Week 4
Forgot your norm? Affine space
The discussion of Euclidean space has shown that the position
of the origin of a coordinate system is in fact a choice we can freely
make. This motivates the definition of affine spaces, which I will discuss
this week. I introduce the notion of dimension, affine linear subspace,
affine linear combinations, and affine linear span, and prove a formula
for the dimension of intersections of affine linear subspaces.
Next, I will discuss affine transformations, which generalize the Euclidean
motions that we studied in week 2. You will see how this generalization
corresponds to a greater freedom of coordinate choices that we allow
ourselves in affine space as opposed to Euclidean space: we now work with
affine frames of reference.
As an application of affine geometry,
we can prove that the three medians of a triangle meet
in its centroid. In fact, the point of this proof, which you may well
know already, is to realize that the centroid is an affine quantity.
Note that the concept of affine space can be viewed as first success
of Klein's viewpoint over the Euclidean one.
Helpful literature:
-
M. Reid, Geometry and Topology, chapters of forthcoming
textbook by M. Reid and B. Szendröi, available from General Office:
Section 1.
- M. Berger,
Geometry I,
Springer:
Sections 2.1-2.4.
Note:
Only topics covered in the lectures and exercises
will be asked in the exam.
The list of helpful literature is meant as an aid, also to raise further
interest, but not as an obligation.