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DEFINITION

Let ''V'', ''W'' and ''X'' be three Vector Space s over the same base Field ''F''. A bilinear map is a Function
B

such that for any ''w'' in ''W'' the map
v

is a Linear Map from ''V'' to ''X'', and for any ''v'' in ''V'' the map
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 ''MR'' 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


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 Number s 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 R3 is a bilinear map R3 × R3 → R3.

  • 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