Free online courseAdvanced Linear Algebra

Course Description

Delve deep into Advanced Linear Algebra with this comprehensive online course from Andrews University, tailored to elevate your understanding of complex algebraic concepts. This course is essential for anyone seeking to expand their knowledge in algebra and related fields, incorporating a detailed exploration of vector spaces, transformations, and matrix functions.

Discover the intricacies of vector spaces, exploring spans, bases, and dimensions that are fundamental to building a solid algebraic foundation. Progress to change of basis and delve into the mechanics of linear transformations and their properties. Engage with advanced topics such as the half derivative and explore the concept of isomorphic vector spaces.

The course further investigates linear and bilinear forms, touching on the vital concepts of inner products, norms, and orthogonality. Master techniques like the Gram-Schmidt process and understand the importance of an orthogonal basis in solving complex algebraic problems.

Advanced learners can challenge themselves with units on the adjoint of linear transformations and Schur's unitary triangularization. The curriculum extends to normal matrices and the real spectral theorem, providing insights into positive definite matrices and their applications.

Explore polar and singular value decompositions, followed by an analysis of Frobenius and operator norms. Take on the stimulating task of finding blocks of the Jordan decomposition and understanding the broader framework of the Jordan decomposition generalized eigenspace.

Courses on functions of matrices, including exponential and trigonometric functions, prepare students for practical applications across various disciplines. Gain an edge in mathematics and related fields with topics on tensors, Markov chains in finance, epidemiology, and page ranking. The inclusion of linear algebra in neural networks and spectral clustering emphasizes real-world applications, bridging theoretical concepts with practical solutions.