Bilinear Forms

A bilinear form is a mathematical function that takes two vector inputs and produces a single number, with the key property that it's linear in each input independently. This means if you change one input slightly, the output changes proportionally, while keeping the other fixed. Think of it like a special kind of "product" that takes two vectors and measures some consistent relationship between them, such as length, angle, or projection. Bilinear forms are fundamental in areas like geometry and physics, helping analyze shapes, spaces, and how different directions relate within a given context.