Advanced Linear Algebra

This section covers advanced topics in linear algebra, and serves as a continuation of the Linear Algebra section. It's intended for a more mathematically mature audience, and assumes a solid understanding of the basics of linear algebra. Generally, this section will cover some "graduate-level" topics in linear algebra, like dual spaces, tensor products, and more.

How do We Teach Linear Algebra?

Linear algebra is a fundamental subject in mathematics, and is taught at various levels, from high school to undergraduate and graduate levels. The way linear algebra is taught can vary significantly depending on the level of the course and the audience.

Firstly, many people emphasize on the basis-free, theoretical approach to linear algebra. This approach focuses on the abstract properties of vector spaces, linear transformations, and other structures. Vector spaces are defined axiomatically, and theorems are proven about these abstract structures. In fact, vector spaces are a special case of a more general algebraic structure called a module, and many results in linear algebra can be generalized to modules over a ring.

On the other hand, when we actually apply linear algebra to solve problems, we will need to work with matrices and vectors in a more concrete way. So some courses instead focus on the computational aspects of linear algebra, such as matrix operations, solving systems of linear equations, and finding eigenvalues and eigenvectors. These courses are often more applied and less theoretical, and are more focused on applying linear algebra to solve other problems.