# What is a Dynkin Diagram

## 10 Dynkin diagrams and classification

### Linear Algebra I (WS 13/14)

Linear Algebra I (WS 13/14) Alexander Lytchak Based on a template by Bernhard Hanke November 15, 2013 Alexander Lytchak 1/12 Reminder A mapping f: V W between real vector spaces is linear if

More

### Linear Algebra I Summary

Prof. Dr. Urs Hartl WiSe 10/11 Linear Algebra I Summary 1 Vector spaces 1.1 Sets and mappings Injective, surjective, bijective mappings 1.2 Groups 1.3 Fields 1.4 Vector spaces definition

More

### Solutions to the exam on Lie algebras

University of Cologne summer semester 2017 Mathematical Institute July 19, 2017 Prof. Dr. P. Littelmann Solutions to the exam on Lie algebras This is not a sample solution, but an aid to the solution

More

### Linear Algebra II 8. Exercise sheet

Linear Algebra II 8. Exercise sheet Department of Mathematics SS 11 Prof. Dr. Kollross 1./9. June 11 Susanne Kürsten Tristan Alex Group exercise Exercise G1 (mini test) Let V be a Euclidean or unitary vector space.

More

### Chapter 3 Linear Algebra

Chapter 3 Linear Algebra Table of Contents VECTORS ... 3 VECTOR SPACES ... 3 LINEAR INDEPENDENCE AND BASES ... 4 MATRICES ... 6 CALCULATION WITH MATRICES ... 6 INVERTABLE MATRIX ... 6 RANK OF A MATRIX AND

More

### Linear Algebra I (WS 13/14)

Lineare Algebra I (WS 13/14) Alexander Lytchak Based on a template by Bernhard Hanke November 29, 2013 Alexander Lytchak 1/13 repetition The rank of a linear representation is equal to the column rank of the depicting one

More

### The homology groups of a simplicial complex

Section 12 The Homology Groups of a Simplicial Complex We will now define the homology groups H i (S), i 0 of a simplicial complex S. It's worth a moment to compare it with that

More

### 1 Basics of representation theory

Seminar Groups in Physics SS 06 Lecture 1 Groups and their Representation Matthias Nagl 1 Basics of Representation Theory This lecture will only deal with linear representations of finite groups in

More

### 1 Euclidean and unitary vector spaces

1 Euclidean and Unitary Vector Spaces In this section we consider real and complex vector spaces with a scalar product. This allows us to define the length of a vector and (in the case of a real

More

### Linear representations of symmetrical groups

Linear representations of symmetrical groups 150 232 (Holtkamp) 2h., Wed 12.00-14.00, NA 2/24 1 example 1. Free monoid over alphabet X example 2. S 1, S 2, S 3, ... sentence 1. ( Bijection between partitions

More

### General information about Lie algebras

Chapter I General information about Lie algebras Sophus Lie 1842 1899 Wilhelm Killing 1847 1923 Elie Cartan 1869 1951 Hermann Weyl 1885 1955 1 Introduction Most students are probably more familiar with examples

More

### Tutorial 4. 1 Bilinear Shapes. Definition. Let U, V, W be vector spaces. A mapping Φ: V W U is called bilinear: Remark. This is equivalent to:

1 Bilinear Shapes Tutorial 4 Definition. Let U, V, W be vector spaces. A mapping Φ: VWU is called bilinear: Φ (αv + w, x) = α Φ (v, x) + Φ (w, x) and Φ (v, βx + y) = β Φ (v, x) + Φ (v, y) remark. This is

More

### 3 Definition: 1. Exercise sheet for the lecture Linear Algebra I. in the winter semester 2003/2004 with Prof. Dr. S. Goette

1. Exercise sheet for the lecture Submission on Thursday, October 30th, 2003 1 Find 2 Let real numbers a, b, c such that a (2, 3, 1) + b (1, 2, 2) + c (2, 5 , 3) = (3, 7, 5). (V ,,) a Euclidean vector space. Demonstrate

More

### 9 vector spaces with scalar product

9 Dot product pink: Lineare Algebra 2014/15 page 79 9 Vector spaces with dot product 9.1 Normalized fields Let K be a field. Definition: A norm on K is a mapping: K R 0, x x with the following

More

### Vector spaces and linear maps

Vector spaces and linear mappings Patricia Doll, Selmar Binder, Lukas Bischoff, Claude Denier ETHZ D-MATL SS 07 11.04.2007 1 Vector spaces 1.1 Definition of the vector space (VR) 1.1.1 Basic operations Um

