3
and hence
a
b
c
d
=
α
γ
β
δ
a
b
c
d
α
β
γ
δ
.
This can be also expressed by saying that the form
f
is obtained from the form
f
by using the change of variables
x
→
αx
+
βy,
y
→
γx
+
δy.
We write this in the form
f
=
Mf.
According to Lagrange two binary quadratic forms
f
and
g
are called
equivalent
if one transforms to another under the change of variables as above defined by
an integral matrix with determinant
±
1. An equivalence class is called the
class
of forms
. Obviously, for any
n
∈
Z
, the set of integral solutions of the equations
f
(
x, y
) =
n
depends only on the class of forms to which
f
belongs. Also it is
clear that two equivalent forms have the same discriminant.
1.2
As we saw before any lattice Λ determines a class of forms expressing the
distance from a point in Λ to the origin. Conversely, given a positive definite
binary form
f
=
ax
2
+ 2
bxy
+
cy
2
we can find a lattice Λ corresponding to
this form. To do this we choose any vector
v
of length
√
a
and let
w
be the
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 Fall '09
 ONTONKONG
 Linear Algebra, Determinant, Quadratic form, positive definite

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