Editing Bilinear map (section)

==Properties==
An immediate consequence of the definition is that {{nowrap|1=''B''(''v'', ''w'') = 0<sub>''X''</sub>}} whenever {{nowrap|1=''v'' = 0<sub>''V''</sub>}} or {{nowrap|1=''w'' = 0<sub>''W''</sub>}}. This may be seen by writing the [[zero vector]] 0<sub>''V''</sub> as {{nowrap|0 ⋅ 0<sub>''V''</sub>}} (and similarly for 0<sub>''W''</sub>) and moving the scalar 0 "outside", in front of ''B'', by linearity.

The set {{nowrap|''L''(''V'', ''W''; ''X'')}} of all bilinear maps is a [[linear subspace]] of the space ([[viz.]] [[vector space]], [[Module (mathematics)|module]]) of all maps from {{nowrap|''V'' × ''W''}} into ''X''.

If ''V'', ''W'', ''X'' are [[finite-dimensional]], then so is {{nowrap|''L''(''V'', ''W''; ''X'')}}. For <math>X = F,</math> that is, bilinear forms, the dimension of this space is {{nowrap|dim ''V'' × dim ''W''}} (while the space {{nowrap|''L''(''V'' × ''W''; ''F'')}} of ''linear'' forms is of dimension {{nowrap|dim ''V'' + dim ''W''}}). To see this, choose a [[Basis (linear algebra)|basis]] for ''V'' and ''W''; then each bilinear map can be uniquely represented by the matrix {{nowrap|''B''(''e''<sub>''i''</sub>, ''f''<sub>''j''</sub>)}}, and vice versa. 
Now, if ''X'' is a space of higher dimension, we obviously have {{nowrap|1=dim ''L''(''V'', ''W''; ''X'') = dim ''V'' × dim ''W'' × dim ''X''}}.