HOMEWORKS
Homework 1
For which natural numbers \(n\) there exists an affine space containing exactly
\(n\) points?
Homework 2
Let \(A = \{(x,y,z) \in \mathbb R^3 \>|\> x^2 + y^2 - z = 0 \}\). Determine
whether the pair \((A, \mathbb R^2)\) forms an affine space with respect to the
operation
\[
(x,y,z) + (u,v) = (x+u, y+v, (x+u)^2 + (y+v)^2) .
\]
Homework 3
Decide whether each of the following maps is an affine map or not:
a) \(\mathbb C \to \mathbb C, z \mapsto \overline z\)
b) \(\mathbb R^2 \to \mathbb R^2, (x,y) \mapsto (x,-y)\)
c) \(\mathbb R^3 \to \mathbb R^2, (x,y,z) \mapsto (1+x, \sqrt{\pi})\)
In each of these examples, the affine space is assumed to be of the form
\((V,V)\), where \(V\) is a vector space. In the case a) consider \(\mathbb C\)
as an affine space over \(\mathbb C\) and also over \(\mathbb R\).
Homework 4
Whether Theorem 2 from p.15 from the slides
remains true if we will change in its formulation the order of the composition? In other
words, whether the following statement is true:
Let \(f,g: (A,V) \to (B,W)\) be two affine maps. Then \(Df = Dg\) if and only if
\(g = f \circ t_v\) for some \(v \in V\).
Homework 5
Prove that if \(\sum_{i=1}^n x_i = 0\), then for any points \(a, a_1,\dots,a_n\)
of an affine space, the vector \(\sum_{i=1}^n x_i (a_i - a)\) does not depend on
\(a\).
Homework 6
Prove the "general distributive law" for barycentric combinations: for any
points \(a_{ij}\) of an affine space, and any elements of the ground field
\(x_j\), \(y_{ij}\) it holds
\[
\sum_{j=1}^n x_j \Big(\sum_{i=1}^{m_i} y_{ij} a_{ij}\Big) =
\sum_{j=1}^n \sum_{i=1}^{m_j} (x_j y_{ij}) a_{ij}
\]
as soon at it makes sense (see Mac Lane-Birkhoff, p.566).
Homework 7
Investigate the question whether we can prove statements about barycentric
combinations by induction on the number of summands. In particular, it is enough
in the formulation of the theorem about characterization of affine maps as
maps preserving barycentic combinations (p.20 of the slides) to require only
that \(f(x a + (1-x) b) = x f(a) + (1-x)f(b)\) for any \(a,b \in A\) and
\(x \in K\)?
Homework 8
Prove that an affine map \(f\) has a unique fixed point if and only if \(Df\)
does not have eigenvalue \(1\).
Homework 9
Is it true that if the affine groups of two affine spaces are isomorphic, then
the affine spaces themselves are isomorphic?
Homework 10
Does every group have an affine extension? How to determine whether a given
group is an affine extension of some (other) group, or not?
Homework 11
Let \(G\) be a subgroup of \(GL(V)\), and \(W\) a vector subspace of \(V\). For
which \(G\) and \(W\) the affine extension of \(G\) coincides with translations
by elements of \(W\)?
Homework 12
Prove that the group \(O(V)\) is the semidirect product of \(SO(V)\) and some
other group. Which "other" group is this? Give a geometric interpretation of
this fact.
Homework 13
Let \(n \ge 3\) be a nonnegative integer, \(\lambda,\mu,\eta \in \mathbb R\),
\(p: \mathbb R^n \to \mathbb R^n\) is the standard orthogonal projection in
\(\mathbb R^n\) on \(\mathbb R^3\), and \(f: \mathbb R^3 \to \mathbb R^3\) is the map given by
\((x_1,x_2,x_3) \mapsto (\lambda x_2, \mu x_3, \eta x_1)\). For which \(n\) and
\(\lambda,\mu,\eta\) the composition \(f \circ p\) is a motion?
Created: Tue Feb 13 2024
Last modified: Mon Jun 17 14:41:53 CEST 2024