Graphics Reference
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1.9.1. Example. The dot product on R n is a bilinear map. More generally, an inner
product is a bilinear map.
1.9.2. Example. The determinant function on R 2 (where we think of either the rows
or columns of the matrix as vectors in R 2 ) is a bilinear map.
Let A be an n ¥ n matrix. The map f : R n
¥ R n
1.9.3. Example.
Æ R defined by
(
) =
T
f
v, w
v
A
w
is a bilinear map.
Let T be a linear transformation on R n . The map f : R n
¥ R n
1.9.4. Example.
Æ R
defined by
(
) =∑ ()
f
v, w
v
T
w
is a bilinear map.
Definition. Let f be a bilinear map on a vector space V . Let B = ( v 1 , v 2 ,..., v n ) be an
ordered basis for V and let a ij = f( v i , v j ). The matrix A = (a ij ) is called the matrix for f
with respect to the basis B. The determinant of A is called the discriminant of f with
respect to the basis B.
The matrix for a bilinear map clearly depends on the chosen basis. However, the
following is true:
1.9.5. Proposition. If B¢=( v 1 ¢, v 2 ¢,..., v n ¢) is another ordered basis for V and if A¢
is the matrix of the bilinear map f with respect to B¢, then
CAC T
A
¢=
,
where C = (c ij ) is the matrix relating the basis B to the basis B¢, that is,
n
 1
¢ =
v
c
v
.
i
ij
j
j
=
Proof.
This can be checked by a straightforward computation.
Definition. A real n ¥ n matrix A is said to be congruent to a real n ¥ n matrix B if
there exists a nonsingular matrix C such that A = CBC T .
It is easy to show that the congruence relation is an equivalence relation on the
set of all n ¥ n real matrices. We can rephrase Proposition 1.9.5.
1.9.6. Corollary. The matrix of a bilinear map is unique up to congruence, so that
the study of bilinear maps is equivalent to the study of congruence classes of matrices.
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