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**Definition 1.1**
The

-dimensional

*real projective space*
is the set of all
lines in

passing through the origin.

Note that if and are two nonzero vectors lying on the same line
through the origin, then they are multiples of each other. This remark
allows us to redefine
as the quotient of
by the equivalence relation
(or ) if and are multiples of each other.

**Exercise 1.1** (00)
Redefine projective space as the quotient of the unit sphere by some
equivalence relation. Use this to show that the map

from

to itself
can be used to identify

with a circle.

**Exercise 1.2** (*05)
If

is a vector space over a field

define the associated
projective space

. If the dimension of

equals

and

has

elements, how many elements does

have?

Another way to understand
is to think of it as the real line plus
a *point at infinity.* Indeed, if we identify every line through the
origin with its slope, we have that all points in
except the
vertical line can be represented by a number. The vertical line is
the point at infinity. It is important to keep in mind that this is not
the only way to introduce coordinates on the real projective line: one
could also intersect each line through the origin with the line
and measure the -coordinate of the intersection. Now the
point at infinity is the horizontal line through the origin.

**Exercise 1.3** (05)
If a line through the origin is neither vertical nor horizontal, the
constructions above give us two different ways to represent it as
a number. If in the first representation this number is

, what
is the formula for the number in the second representation?

**Exercise 1.4** (00)
Show that the action of

on

defined by

is transitive. Show that a matrix

fixes all points of

if and only if
it is a multiple of the indentity.

**Definition 1.2**
The

*projective group*
, is the quotient of

by the equivalence relation: two matrices are equivalent
if they are multiple of each other.

**Exercise 1.5** (00)
Use exercise

1.4 to define a transitive action of

on

. Show that if an element of the projective
group fixes all points in projective space, then it is the identity.

**Exercise 1.6** (00)
Let

be the slope of the line

passing through
the origin and let

be an invertible matrix. Verify that the slope of the line

equals

.

This exercise tells us that in suitable coordinates projective
transformations have the form
.

As explained in [2], projective geometry arose from the
artists' needs to represent the three-dimensional world on a
two-dimensional canvas. An artist in Flatland just has to worry about
representing a two-dimensional world on a one-dimensional canvas. The
figure below shows a Flatland artist copying a one-dimensional image
onto a canvas. Notice how distances are distorted.

This type of correspondence between the points in
two lines is called a *perspective*.

**Exercise 1.7** (15)
Could you explain a Flatland artist the relevance of the projective line and
projective transformations to his work? Start by explaining why perspectives
are projective transformations.

**Exercise 1.8** (15)
Characterize perspectives among all projective transformations and
show that a composition of perspectives is not necessarily a perspective.

** Next:** The fundamental theorems
** Up:** The Real Projective Line
** Previous:** The Real Projective Line
Juan Carlos Alvarez
2000-10-27