Editing Bilinear map (section)

== Definition ==

=== Vector spaces ===
Let <math>V, W </math> and <math>X</math> be three [[vector space]]s over the same base [[Field (mathematics)|field]] <math>F</math>. A bilinear map is a [[Function (mathematics)|function]]
<math display=block>B : V \times W \to X</math>
such that for all <math>w \in W</math>, the map <math>B_w</math>
<math display=block>v \mapsto B(v, w)</math>
is a [[linear map]] from <math>V</math> to <math>X,</math> and for all <math>v \in V</math>, the map <math>B_v</math>
<math display=block>w \mapsto B(v, w)</math>
is a linear map from <math>W</math> to <math>X.</math> In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed.

Such a map <math>B</math> satisfies the following properties.

* For any <math>\lambda \in F</math>, <math>B(\lambda v,w) = B(v, \lambda w) = \lambda B(v, w).</math>
* The map <math>B</math> is additive in both components: if <math>v_1, v_2 \in V</math> and <math>w_1, w_2 \in W,</math> then <math>B(v_1 + v_2, w) = B(v_1, w) + B(v_2, w)</math> and <math>B(v, w_1 + w_2) = B(v, w_1) + B(v, w_2).</math>

If <math>V = W</math> and we have {{nowrap|1=''B''(''v'', ''w'') = ''B''(''w'', ''v'')}} for all <math>v, w \in V,</math> then we say that ''B'' is ''[[Symmetric function|symmetric]]''.  If ''X'' is the base field ''F'', then the map is called a ''[[bilinear form]]'', which are well-studied (for example: [[scalar product]], [[inner product]], and [[quadratic form]]).

=== Modules ===
The definition works without any changes if instead of vector spaces over a field ''F'', we use [[Module (mathematics)|module]]s over a [[commutative ring]] ''R''. It generalizes to ''n''-ary functions, where the proper term is ''[[Multilinear map|multilinear]]''.

For non-commutative rings ''R'' and ''S'', a left ''R''-module ''M'' and a right ''S''-module ''N'', a bilinear map is a map {{nowrap|''B'' : ''M'' × ''N'' → ''T''}} with ''T'' an {{nowrap|(''R'', ''S'')}}-[[bimodule]], and for which any ''n'' in ''N'', {{nowrap|''m'' ↦ ''B''(''m'', ''n'')}} is an ''R''-module homomorphism, and for any ''m'' in ''M'', {{nowrap|''n'' ↦ ''B''(''m'', ''n'')}} is an ''S''-module homomorphism.  This satisfies

:''B''(''r'' ⋅ ''m'', ''n'') = ''r'' ⋅ ''B''(''m'', ''n'')
:''B''(''m'', ''n'' ⋅ ''s'') = ''B''(''m'', ''n'') ⋅ ''s''

for all ''m'' in ''M'', ''n'' in ''N'', ''r'' in ''R'' and ''s'' in ''S'', as well as ''B'' being [[Additive map|additive]] in each argument.