Advanced Linear Algebra by Andrews University

About this course

Strengthen the mathematical foundation that powers modern science, engineering, and data-driven work. This free online course in advanced linear algebra helps you move beyond routine matrix manipulation and into the ideas that make the subject truly useful: structure, abstraction, and the ability to recognize the same pattern appearing across different problems. You will gain confidence working with vector spaces and subspaces, spanning sets, bases, and dimension, and you will learn how these concepts organize everything that follows.

The course develops a clear, intuitive understanding of linear transformations and how to represent them with matrices, including how changes of basis alter the same underlying map. Along the way, you will sharpen your ability to reason about independence, coordinate systems, and what it means for two spaces to be isomorphic. This perspective is essential when switching between concrete settings like Rn and more conceptual ones such as polynomial spaces, where transformations like differentiation become a powerful example of linear structure.

You will also explore the geometry that linear algebra captures through inner products, norms, orthogonality, and orthonormal bases. Ideas such as Gram-Schmidt and adjoints connect computation with meaning, preparing you for deeper results about eigenvalues and matrix structure. Topics like Schur triangularization, normal matrices, the real spectral theorem, and positive definiteness show how symmetry and stability lead to cleaner decompositions and reliable interpretations in applications.

As the course progresses, you will encounter core tools used across applied mathematics and computing: polar decomposition, singular value decomposition, operator norms, and Jordan form, including generalized eigenspaces and when diagonalization is possible. You will also learn how to make sense of functions of matrices such as the matrix exponential. To connect theory with impact, the course highlights how linear algebra ideas appear in tensors, Markov chains, ranking systems, finance and epidemiology models, neural networks, and spectral clustering. Short exercises help you test understanding, reinforce key assumptions, and build the problem-solving habits needed for exams, research, and real-world projects.