Projective Geometry - Lecture Notes Chapter 2

2.1 The Projective Space P(V)

We start by constructing a particular projective space starting from a vector space. We will later show that for dimension at least three, this example is the only example (however, for dimension 2 there are many other examples of projective planes). The vector spaces that we start from are a bit more general than those typically studied in a linear algebra course, in that we will permit the scalars to come from an algebraic structure more general than a field.

Def: A division ring (F,+,.) is a set F with two binary operations, + and ., such that (F,+) is an abelian group with identity 0, (F\{0},.) is a group (not necessarily commutative) and both right and left distributive laws hold.

A division ring with commutative multiplication is called a field. We shall call a division ring with non-commutative multiplication a skewfield. (This last definition is not universally agreed upon, but it is quite common.)

Example: Consider the set H of elements of the form a + bi + cj + dk, where a,b,c and d are real numbers and i,j,and k are special symbols such that i2 = j2 = k2 = -1, ij = -ji = k, jk = -kj = i and ki = -ik = j. Under the usual rules for addition and multiplication (treating the i, j and k as indeterminants subject to the above multiplication rules) H is a skewfield, known as the quaternions.

Since we will dealing with a possibly non-commutative multiplication, we need to be more careful in defining a vector space. The distinction that we make in the next definition is also appropriate when the scalars form a field, but in that case there is no significant consequence so the distinction is not usually pointed out.

Def: V is a left vector space over the division ring F if when v is a vector and a an element of F, then av is a vector (and va may not be a defined object). Right vector spaces can be defined analogously.

As we will only deal with left vector spaces, they shall be referred to simply as vector spaces.

Given a vector space, there is concept of dimension of the vector space (for finite dimensional vector spaces, this is the number of elements in a basis) and this is NOT the same as the concept of dimension in a geometry. It is traditional to live with both ideas and just be careful of the context in which the word dimension is used. When the need to distinguish them is present, we refer to the vector space (or algebraic) dimension versus the projective (or geometric) dimension. We will reserve the notation "dim" for the projective dimension, and use "dimV" for the vector space dimension.

Def: Let V be a vector space over a division ring F with dimV = d+1 3. Define the rank 2 geometry P(V) by:

Theorem 2.1.1: Let V be a vector space over a division ring F with dimV = d+1 3. Then P(V) is a projective space which we refer to as the projective space coordinatized by F.

Lemma 2.1.2: (a) Let V' be a subspace of the vector space V. Then P(V') is a subspace of P(V).
(b) Let U be a subspace of the projective space P(V). Then there exists a vector subspace V' of V such that P(V') = U.

Corollary 2.1.3: The projective space P(V) has dimension d.

Lemma 2.1.4: The line of P(V) through the points <v> and <w> consists of the point <w> and the points <v + aw> (a in F). In particular, if F is a finite field with q elements, then any line of P(V) has exactly q+1 points. In this case P(V) has order q.

If V is a (d+1)-dimensional vector space over the division ring F then we denote the projective space P(V) by PG(d,F). If F is a finite field of order q we also write PG(d,F) = PG(d,q).

The significance of the above lemma follows from an algebraic result.

Theorem (Wedderburn): Every finite division ring is a field. (I.e., there are no finite skewfields).

Thus, all finite examples of this construction require vector spaces over finite fields. To review the construction of finite fields and their basic properties you can examine the web page on finite fields.