A bilinear form is a mathematical function that takes two vectors as inputs and produces a scalar output. It is linear in each argument, meaning that if you change one vector while keeping the other fixed, the output changes linearly. Bilinear forms are often represented in the context of vector spaces and can be expressed using matrices. Bilinear forms are commonly used in various fields, including linear algebra, geometry, and physics. They help define concepts such as inner products and quadratic forms, which are essential for understanding angles, lengths, and other geometric properties in multi-dimensional spaces.