Group theory arises throughout mathematics, but this is not at all clear to someone learning it for the first time. Many of the applications, are in areas such as number theory, combinatorics, geometry, and topology, which are more advanced and typically learned later. In this lecture, we’ll see some of the basic theory of homomorphisms that we’ve learned such far arise in more remedial material, such as linear algebra, differential equations, calculus, logarithms, and even trigonometry. Though this theory is not needed to understand these subjects, seeing how it arises behind the scenes should lead to a deeper appreciation for how ingrained group theory is in mathematics.
Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule/visualalgebra/
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CHAPTERS
0:00 Introduction
0:55 Topic overview
2:16 Linear algebra: systems of equations
6:30 Differential equations: solutions to inhomogeneous linear ODEs
16:34 Calculus: constants of integration
20:07 Precalculus: exponentials and logarithms
24:41 Trigonometry: angle addition formulas