6LAG4 Linear Algebra 4, summer semester 2023/2024

A BRIEF SYNOPSIS
(each class lasted approximately 3h unless specified otherwise)

Class 1 February 13, 2024 (2h 20min)
Organizational issues. Refresher: linear spaces as central objects in linear algebra, their importance, idea of linearity. Affine space: definition, examples and non-examples, elementary properties. Affine maps.
(Slides, pp.3-14; Kostrikin-Manin, pp.195-199; Mac Lane-Birkhoff, pp.564-565; Postnikov, pp.46-48).

Class 2 February 20, 2024 (2h 20min)
Composition of affine maps is an affine map. Affine maps with equal linear parts. Affine coordinates, barycentric combination, barycentric coordinates, characterization of affine maps as maps preseving barycentric combinations.
(Slides, pp.15-20; Kostrikin-Manin, pp.197-203; Mac Lane-Birkhoff, pp.566-569; Postnikov, pp.49-50; Vinberg, pp.240-241).

Class 3 February 27, 2024
Isomorphisms and automorphisms of affine spaces. Any affine space is isomorphic to the space of the form (V,V). Refresher: general linear group, semidirect product. Affine group.
(Slides, pp.21,23; Kostrikin-Manin, pp.198-199,203-204; Mac Lane-Birkhoff, pp.568-572; Postnikov, pp.48-49).

Class 4 March 5, 2024
Discussion of homeworks 1-7. Affine group (continuation). Affine extension of a group. Euclidean affine spaces.
(Slides, pp.30-31; Kostrikin-Manin, p.204; Mac Lane-Birkhoff, p.570 onwards).

Class 5 March 19, 2024
Discussion of homework 8. Motions. The group of motions of an Euclidean affine space is isomorphic to the affine extension of the orthognal group. Geometric interpretation of motions in low-dimensional spaces.
(Slides, pp.31-32; Kostrikin-Manin, p.204 onwards; Mac Lane-Birkhoff, p.586 onwards; Postnikov, pp.144-146).

Class 6 March 26, 2024
Discussion of homeworks 9-11. Affine subspace, parallel subspaces. Characterization of affine subspaces as sets of zeroes of affine maps.
(Slides, pp.25-26,28; Kostrikin-Manin, pp.207-208,210).

Class 7 April 2, 2024
Discussion of homeworks 12-13. Counting the number of non-parallel lines in an affine space. Affine span. Computation of distance between two affine subspaces.
(Slides, pp.27-28; Kostrikin-Manin, pp.209,211-214; Reventos Tarrida, pp.159-162).

Class 8 April 9, 2024
Computation of distance between two affine subspaces (continuation), examples. Linear programming.
(Slides, pp.35-37; Kostrikin-Manin, pp.212-213,215-218; Reventos Tarrida, pp.159-162).

Class 9 April 16, 2024
Projective space: definitions, realizations, examples. The Fano plane. Counting subspaces in the finite vector spaces. Projective subspaces, projective span.
(Slides, pp.45-49; Kostrikin-Manin, pp.222-226,241; Mac Lane-Birkhoff, pp.592-595; Onishchik-Sulanke, pp.2-3,12).

Class 10 April 23, 2024
Counting in finite vector spaces (continuation). Projective span (continuation). Formula for dimensions of intersection and projective span of two projective subspace, its consequences. Projective duality.
(Slides, pp.49-52; Kostrikin-Manin, pp.226,229; Onishchik-Sulanke, pp.65-71).

Class 11 May 7, 2024
Projective group.
(Slides, pp.54-55; Kostrikin-Manin, pp.233-236; Mac Lane-Birkhoff, pp.595-596; Vinberg, p.285).

HOMEWORKS

6LAG4 Linear Algebra 4, winter semester 2022/2023

A BRIEF SYNOPSIS

Class 1 September 21, 2022
Organizational issues. Refresher: linear spaces as central objects in linear algebra, their importance, idea of linearity. Affine space: definition, examples and non-examples, elementary properties. Affine maps.
(Slides, pp.3-15; Kostrikin-Manin, pp.195-200, Mac Lane-Birkhoff, pp.564-565; Postnikov, pp.46-48).

Class 2 October 5, 2022
The functor from the category of affine spaces to the category of vector spaces. Affine coordinates, barycentric combination, barycentric coordinates, characterization of affine maps as maps preseving barycentric combinations. Isomorphism of affine spaces. Any affine space is isomorphic to the space of the form (V,V).
(Slides, pp.17-21; Kostrikin-Manin, pp.197-203; Mac Lane-Birkhoff, pp.566-569; Postnikov, pp.49-50; Vinberg, pp.240-241).

Class 3 November 2, 2022
General linear group, orthogonal group, semidirect product, affine group, affine extension of a group. Motions.
(Slides, p.23,31-32; Kostrikin-Manin, p.203 onwards; Mac Lane-Birkhoff, p.570 onwards; Postnikov, pp.144-146).

