Free online courseLinear algebra

Course Description

Linear algebra is the language behind modern science and technology: it explains how search engines rank pages, how graphics and 3D scenes are rendered, how data is compressed, and why many algorithms in engineering, economics, and AI work at all. This free online course helps you build that language step by step, turning abstract symbols into practical tools you can use with confidence.

You will develop a clear understanding of vectors and matrices as ways to represent information, and of systems of equations as structured problems you can solve efficiently. Along the way, you will learn to reason about what solutions mean geometrically, when they exist, and how ideas like rank, independence, and dimension reveal the real degrees of freedom inside a problem.

The course connects computation with insight. You will work with core operations such as elimination and matrix multiplication while also learning how subspaces organize complicated spaces into simpler parts. Concepts like column space, null space, and orthogonality become useful guides for deciding what a matrix can and cannot do, and for recognizing when projection and least squares provide the best possible answer even when an exact solution is impossible.

As you progress, determinants and inverses stop being memorized procedures and become meaningful tests of invertibility and structure. From there, eigenvalues and eigenvectors open the door to understanding stability, long-term behavior, and diagonalization—ideas that appear in differential equations and iterative processes such as Markov chains.

Later topics tie the subject to real applications, including symmetric matrices, positive definiteness, similarity, and powerful factorizations such as the Singular Value Decomposition and the Fast Fourier Transform. Short exercises throughout help you check comprehension immediately and strengthen your intuition so you can apply linear algebra in exams, further math courses, or technical projects.