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.

Bi*lin"e*ar (?), a. Math.

Of, pertaining to, or included by, two lines; as, bilinear coordinates.

 

© Webster 1913.

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