Organising assistent: Hans Forrer

Lecture: Wednesday, 13-15, HG F1

Exercises: Tuesday, 10-11

Organising assistent: Ana-Maria Matache

Lecture: Wednesday, 10-12, HG E3

Exercises: Wednesday, 13-14

Assistent |
Room |
Students |

Jochen Maurer | HG D3.2 | Alt - Blätter, Christian |

Ana-Maria Matache | HG D5.1 | Blättler, Dimitri - Eichholz |

Susanne Zimmermann | HG E21 | Eichhorn - Grob |

Marina Savalieva | HG E33.1 | Guarisco - Johansson |

Sandra Nägele | HG F26.1 | Joho - Leuenberger |

Doris Folini | HG G26.1 | Leuzinger - Mühlemann |

Markus Brun | LFW E13 | Müller - Rauchenstein |

Guido Giese | ML J37.1 | Reichmuth - Schneebeli |

Gregor Schmidlin | ML H41.1 | Schneider - Terribilini |

Fausta Leonardi | ML H43 | Tobler - Zwyssig |

Locations: |
Room |
Students |

HG E5 | A - F | |

HG E7 | G - M | |

HG E1.1 | N - R | |

HG F3 | S - Z |

9.30-11.00 in front of HG G53 on the following days:

August 10, 12, 17, 19, 24, 26, 30, 31

September 1, 2, 3, 6, 7, 8, 9, 10, 14, 15, 16, 17, 21, 23

See Nipp/Stoffer, Chapters 1 to 9

- Systems of linear equations, Gaussian elimination
- Examples
- Derivation of the algorithm
- Elimination and back-substitution
- Consistency conditions
- Consequences of the final elimination scheme
- Computational costs

- Matrices
- Definition and special matrices
- Calculating with matrices
- Transpose of a matrix
- The inverse of a quadratic matrix
- Gauss-Jordan algorithm
- Orthogonal matrices, Givens' rotations
- Elementary matrices for Gaussian elimination
- LR factorization
- Roundoff errors
- Pivot strategies

- Determinants
- Definition and geometric interpretation
- Basic properties
- Efficient calculation of determinants
- Determinants and systems of linear equations

- Vector spaces
- Definition and examples
- Spaces of polynomials and continuous functions
- Some consequences of the vector space axioms
- Linear subspaces
- Linear independence, basis and dimension
- Coordinates
- Normed vector spaces
- Hölder's inequality
- Minkowski's inequality
- The scalar product for real and complex vector spaces
- The Schwarz inequality
- Gram-Schmidt orthogonalization

- Method of least squares
- Normal equations
- QR factorization

- Linear maps
- Linear maps and matrices
- Linear maps and the scalar product
- Endomorphisms of vector spaces
- Coordinate transformations
- Norm of a matrix
- Characterisation of orthogonal maps

- The eigenvalue problem
- Eigenvalues and characteristic polynomials
- Eigenvectors
- Algebraic and geometric multiplicity
- Eigenvalue problem for symmetric matrices
- Powers of symmetric matrices
- 2-norm of a matrix
- Condition numbers

- Applications of the eigenvalue problem
- Coupled systems of linear differential equations with constant coefficients
- First and second order systems
- Turning a higher order differential equation into a first order system
- Curves and surfaces of second order

- Numerics of the eigenvalue problem
- Forward and backward vector iteration
- Jacobi rotations
- Jacobi's method for symmetric matrices

- Propositional and first order predicate logic
- Truth tables
- Connectives (and, or, negation, implication, equivalence)
- Associativity and commutativity
- Double negation law
- De Morgans's laws
- Distributivity
- Existential and universal quantifier
- Negation laws

- Set theory
- Cantor's definition of a set
- Antinomies
- Zermelo-Fraenkel axioms
- Operations on sets
- Relations
- Functions (Biggs, 2.1)
- Surjections, injections, bijections (Biggs, 2.2)
- Finite and infinite sets (Biggs, 2.5)
- Example: infinitely many primes (Biggs, 2.5)
- Cardinality
- Power set has a larger cardinality

- Combinatorics
- Addition principle (Biggs, 3.1)
- Counting sets of pairs and tuples (Biggs, 3.2)
- Functions, words and selections (Biggs, 3.4)
- Injections as ordered selections without repetition (Biggs, 3.5)
- Binomial numbers (Biggs, 4.1)
- Unordered selections with repetition (Biggs, 4.2)
- The binomial theorem (Biggs, 4.3)
- The sieve principle (Biggs, 4.4)
- Partitions of a set (Biggs, 5.1)
- Classification and equivalence relations (Biggs, 5.2)
- Distributions and multinomial numbers (Biggs, 5.3)

- Graph theory
- Graphs and their representation (Biggs, 8.1)
- Isomorphism of graphs (Biggs, 8.2)
- Valency (Biggs, 8.3)
- Complete graphs, cyclic graphs, Petersen graph, hypercubes
- Paths and cycles (Biggs, 8.4)
- Trees (Biggs, 8.5)
- Colouring the vertices of a graph (Biggs, 8.6)
- The greedy algorithm for vertex-colouring (Biggs, 8.7)
- Characterisation of bipartite graphs (Biggs, 8.7)
- Digraphs and tournaments (Biggs, 11.1)
- Networks and critical paths (Biggs, 11.2)
- Flows and cuts (Biggs, 11.3)
- The max-flow min-cut theorem (Biggs, 11.4)
- Breadth-first search (Biggs, 9.5)
- The labelling algorithm for network flows (Biggs 11.5)

- Modular arithmetic
- Congruences (Biggs, 6.1)
- Z
_{m}and its arithmetic (Biggs, 6.2) - Invertible elements of Z
_{m}(Biggs, 6.3) - Euler's function (Biggs, 3.3)
- Theorems of Euler and Fermat (Biggs, 6.3)

- Algebraic structures
- The axioms of a group (Biggs 13.1)
- Examples of groups (Biggs 13.2)
- Homomorphisms and isomorphisms of groups (Biggs, 13.5)
- Rings (Biggs, 15.1)
- Invertible elements of a ring (Biggs, 15.2)
- Fields (Biggs, 15.3)
- Polynomials (Biggs, 15.4)
- Ideals in Rings (Biggs, 17.5)

- Error-correcting codes
- Words, codes, errors, Hamming distance, decoding (Biggs, 17.1)
- Linear codes (Biggs, 17.2)
- Construction of linear codes (Biggs, 17.3)
- Correcting errors in linear codes (Biggs, 17.4)
- Perfect codes, Hamming codes (Biggs, 17.4)
- Cyclic codes (Biggs, 17.5)

- Kaspar Nipp and Daniel Stoffer, Lineare Algebra, Eine Einführung für Ingenieure unter besonderer Berücksichtigung numerischer Aspekte, 3. Auflage, vdf Hochschulverlag AG an der ETH Zürich, 1994.
- Norman L. Biggs: Discrete Mathematics, revised edition, Oxford University Press, 1989.

Please send comments and suggestions to Uwe Schmock, email: schmock@fam.tuwien.ac.at. Last update: June 14, 2003 |