LECTURE : BILINEAR FORMS

Today we will focus on extra structure we can put on a vector space that render certain constructions more natural.

Definition. A \underline{bilinear form} ⟨−, −⟩ on V is a function

⟨−, −⟩: V × V → 𝔽

that is \underline{bilinear}:

⟨av⃗ + bu⃗, w⃗⟩ = a⟨v⃗, w⃗⟩ + b⟨u⃗, w⃗⟩ and
⟨v⃗, au⃗ + bw⃗⟩ = a⟨v⃗, u⃗⟩ + b⟨v⃗, w⃗⟩.

It is important to note that ⟨−, −⟩ is a function on V × V as a set and not a linear map on the direct sum V ⊕ V. This makes this definition a little unsettling: we consider structure on vector spaces that does not stay in the category of vector spaces.

We should think of ⟨−, −⟩ as a generalization of the dot product on ℝ^{n}. This provides a fantastic example.

Example. (1) The dot product on ℝ^{n} is a bilinear form.

(2) The generalization of the dot product on 𝔽^{n} is a bilinear form.

(3) If a_{1}, …, a_{n} are elements of ℝ (or 𝔽), then ⟨v⃗, w⃗⟩ = a_{1}v_{1}w_{1} + ⋯ + a_{n}v_{n}w_{n} is a bilinear form.

The last part is very helpful, and it helps us tie bilinear forms to matrices.