BILINEAR MAP

In mathematics, a 'bilinear map' is a function which is linear in both of its arguments. An example of such a map is multiplication of integers.

Contents

Definition

Properties

Examples

See also

Definition

Let ''V'', ''W'' and ''X'' be three vector spaces over the same base field ''F''. A bilinear map is a function
:''B'' : ''V'' × ''W'' → ''X''
such that for any ''w'' in ''W'' the map
:''v'' ↦ ''B''(''v'', ''w'' )
is a linear map from ''V'' to ''X'', and for any ''v'' in ''V'' the map
:''w'' ↦ ''B''(''v'', ''w'' )
is a linear map from ''W'' to ''X''.
In other words, if we hold the first entry of the bilinear map fixed, while letting the second entry vary, the result is a linear operator, and similarly if we hold the second entry fixed.
If ''V'' = ''W'' and we have ''B''(''v'',''w'' ) = ''B''(''w'',''v'' ) for all ''v'',''w'' in ''V'', then we say that ''B'' is ''symmetric''.
The case where ''X'' is ''F'', and we have a 'bilinear form', is particularly useful (see for example scalar product, inner product and quadratic form).
The definition works without any changes if instead of vector spaces we use modules over a commutative ring ''R''. It also can be easily generalized to ''n''-ary functions, where the proper term is ''multilinear''.
For the case of a non-commutative base ring ''R'' and a right module ''M_R'' and a left module ''_RN'', we can define a bilinear map ''B'' : ''M'' × ''N'' → ''T'', where ''T'' is an abelian group, such that for any ''n'' in ''N'', ''m'' ↦ ''B''(''m'', ''n'' ) is a group homomorphism, and for any ''m'' in ''M'', ''n'' ↦ ''B''(''m'', ''n'' ) is a group homomorphism, and which also satisfies
:''B''(''mt'', ''n'' ) = ''B''(''m'', ''tn'' )
for all ''m'' in ''M'', ''n'' in ''N'' and ''t'' in ''R''.

Properties

A first immediate consequence of the definition is that $B(x,y)=o$
whenever ''x''=o or ''y''=o. (This is seen by writing the null vector ''o'' as 0·''o'' and moving the scalar 0 "outside", in front of ''B'', by linearity.)
The set ''L(V,W;X)'' of all bilinear maps is a linear subspace of the space (viz vector space, module) of all maps from ''V''×''W'' into ''X''.
If ''V'',''W'',''X'' are finite-dimensional, then so is ''L(V,W;X)''. For ''X=F'', i.e. bilinear forms, the dimension of this space is dim''V''×dim''W'' (while the space ''L(V×W;K)'' of ''linear'' forms is of dimension dim''V''+dim''W''). To see this, choose a basis for ''V'' and ''W''; then each bilinear map can be uniquely represented by the matrix $B(e\_i,f\_j)$, and vice versa.
Now, if ''X'' is a space of higher dimension, we obviously have dim''L(V,W;X)''=dim''V''×dim''W''×dim''X''.

Examples

★ Matrix multiplication is a bilinear map M(''m'',''n'') × M(''n'',''p'') → M(''m'',''p'').

★ If a vector space ''V'' over the real numbers 'R' carries an inner product, then the inner product is a bilinear map ''V'' × ''V'' → 'R'.

★ In general, for a vector space ''V'' over a field ''F'', a bilinear form on ''V'' is the same as a bilinear map ''V'' × ''V'' → ''F''.

★ If ''V'' is a vector space with dual space ''V
★ '', then the application operator, ''b''(''f'', ''v'') = ''f''(''v'') is a bilinear map from ''V''
★ × ''V'' to the base field.

★ Let ''V'' and ''W'' be vector spaces over the same base field ''F''. If ''f'' is a member of ''V''
★ and ''g'' a member of ''W''
★ , then ''b''(''v'', ''w'') = ''f''(''v'')''g''(''w'') defines a bilinear map ''V'' × ''W'' → ''F''.

★ The cross product in 'R'³ is a bilinear map 'R'³ × 'R'³ → 'R'³.

★ Let ''B'' : ''V'' × ''W'' → ''X'' be a bilinear map, and ''L'' : ''U'' → ''W'' be a linear operator, then (''v'', ''u'') → ''B''(''v'', ''Lu'') is a bilinear map on ''V'' × ''U''

★ The null map, defined by $B(v,w)\; =\; o$ for all (''v'',''w'') in ''V''×''W'' is the only map from ''V''×''W'' to ''X'' which is bilinear and linear at the same time. Indeed, if (''v,w'')∈''V''×''W'', then if ''B'' is linear, $B(v,w)=\; B(v,o)+B(o,w)=o+o$ if ''B'' is bilinear.

See also

★ Tensor product

★ Multilinear map

★ Sesquilinear form

★ Bilinear filtering