Multilinear Transformations#

Definition: Multilinear Transformation

Let \(V_1, \dotsc, V_m, W\) be vector spaces over the same field \(F\).

A function \(f: V_1 \times \cdots \times V_m \to W\) is multilinear if for each \(i \in \{1, \dotsc, m\}\) and all \(\mathbf{v}_j \in V_j\) (\(j \ne i\)), the function \(f(\mathbf{v}_1,\dotsc,\mathbf{v}_{i-1},\cdot,\mathbf{v}_{i+1},\dotsc,\mathbf{v}_m): V_i \to W\) is linear.

Example: Matrix Product

The matrix product \(f: F^{m \times n} \times F^{n \times p} \to F^{m \times p}\) is bilinear.