This course covers the fundamentals of linear algebra including matrices and systems of linear equations, determinants, vectors spaces, linear transformations, orthogonality, and eigenvalues.
Throughout this lecture series, I introduce the relevant topics (listed above) in a total of six chapters and each lecture corresponds to a single section for each chapter.
Each lecture introduces new topics through a sequence of definitions with follow-ups on theory and examples to make the concepts more concrete. Occasionally, there will be coding throughout the series that is used to apply the theory to realistic contexts in science.
In the first section, I define matrices along with special types and operations that can be performed on matrices. I will then speak about row operations between matrices and solving linear equations using Gaussian elimination.
In the second section, I speak about determinants for square matrices along with calculations using the classical adjoint and cofactor expansion.
In the third section, I discuss the eight axioms that must hold for a vector space. Additionally, I speak about the minimally generating set of a vector space, the dimension of a vector space, and properties of linear combinations of vectors such as linear independence. I will then speak about row and column spaces, which will be used to determine the consistency of solving linear equations.
In the fourth section, I discuss the Linear Transformation that acts as a function between vector spaces.
In the fifth section, I discuss Inner Product spaces with an emphasis on orthogonality, which is a property of inner products. In this section, I introduce Gram-Schmidt Orthogonalization, a method to construct an orthonormal set of vectors.
In the last section, I introduce the eigenvalue problem which bridges the gap between linear operators, vector spaces, determinants, and calculus. The eigenvalue problem is the most common problem utilized in applications.
The course is conducted through a series of lectures with downloadable pdf beamers and is based on Stephen J. Leon's: Linear Algebra with Applications 9th edition.
