Page 348 - A Course in Linear Algebra with Applications

332 Chapter Nine: Advanced Topics

9.3 Bilinear Forms

Roughly speaking, a bilinear form is a scalar-valued linear
function of two vector variables. One type of a bilinear form
which we have already met is an inner product on a real vector
space. It will be seen that there is a close connection between
bilinear forms and quadratic forms.
Let V be a vector space over a field of scalars F and write

VxV

for the set of all pairs (u, v) of vectors from V. Then a bilinear
form on V is a function

f:VxV^F,

that is, a rule assigning to each pair of vectors (u, v) a scalar
/(u,v), which satisfies the following requirements:
/
(i) (ui + u 2 ,v) = (ui,v) + (u 2 ,v);
/
/
/
(ii) (u, vi + v 2 ) = (u, vi) + (u, v 2 );
/
/
/
(iii) (cu,v) = c/(u,v);
/
(iv) (u,cv) = c/(u,v).
These rules must hold for all vectors u, ui, 112, v, vi, v 2 in V
and all scalars c in F. The effect of the four defining properties
is to make (u, v) "linear" in both the variables u and v.
/
As has been mentioned, an inner product < > on a real
vector space is a bilinear form / in which
/(u, v) = < u, v > .

Indeed the defining properties of the inner product guarantee
this.
A very important example of a bilinear form arises when-
ever a square matrix is given.