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**Definition 1.1**
If

is a vector subspace, the set of all lines
in

passing through the origin is a

*projective subspace* of

that we shall denote by

. If the dimension of

equals

, we say that

is a

*-dimensional projective subspace.*
A projective subspace of dimension one is called a

*projective line*
and a projective subspace of codimension one is called a

*projective hyperplane.*

**Exercise 1.1** (00)
Show that projective transformations of

send

-dimensional
projective subspaces to

-dimensional projective subspaces.

**Exercise 1.2** (* 05)
Let

be a four-dimensional vector space over a field

with

elements. How many projective lines are then in

? How many
projective hyperplanes?

**Exercise 1.3** (10)
Verify that projective lines in

have the following two properties:

- Any two projective lines intersect at one point.
- Any two distinct points are joined by a unique projective line.

Figure 1 shows that the set of lines in
that pass
through the origin and are not horizontal can be identified with the
set of points on the plane . Note also how a
straight line on the plane corresponds to a projective line on
.

Figure 1.

**Exercise 1.4** (00)
Show that

minus a projective line can be identified with

.

The analytic version of the figure above is as follows:

**Exercise 1.5** (00)
Let

be the set of points in the projective plane whose homogeneous
coordinates

have

(these correspond to lines in

that pass through the origin and are not horizontal). Show that the map

given by

is a bijection.

**Exercise 1.6** (10)
Consider now

and define

to be the set of
points whose homogeneous coordinates

have

. Show that

- The union of the
, is all of
.
- The map
given by
is a bijection.
- Compute the map
defined on
.

The maps
are called
*charts* and they can be used to work on projective space
as if it were
. When we do this, the projective hyperplane
given by all points whose homogenous coordinates are of the form
is called the
*hyperplane at infinity* (visualize this in figure 1). These
are exactly the points that are not in and are not taken into
account by our representation. As the next exercise shows, we can
make any projective hyperplane be our hyperplane at infinity.

**Exercise 1.7** (05)
Deduce from the previous exercise that if

is
a projective hyperplane, then

can be
identified with

. Give a picture proof in the case

.

We now describe the action of
using the charts of
. To keep things simple, we only consider the chart
.

If is an invertible
matrix, then its action
on
in homogeneous coordinates is given simply by
. If
, we
identify it with the point
.
The matrix sends this point to the point

If we use to identify this point with a point in
,
we obtain
, where
If it happens that the denominator equals zero, we say that the point
*is sent to infinity.*

** Next:** Projective geometry vs. affine
** Up:** The Real Projective Plane
** Previous:** The Real Projective Plane
Juan Carlos Alvarez
2001-01-22