Free online courseUndergraduate Linear Algebra Full Course

Course Description

Linear algebra is the language behind data science, engineering, physics, computer graphics, economics, and countless modern technologies. This free online course helps you build real fluency in that language, moving step by step from the geometric intuition of vectors to the powerful abstraction of vector spaces and linear transformations. Instead of memorizing disconnected rules, you will learn how the ideas fit together so you can recognize patterns, choose the right method, and explain your reasoning with confidence.

You will start by developing a solid foundation with vector operations and the dot product, including projections and the geometric meaning of algebraic formulas. From there, you will work with matrices as tools for organizing information and performing computations efficiently. As the course progresses, you will see how matrix multiplication encodes composition, why certain properties behave differently from real-number arithmetic, and how these differences matter when solving real problems.

A key focus is solving linear systems using Gaussian elimination and reduced row echelon form, with careful attention to what each row operation does and what solution outcomes are possible. You will connect system equivalence, rank, and row space to a clearer understanding of when solutions exist and how many degrees of freedom you truly have. This prepares you to handle both computational tasks and the conceptual questions that often appear in exams.

The course also develops determinant intuition, not as a mysterious formula, but as a quantity tied to structure and invertibility, and as a practical tool alongside row reduction. From there, you will tackle eigenvalues and diagonalization, learning how to test whether a number is an eigenvalue and what conditions allow a matrix to be diagonalized, with an eye toward simplifying complex computations.

In the final stretch, the perspective widens to vector spaces, subspaces, span, linear independence, and bases, giving you a rigorous framework for understanding dimension and coordinate systems. You will learn how vectors are represented relative to different bases, how transition matrices work, and how linear transformations can be represented by matrices in chosen coordinate systems. The course culminates with the Dimension Theorem and isomorphisms, tying together kernel, image, and structure in a way that makes later math and applications far more approachable.

Throughout, exercises reinforce each concept and help you diagnose gaps early, making this a strong option for self-study, course review, or preparation for more advanced subjects. By the end, you should be able to compute accurately, reason abstractly, and interpret results in a way that makes linear algebra genuinely useful rather than purely procedural.