# Bilinear map/Definition

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< Bilinear map

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Given the vector spaces [ilmath](U,F),(V,F)[/ilmath] and [ilmath](W,F)[/ilmath] - it is important they are over the same field - a *bilinear map*^{[1]} is a function:

- [math]\tau:(U,F)\times(V,F)\rightarrow(W,F)[/math] or
- [math]\tau:U\times V\rightarrow W[/math] (in keeping with mathematicians are lazy)

Such that it is linear in both variables. Which is to say that the following "Axioms of a bilinear map" hold:

For a function [math]\tau:U\times V\rightarrow W[/math] and [math]u,v\in U[/math], [math]a,b\in V[/math] and [math]\lambda,\mu\in F[/math] we have:

- [math]\tau(\lambda u+\mu v,a)=\lambda \tau(u,a)+\mu \tau(v,a)[/math]
- [math]\tau(u,\lambda a+\mu b)=\lambda \tau(u,a)+\mu \tau(u,b)[/math]
- ↑ Advanced Linear Algebra - Steven Roman - Third Edition - Springer Graduate texts in Mathematics