Linear Algebra

The course begins with an introduction to vectors and vector spaces. Students learn the definitions and properties of vectors, vector spaces, and subspaces. This section also covers linear combinations, span, and basis, providing a foundation for understanding dimension and rank.

The curriculum then focuses on matrices, exploring matrix operations such as addition, multiplication, and transposition. Students study different types of matrices, including square, diagonal, identity, and zero matrices. The section also covers the inverse of a matrix and various matrix factorizations, such as LU and QR decomposition.

Students learn to solve systems of linear equations, focusing on the existence and uniqueness of solutions. Techniques such as Gaussian elimination and Gauss-Jordan elimination are covered, along with row reduction and echelon forms. This section also distinguishes between homogeneous and non-homogeneous systems.

The curriculum includes a detailed study of determinants, explaining their definition and properties. Students learn to compute determinants using methods such as Laplace expansion and row reduction. The applications of determinants, including volume calculation and Cramer’s rule, are also explored.

This section delves into the definition and properties of eigenvalues and eigenvectors. Students learn about the characteristic polynomial and the process of diagonalization of matrices. The applications of these concepts to differential equations and dynamical systems are also highlighted.

Orthogonality is another key topic, where students study inner products, norms, and distances. The curriculum covers orthogonal and orthonormal sets, the Gram-Schmidt process, and orthogonal projections. Practical applications such as least squares problems are also included.

Students explore linear transformations, learning their definitions and various examples. The section covers the kernel and image of a linear transformation, matrix representation of linear transformations, and the concept of change of basis and similarity

Finally, the curriculum emphasizes the real-world applications of linear algebra. Topics include applications in computer graphics, data science, and machine learning, as well as Markov chains and linear programming. Advanced courses might also cover topics like Jordan Canonical Form and Singular Value Decomposition (SVD), providing a comprehensive understanding of linear algebra and its practical uses.