More

### 5.7 Linear dependency, basis and dimension

8 Chapter 5. Linear Algebra 5.7 Linear Dependency, Basis and Dimension Let v, ..., v n vectors from a vector space v over a fieldK. The set of all linear combinations of v, ..., v n, namely {n

More

### Exam preparation sheet for linear algebra

Exam preparation sheet for linear algebra summer semester 25 Exercise 2 2 Let A 3 3 8 2 4 3 R4 5. 5 2 a) Find the solution set of the linear system of equations Ax b) Is Ax b solvable with b? (Justify

More

### β 1 x: =., and b: =. K n β m

44 Systems of linear equations, notations Consider the system of linear equations () Let A = (α ij) i = ,, mj =, n α x + α x + + α nxn = β α x + α x + + α nxn = β α mx + α mx + + α mn xn = β m the coefficient matrix

More

### {id, if sgn (σ) = 1, τ, if sgn (σ) = 1,

Exercise I1 (4 points) Let (G,) and (H,) groups a) When is a mapping Φ: G H called a group homomorphism? b) Let Φ, Ψ: G H be two group homomorphisms Show that a subgroup

More

### 5 Analytical Geometry

5 Analytical Geometry The basic idea of ​​analytical geometry is to describe geometric objects in spaces using linear algebra. 51 Affine spaces Definition 511 An affine space (AR) over

More

### Sample solution for series 10

D-MATH, D-PHYS, D-CHAB Lineare Algebra II FS 1 Prof. Giovanni Felder, Thomas Willwacher Sample solution for series 1 1. a) As a reminder: An equivalence relation on a set M is a relation which the

More

### 9 Eigenvalues ​​and Eigenvectors

92 9 Eigenvalues ​​and Eigenvectors We saw in the previous chapter that a linear mapping from R n to R n can be described by different representation matrices (depending on the choice

More

### 9 Eigenvalues ​​and Eigenvectors

92 9 Eigenvalues ​​and Eigenvectors We saw in the previous chapter that a linear mapping from R n to R n can be described by different representation matrices (depending on the choice

More

### Glossary for the lecture Lie algebras

Glossary for the lecture Lie algebras 1 Definitions and examples Definition 1.1: Let k be a field. A k-vector space L is called a Lie algebra (over k) if a k-bilinear mapping [,]: L L L, (x, y) [x,

More

### Vector spaces and linear maps

Chapter 11. Vector spaces and linear mappings 1 11.1 Vector spaces Let K be a field. Definition. A vector space over K (K-vector space) is a set V together with a binary operation + a distinguished one

More

### 5 Linear Algebra (Part 3): Dot Product

5 Linear Algebra (Part 3): Dot Product The concept of linear dependency enables the definition of when two vectors are parallel and when three vectors lie in one plane. But that is real

More

7.2 The Adjoint Mapping Definition 7.2.1 A linear mapping f: V K is called a linear functional or linear form. (This definition applies to any K-vector spaces, not just to interior product spaces.)

More

### 1.5 Dual grid and discriminant group

Lattice and Codes c Rudolf Scharlau April 24, 2009 27 1.5 Dual Lattice and Discriminant Group This section is essentially algebraic in nature: It doesn't matter that our lattice is in one

More

### On the cycle notation of permutations

On the cycle notation of permutations Olivier Sète June 16, 2010 1 Basics Definition 1.1. A permutation of {1, 2, ..., n} is a bijective mapping σ: {1, 2, ..., n} {1, 2, ..., n}, i σ (i).

More

### R 3 and U: = [e 2, e 3] that generated by e 2, e 3

Exercise (Let e ​​=, e = subspace (, e = (R and U: = [e, e]) let the continuation generated by e, e be G: = {A GL (, RA e = e and AUU} (a Show that G is a subgroup of GL (, R (b give

More

### Chapter V. Affine Geometry

Chapter V Affine Geometry 1 Affine Spaces Consider a linear system of equations Γ: a 11 x 1 + a 12 x 2 + + a 1n xn = b 1 a 21 x 1 + a 22 x 2 + + a 2n xn = b 2 a m1 x 1 + a m2 x 2 + + a mn xn = b

More

### 2.3 Basis and dimension

Linear Algebra I WS 205/6 c Rudolf Scharlau 65 2.3 Basis and Dimensions In this central section, some basic concepts that are fundamental to all linear algebra are introduced: Linear dependency

More

### 2 The dimension of a vector space

2 The dimension of a vector space Let V be a K vector space and v 1, ..., vr V. Definition: v V is called a linear combination of the vectors v 1, ..., vr if there are elements λ 1, ..., λ r K such that v = λ 1 v 1

