In our introductory lecture, Dr. Snellman covers precalculus fundamentals, focusing on functions as input-output rules and their operations—addition, subtraction, multiplication, division, and composition. Together, we explore inverse functions as tools that reverse original functions, with examples using exponentials and logarithms. The lecture concludes by showing how inverse functions reflect across the line y = x, revealing their symmetrical relationship.

In lecture two, we investigate complex numbers, tracing their origins to the 1500s as solutions to quadratics with negative discriminants, introducing the imaginary unit i. We cover arithmetic operations, the conjugate, modulus, and representations in standard and polar form using Euler’s formula. Finally, we apply polar form to solve polynomial equations, revealing how complex numbers and roots of unity are fundamental to advanced applications in fields like harmonic analysis and signal processing through the fast Fourier transform.

In lecture three, we delve into matrices and vectors as systematic tools for solving systems of linear equations, moving beyond graphical methods to handle multiple variables and equations simultaneously. We explore matrix arithmetic operations, then demonstrate how to represent systems of linear equations in matrix form and solve them using matrix inverses when they exist. Dr. Snellman concludes by connecting these concepts to linear regression, showing how matrices enable us to find the best-fit line through data points, which forms the fascinating foundation for machine learning, data science, and AI applications that are transforming our world today.