Bilinear Forms

This tutorial introduces the quadratic forms associated with symmetric bilinear forms.

In this tutorial we learn that over an arbitrary field, every finite-dimensional bilinear space has an orthogonal decomposition into subspaces of dimension at most two. Over a field of characteristic not two, every finite-dimensional bilinear space possesses an orthogonal basis.

This tutorial explains the connection between bilinear forms over free modules and Gram matrices.

Before embarking on a journey to study quadratic (and hermitian) forms on modules over general rings, we will learn about bilinear forms over rings