More

### Repetition: linear mappings

Repetition: linear mappings Def Let (V, +,) and (U, +,) be two vector spaces A map f: VU is called linear if for all vectors v 1, v 2 V and for every λ R: (a) f (v 1 + v 2) =

More

### Projective spaces and subspaces

Projective Spaces and Subspaces Erik Slawski Introductory Seminar Analytical Geometry with Prof. Dr. Werner Seiler and Marcus Hausdorf Winter Semester 2007/2008 Department 17 Mathematics University of Kassel Contents

More

### 1 calculation with 2 2 matrices

1 Calculate with 2 2 matrices 11 product We calculate the general product of A = For the product, AB = a11 a 12 a 21 a 22 a11 b 11 + a 12 b 21 a 11 b 12 + a 12 b 22 a 21 b 11 + a 22 b 21 a 21 b 12

More

### 13 linear maps

13 Linear mappings Roughly speaking, linear mappings in vector spaces are the same as homomorphisms in groups, namely structure-preserving mappings. In this chapter too, let V, W be vector spaces.

More

### Linear Algebra II 11th Exercise Sheet

Linear Algebra II Exercise Sheet Department of Mathematics SS Prof Dr Kollross 9 / June Susanne Kürsten Tristan Alex Group exercise Task G (Minitest (processing within minutes and without using the

More

### 7 Linear mappings and scalar product

Mathematics II for inf / swt, summer semester 22, page 121 7 Linear maps and scalar product 71 Preliminary remarks Standard scalar product see chapter 21, scalar product abstract see chapter 34 Norm u 2 u, u

More

### Examination linear algebra, B: = (), C: = 1 1 0

1. Let 1 0 2 0 0 1 3 0 A: =, B: = (1 2 3 4), C: = 1 1 0 0 1 0. 0 0 0 1 0 0 1 0 0 0 0 Which of the following Statement is correct? A. A and C are stepped, B is not. B. A and B are stepped,

More

### 1 2. Body enlargements

1 2. Body extensions 1 2. 1. Definition: If K, L is a body and i: K L is a ring homomorphism, then i is injective, we understand K as a subfield of L because of i, we write L K and call L one

More

### Examination linear algebra 2

1. Check the following statements: (1) Two real symmetric matrices are similar if and only if they have the same signature. (2) Every symmetric matrix is ​​congruent to a diagonal matrix,

More

### Invariant cones in Cartan algebras

Seminar Sophus Lie 1 (1991) 55 63 Invariant Cones in Cartan Algebras Ulrike Zimmermann For the determination of semigroups in a Lie group, invariant cones as their tangential objects play a very important role

More

### Linear algebra for physicists 11. Exercise sheet

Linear Algebra for Physicists 11. Exercise Sheet Department of Mathematics SS 01 Prof. Dr. Matthias Schneider./. July 01 Dr. Silke Horn Dipl.-Math. Dominik Kremer Group exercise Exercise G1 (mini test) (a) Which

More

### 6. Normal pictures

SCALAR PRODUCTS 1 6 Normal mappings 61 Memory Let V be an n-dimensional pre-Hilbert space, i.e. an n-dimensional vector space over K (R or C) also provided with a scalar product, ra K The Euclidean

More

### 1 Arranged bodies and arrangement

1 ARRANGED BODIES AND ARRANGEMENT 1 1 Arranged Bodies and Arrangement The next idea we need to interpret is the arrangement. One can show that it is not possible with every body.

More

### 3.3 Scalar products 3.3. SCALAR PRODUCTS 153

3.3. SCALAR PRODUCTS 153 For this we still have to show the uniqueness (independence from the choice of the base or the coordinate system). Let β be a bilinear form and q the corresponding quadratic

More

### 11.2 Orthogonality. Winter semester 2013/2014

University of Konstanz Department of Mathematics and Statistics Winter Semester 2013/2014 Markus Scheighofer Linear Algebra I 11.2 Orthogonality Definition 11.2.1. Let V be a K-vector space with scalar product

More

### Analysis II. Lecture 48. The Hessian Form

Prof. Dr. H. Brenner Osnabrück SS 2015 Analysis II Lecture 48 The Hessian Form We are of course also interested in sufficient criteria for the existence of local extremes. As in the one-dimensional

More

### 6 eigenvalues ​​and eigenvectors

6.1 Eigenvalue, Eigenspace, Eigenvector Definition 6.1. Let V be a vector space and f: V V a linear map. If λ K and v V are given with v 0 and f (v) = λv, then the number λ is called the eigenvalue (EW) of f,

More