Introduction to Geometry

Course Purpose
The purpose of this lecture course is to learn advanced linear algebra and its application to geometry.
Learning Goals
Students will get acquainted with complex metric vector spaces, canonical forms of normal matrices, and their application to geometry of quadratic surfaces. Moreover, students should be able to solve fundamental problems on those topics.
Topic
Session 1Diagonalization of a real symmetric matrix by an orthogonal matrix (review)
Session 2Orthonormal basis of Euclidean space, Schmidt's orthogonalization method, QR decomposition
Session 3Orthogonal complements and projections
Session 4Orthogonal matrices, reflections, and rotations
Session 5Standard Hermitian inner products and unitary matrices on complex vector spaces
Session 6Triangularization by unitary matrices (Schur decomposition)
Session 7Normal form of a normal matrix (Toeplitz theorem)
Session 8Hermitian inner product space (complex metric vector space)
Session 9Hyperplanes and quadratic hypersurfaces in Euclidean space
Session 10Center of a quadratic hypersurface
Session 11Standard form of quadratic hypersurfaces
Session 12Standard form of quadratic hypersurfaces 2
Session 13Improper eigenvectors and Jordan normal forms
Session 14Supplementary information and summary
＊＊This content is based on April 1, 2025. For the latest syllabus information and details, please check the syllabus information inquiry page provided by the university.＊＊