Quadratic Forms and Symmetric Bilinear Forms

This lecture covers the definition of quadratic forms and symmetric bilinear forms on vector spaces over arbitrary fields. It explains the relationship between quadratic forms and symmetric bilinear forms, providing proofs and examples. The lecture also discusses non-degeneracy, uniqueness, and the associated symmetric bilinear form. The instructor demonstrates how to reduce a quadratic form to a sum of squares using the completion of squares method. Additionally, the lecture explores the concept of orthogonality in the context of quadratic and bilinear forms, illustrating the properties and applications of these mathematical structures.