Broken Tailbone Itchy

quadratic forms and orthogonal groups

A body is a division ring in which the Multiplication is commutative. A vector space
over a field K is a K

Links module.

Thus we see that the concept vector space in this sense, nothing new.
They say the fact that we are dealing with a module over a field.
When talking here of a K-vector space, then with K will mean the body.
Let W be a submodule of a K-vector space V. Then W is a subspace

of V. Let U, W subspaces vonV. Then we define their sum U + W as follows:

It is even said that U + W is a subspace of V.
If U + W = V and

we write

and call V the direct sum of U and W.

So let V be a K-vector space. We now consider representations of V to K, which have particularly important structural properties, namely linearity.

A linear form is a mapping from V to K, which is linear, ie for which:

The set of all linear forms on a given vector space V

forms its dual space .

A bilinear is a picture

the linear in each variable is.

A quadratic form on V is a mapping

with the property:

and to apply that

is a bilinear form.
We call this the bilinear form b polar form of Q and say that Q is polarized to b.

A quadratic form is nondegenerate if its polar form b has the property

.

Let V be a vector space over a field K and let B be a bilinear form on V. We call the set

the radical of b.
b is non-degenerate if and if

A vector space V, equipped with a nondegenerate quadratic form Q is called a orthogonal geometry .

A subspace W is called non-degenerate if

A singular vector is a

A non-zero vector u is isotropic if

A subspace is called W completely isotropic if

This means W is a subset of the radical of b is in W.

A pair of isotropic vectors (u, v)

a hyperbolic Pair, and the amount

is hyperbolic line.
Recall P (V) is the projective geometry of V.

Is

and also

we write and say

: V is orthogonal direct sum of U and W.

Why all this? Now there is a nice statement that we want to formulate a lemma. A
Lemma is a phrase that is required as a prerequisite to formulate a mathematical theorem. The proposition to prove the the following lemma helps is the set of Witt, which we consider in this context does not, however, be. We need the lemma as a separate sentence.

Lemma: Let V
provided by Q, an orthogonal geometry. Let W be a non-degenerate two-dimensional subspace of V that contains a bilinear form with respect to the isotropic vector u b. Then W is the product of a hyperbolic pair (u, v), which can be chosen v such that Q (v) = 0 Therefore, there is then

In the proof we give here, because we do not need another lemma.
He can read about in the very recommendable book by Donald E. Taylor, "The geometry of the Classical Groups".

If V is an orthogonal geometry over a field K with q elements and contains a singular vector, we can write to the above lemma:

are being

hyperbolic lines and W contains no singular vector.
The number m is Witt index of V.

sentence Each vector space of dimension at least 3 over a finite field has a singular vector.
(here without proof, proof again at Taylor "The geometry of the Classical Groups" p.138)

the dimension of W under the above conditions are only 0.1 or 2 after that game. Witt's theorem then says that V up to isomorphism (bijective linear maps) by m W and is unique.

We want to write down the hyperbolic lines of generators. That is

Let V and W two K-vector spaces. A picture

is K-linear map, or homomorphism if Be

V is a K-vector space and f is a homomorphism of V into itself
f is a bijective homomorphism, then f is called a automorphism of V.
Since f is bijective, there is an inverse linear transformation of f, ie a linear map g so that the sequence of execution is f and g is the identity.
It is said therefore that f is invertible, or that is f regular.
The set of all regular linear maps of V, with the concatenation of images as a link, called the general linear group of V and we write for GL (V).

That the GL (V) is a group, the reader may verify himself.

Now we can formulate what is the orthogonal group.
Let V be a K-vector space and Q a nondegenerate quadratic form. The
orthogonal group, which is associated with V and Q is the amount