Bilinear maps

Let \(U, V, W\) be vector spaces over a field \(\mathbb{F}\). A bilinear function is a function \[\begin{equation} \require{physics} f: U \times V \rightarrow W: (\vb{u}, \vb{v}) \mapsto f(\vb{u}, \vb{w}) = \vb{w} \thinspace , \end{equation}\] such that \(f\) is linear in both of its arguments. This means that \(\forall \vb{u}_1, \vb{u}_2 \in U; \forall \vb{v}_1, \vb{v}_2 \in V; \forall a, b, c, d \in \mathbb{F}:\) \[\begin{equation} f(a \vb{u}_1 + b \vb{u}_2, c \vb{v}_1 + d \vb{v}_2) = ac \thinspace f(\vb{u}_1, \vb{v}_1) + ad \thinspace f(\vb{u_1}, \vb{v}_2) + bc \thinspace f(\vb{u}_2, \vb{v}_1) + bd \thinspace f(\vb{u}_2, \vb{v}_2) \thinspace . \end{equation}\]

An example of a bilinear map is general matrix multiplication. In the most general case, matrix multiplication is a bilinear map between \(\mathbb{R}^{m \times n}\) and \(\mathbb{R}^{n \times p}\) to \(\mathbb{R}^{m \times p}\).

In the case that \(U=V\), and \(W\) is the field \(\mathbb{F}\) itself, we would talk about a bilinear form.

An example of a bilinear form would be an inner product on \(V\).