Class 4 November 9, 2022
Affine subspace, parallel subspaces, affine span. Characterization of affine subspaces as a set of zeroes of affine maps.
(Slides, pp.25-28; Kostrikin-Manin, pp.207-210).

Class 5 November 16, 2022
Inductive reasonings involving barycentric combination. Distance, computation of distance between two affine subspaces. Linear programming. (Slides, pp.33,35-37; Kostrikin-Manin, pp.212-218; Reventos Tarrida, pp.159-162).

Material for self-study: projective space, projective group (Slides, pp.45-55; Kostrikin-Manin, pp.222-229,233-242).

Class 6 November 30, 2022
Projective spaces, projective subspaces, projective group.

Class 7 December 7, 2022
Affine quadrics (Slides, pp.39-43; Kostrikin-Manin, pp.218-222).

HOMEWORKS

6LAG4 Linear Algebra 4, summer semester 2020/2021

A BRIEF SYNOPSIS

Class 1 February 8, 2021: Organizational issues. Refresher: linear spaces as central objects in linear algebra, their importance. Affine space: definition, examples, elementary properties. (Slides, pp.6-11; Kostrikin-Manin, pp.195-197, Mac Lane-Birkhoff, pp.564-565; Postnikov, pp.46-48).
video

Class 2 February 15, 2021: Affine maps, affine coordinates, definition of barycentric combination. (Slides, pp.13-18; Kostrikin-Manin, pp.197-201; Mac Lane-Birkhoff, p.566; Postnikov, pp.49-50; Vinberg, pp.240-241).

Class 3 February 22, 2021: Barycentric combination (continuation): geometrical and phyiscial interpretations, characterization of affine maps as maps preseving barycentric combinations. Isomorphism of affine spaces. Any affine space is isomorphic to the space of the form (V,V). (Slides, pp.19-21; Kostrikin-Manin, pp.198,202-203; Mac Lane-Birkhoff, p.568(Theorem 1)).

Class 4 March 1, 2021: Affine group. Definition of affine subspace, parallel subspaces. (Slides, pp.23-26; Kostrikin-Manin, pp.203-204,207-208; Mac Lane-Birkhoff, pp.570-573).

Class 5 March 8, 2021: Two affine subspaces are parallel iff they are obtained by a translation of each other. Affine span. (Slides, pp.26-27; Kostrikin-Manin, pp.208-209(including Proposition 3.3)).

Class 6 March 15, 2021: Characterization of affine subspaces: as set of zeros of a set of linear maps; as sets containing an affine span of every two points. Definition of affine Euclidean space and of motion. (Slides, pp.28,30-31; Kostrikin-Manin, pp.204,210,213-214; Mac Lane-Birkhoff, pp.586,588).

Class 7 March 22, 2021: Affine extension of a group. Orthogonal group. The group of motions. Motions in 1- and 2-dimensional spaces. (Slides, pp.31-32; Kostrikin-Manin, pp.204-207; Postnikov, pp.144-146).
Material for self-study: proof of Theorem: for any motion \(f\) of an affine Euclidean space \((A,V)\) there are \(a \in A\), \(v \in V\), and a motion \(g\) such that \(Df(v) = v\), \(g(a) = a\), and \(f = t_v \circ g\) (Kostrikin-Manin, Theorem 2.6 at pp.205-206).

Class 8 March 29, 2021: Angles in affine Euclidean space. Distance, computation of distance between two affine subspaces. Linear programming. (Slides, pp.33,35-37; Kostrikin-Manin, pp.212-213,215-217; Reventos Tarrida, pp.159-162).

Class 9 April 12, 2021: Linear programing (end; proof of the theorem that the maximum of the objective function is attained at a vertex of the polyhedron). Projective spaces. (Slides, pp.37,45-46; Kostrikin-Manin, pp.217,222-225; Mac Lane-Birkhoff,pp.592-594; Onishchik-Sulanke, pp.2-3; Postnikov, pp.311-312; Vinberg, pp.280-282).

Class 10 April 19, 2021: The Fano plane. Projective subspaces. Projective span. (Slides, pp.47,49; Kostrikin-Manin, pp.225-226; Mac Lane-Birkhoff, pp.594-595; Onishchik-Sulanke, p.12).

Class 11 April 26, 2021: Formula for dimensions of intersection and projective span of two projective subspace, its consequences. Projective duality. (Slides, pp.49-52; Kostrikin-Manin, pp.226-229; Onishchik-Sulanke, pp.65-71; Postnikov, pp.302-311).

Class 12 May 3, 2021: Projective group. Projective quadrics. (Slides, pp.54-56 (except of the theorem on p.55); Kostrikin-Manin, pp.233-234; Mac Lane-Birkhoff, pp.595-597; Onishchik-Sulanke, pp.20,22-23; Vinberg, p.285).


HOMEWORKS

Created: Thu Feb 4 2021
Last modified: Sun Feb 1 2026 13:43:33 CET