion resource.


Through these essays, I hope to consolidate my understanding of these foundational concepts while if possible, offering additional clarity to others with what I hope to be an intuitive-based approach to learning math. If there are any mistakes or opportunities for me to elaborate further, please share and I can make the necessary amendments.


Linear equations and systems of linear equations have a variety of real-world applications in the domains of Finance, Engineering, Chemistry, Computer Science, Statistics, and Physics and beyond. In Chemistry, linear equations are used to balance chemical reactions and calculate the quantities of reactants and products. This cornerstone of Linear Algebra also appears in Physics where linear equations are used within Kinematics and Thermodynamics to describe the motion of objects, helping calculate distances, speeds, and accelerations and model heat transfer and energy flow in physical systems respectively. The financial field relies on linear equations and systems for budgeting and portfolio analysis while engineers might use the same tools to conduct structural analysis to model forces and stresses in buildings. Linear Algebra is ubiquitous; everyone can appreciate it to some degree.

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