HOMEWORKS
Homework 13
Find the affine span of the points (1,2) and (2,3) in the affine space
\((\mathbb R^2, \mathbb R^2)\).
Homework 14
Characterize affine spaces which have at most 3 non-parallel lines (i.e.,
one-dimensional subspaces).
Homework 15
Let \(V\) be the vector space of all polynomials with real coefficients with the
length of the vectors defined as
\[
|f| = \sqrt{\Big(\int_{-1}^1 f(x)^2 dx\Big)^2}
\]
and let \((V,V)\) be the corresponding affine Euclidean space. Find in this
space the distance between the point \(x^n\) and the subspace consisting of
polynomials of degree \(< n\).
Homework 16
Let \(\kappa\) be the cardinality of the set of solutions of a linear
programming problem (i.e., the set of points of the polyhedron where the
objective function attains maximum). Describe all possible values of \(\kappa\).
Homework 17
Give a counterexample to the exclusion-inclusion formula for the dimension of
intersections and spans of three projective subspaces.
Homework 18
Find the number of lines in the 3-dimensional projective space over \(GF(3)\).
Created: Thu Feb 4 2021
Last modified: Tue Mar 19 14:25:34 CET 2024