Page 349 - A Course in Linear Algebra with Applications

9.3: Bilinear Forms 333

Example 9.3.1
Let A be an n x n matrix over a field F. A function
n n
/ : F x F —> F is defined by the rule
T
f(X, Y) = X AY.

That / is a bilinear form on F n follows from the usual rules of
matrix algebra. The importance of this example stems from
the fact that it is typical of bilinear forms on finite-dimensional
vector spaces in a sense that will now be made precise.

Matrix representation of bilinear forms
Suppose that f : V xV —>• F is a bilinear form on a vector
space V of dimension n over a field F. Choose an ordered basis
B = {vi,..., v n } of V and define a^- to be the scalar (VJ, Vj).
/
Thus we can associate with / the n x n matrix

A= [ aij].

Now let u and v be arbitrary vectors of V and write them
V
c v
in terms of the basis as u = YM=\ ^ * an< ^ v = S?=i j i '
then the coordinate vectors of u and v with respect to the
given basis are

Cl
[u] B = I : and [v] B =
\bn/

/
The linearity properties of / can be used to compute (u, v)
in terms of the matrix A.
n n n n
bi Vi
c w
c v
/(u, v) = f(^2 b , J2 i i) = X) f( > Yl i i">
iVl
i = l j = l i = l j'=l
n n
i=l i = l