Let

*V,W,L* be

vector spaces over a

field *k*. A function

*f:VxW-->L*
is

**bilinear** if it is linear in each variable. That is

*
f(av+bv',cw+dw')=acf(v,w)+adf(v,w')+bcf(v',w)+bdf(v',w') (*)
*

for

*a,b,c,d* in

*k* and

*v* in

*V*
and

*w* in

*W*.

Equivalently, for each *v* in *V* and each
*w* in *W* the induced functions
*W-->L* defined by *x|-->f(v,x)*
and *V-->L* defined by *y|-->f(y,w)*
are linear transformations.

More generally, if *V,W,L* are modules over a commutative ring
*k* then we can use (*) to define bilinearity in this
more general context.