Bilinear form

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a bilinear form is a <a href="/facts/Bilinear_map/LH1lHVeP">bilinear map</a> V × V → K on a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> V (the elements of which are called <a href="/facts/Vector_(mathematics)/U8mM7A5F">vectors</a>) over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> K (the elements of which are called <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalars</a>). In other words, a bilinear form is a function B : V × V → K that is <a href="/facts/Linear_map/Q1PBKNVV">linear</a> in each argument separately:

<ul><li>B(u + v, w) = B(u, w) + B(v, w)     and     B(λu, v) = λB(u, v)</li>
<li>B(u, v + w) = B(u, v) + B(u, w)     and     B(u, λv) = λB(u, v)</li></ul>
The <a href="/facts/Dot_product/tNz8MLjT">dot product</a> on 
 
 
 
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} ^{n}}
 
 is an example of a bilinear form which is also an <a href="/facts/Inner_product_space/HoyjElEy">inner product</a>. An example of a bilinear form that is not an inner product would be the <a href="/facts/Four-vector/89GpELMA">four-vector</a> product.
The definition of a bilinear form can be extended to include <a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a> over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a>, with <a href="/facts/Linear_map/Q1PBKNVV">linear maps</a> replaced by <a href="/facts/Module_homomorphism/SCSzvgxx">module homomorphisms</a>.
When K is the field of <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> C, one is often more interested in <a href="/facts/Sesquilinear_form/Kp6qUA8v">sesquilinear forms</a>, which are similar to bilinear forms but are <a href="/facts/Conjugate_linear/GTXzWMKc">conjugate linear</a> in one argument.

Bilinear form open-in-new

Bilinear form