Bilinear Transformations and Forms

Abstract

Let V, W, and Y be vector spaces over a field F. A bilinear transformation from V × W to Y is a function f:V × W → Y which satisfies the following conditions:

• (1) For each given vector w ∈ W, the function from W to Y given by v ↦ f(v, w) is a linear transformation.

• (2) For each given vector v ∈ V, the function from W to Y given by w ↦ f(v, w) is a linear transformation.