Page 333 - DMTH502_LINEAR_ALGEBRA

Unit 30: Bilinear Forms and Symmetric Bilinear Forms

The discussion in Example 2 can be generalized so as to describe all bilinear forms on a finite- Notes
dimensional vector space. Let V be a finite-dimensional vector space over the field F and let
= { , ..., } be an ordered basis for V. Suppose f is a bilinear form on V. If
1 n
= x + ... + x and = y + ... + y
1 1 n n 1 1 n n
are vectors in V, then

f( , ) = f x i i ,
i

= x i ( f i , )
i

= x f i , y a
j j
i
i j
x y ( f , )
= i i i j
i i
If we let A = f( , ), then
ij i j
f( , ) = A x y
ij i i
i i
t
= X AY
where X and Y are the coordinate matrices of and in the ordered basis . Thus every bilinear
form on V is of the type

t
f( , ) = [ ] A[ ] … (3)
for some n n matrix A over F. Conversely, if we are given any n n matrix A, it is easy to see
that (3) defines a bilinear form f on V, such that A = f( , ).
ij i j
Definition: Let V be a finite-dimensional vector space, and let = { , ..., } be an ordered basis
1 n
for V. If f is a bilinear form on V, the matrix of f in the ordered basis is the n n matrix A with
entries A = f( , ). At times, we shall denote this matrix by [f] .
ij i j
Theorem 1: Let V be a finite-dimensional vector space over the field F. For each ordered basis
of V, the function which associates with each bilinear form on V its matrix in the ordered basis
is an isomorphism of the space L(V, V, F) onto the space of n n matrices over the field F.
Proof: We observed above that f [f] is a one-one correspondence between the set of bilinear
forms on V and the set of all n n matrices over F. That this is linear transformation is easy to
see, because
(cf + g) ( , ) = cf( , ) + g( , )
i j i j i j
for each i and j. This simply says that
[cf + g] = c[f] + [g] .
Corollary: If = { , ..., } is an ordered basis of V, and * = {L , ... L } is the dual basis for V*, then
1 n 1 n
2
the n bilinear forms
f ( , ) = L ( ) L ( ), 1 i n, 1 j n
ij i j
form a basis for the space L(V, V, F). In particular, the dimension of L(V, V, F) is n .
2
Proof: The dual basis {L , ... L } is essentially defined by the fact that L ( ) is the ith coordinate of
1 n i
in the ordered basis (for any in V).

LOVELY PROFESSIONAL UNIVERSITY 327