**Spring 2013: Math 401 (Applications of Linear Algebra)**

**Schedule**: Monday, Wednesday, Friday, 11:00am-11:50am (**MTH 0407**).**Textbook**:*Linear Algebra and its Applications*, 4th Edition, by G. Strang. Published by Thomson Publishing.*ISBN: 0030105676.***Office Hours**: Monday, Wednesday, 1:00am-2:00am.**Description of the course**: Various applications of linear algebra: theory of finite games, linear programming, matrix methods as applied to finite Markov chains, random walk, incidence matrices, graphs and directed graphs, networks, transportation problems.**Grades**: 20% for homeworks, 20% for numerical projects, 10% for quizzes, 15% for 1st test, 15% for 2nd test, 20% for final exam. For each homework assignment, only a random selection of exercises will be graded. Letter grades will be based on the accumulated points at the end of the semester, according to the following scheme: 90%-A; 80%-B; 70%-C; 50%-D; less than 50%-F.**Homework**: There will be homework assignments (mostly biweekly) and 2 short numerical projects to complete. The lowest homework grade will be dropped. No late homework will be accepted (but early ones are welcome).**Examinations**: There will be 2 tests plus a final exam.**Expectations**: Students are expected to understand some important, basic concepts of linear algebra. This implies both a good knowledge of the theory and the ability to use it to solve practical problems or implement the methods numerically.

**Examinations**

Type | Content | Due date |
---|---|---|

Quizz 1 |
Section 1.1 to 1.5 (without algorithmic complexity) | 02/01/2013 |

Homework 1 |
Section 1.2: 3, 10; section 1.3: 12, 24; section 1.4: 4, 7, 20, 42 (with proper justifications); section 1.5: 4, 29. | 02/11/2013 |

Quizz 2 |
Section 1.5 to 1.7 (without permutations) | 02/15/2013 |

Homework 2 |
Chapter 1 review exercices (p.72 and following): 1.1, 1.12, 1.15, 1.23 and 1.28 | 02/25/2013 |

Midterm1 |
First chapter of the book | 03/04/2013 |

Homework 3 |
Section 2.4: 29. Section 2.5: 6, 8, 10, 12, 17 | 04/10/2013 |

Midterm2 |
Everything we did about chapter 1 and 2 | 04/17/2013 |

Final |
Everything we did since the beginning | 05/14/2013 |

**Numerical project assignment**

The projects consists in writing a code implementing the corresponding algorithm. Students can work in small teams (up to 3 people) and choose any mathematics related language (except Fortran). There are two deadlines for each project assignment. The latest one is the date at which the project must be completed and submitted to me. The first one is the date before which students must decide on their project and the composition of their team. Each team has to send me an e-mail before the first date (with all members of the team in copy) indicating their choice and the composition of the team.

The final project has to be submitted by e-mail to my address. Each submission should recall in the e-mail the names of the students having worked on it and the subject chosen. To each e-mail should be attached the source code and a short report (Latex, word, pdf…). The report should describe summarily how the algorithm was implemented, how can I test it and the main conclusion (typically a few pages are more than enough).

Choice before March 11th, completion before April 1st | |
---|---|

Subject 1 |
Write a program which to a given square matrix A as entry returns its LU decomposition or an error when this is not possible. The use of existing language-dependent routines performing the same role is of course prohibited. |

Subject 2 |
Write a program which given an integer n and a function f solve the n points finite-difference approximation (as studied in class) of the Laplace’s equation -u”(x) = f(x), u(0) = u(1) = 0 using the LU decomposition of the Laplace matrix. You can compare when possible the approximate and the exact solution of the problem. |

Subject 3 |
Find about the Gauss-Seidel method to solve linear systems. Write a program which given a square matrix A, a vector y, an initial vector z and a number n as entries returns the vector x approximate solution to Ax=y after n steps of the method starting from z. The use of existing language-dependent routines performing the same role is prohibited as always. |

**Numerical project assignment 2
**

The projects consists in writing a code implementing the corresponding algorithm. Students can work in small teams (up to 3 people) and choose any mathematics related language. There are two deadlines for each project assignment. The latest one is the date at which the project must be completed and submitted to me. The first one is the date before which students must decide on their project and the composition of their team. Each team has to send me an e-mail before the first date (with all members of the team in copy) indicating their choice, the language used and the composition of the team.

The final project has to be submitted by e-mail to my address. Each submission should recall in the e-mail the names of the students having worked on it and the subject chosen. To each e-mail should be attached the source code and a short report (Latex, word, pdf…). The report should describe summarily how the algorithm was implemented, how can I test it and the main conclusion (typically a few pages are more than enough).

Choice before May 3rd, completion before May 9th | |
---|---|

Subject 1 |
Write a program which to a given set of k points of the cartesian plane (x_{1},y_{1}), …, (x_{k},y_{k}) (or equivalently two k-dimensional vectors X, Y) and to m values b_{1}, …, b_{n} return the least square approximation of this cloud of points by a polynomial of degree n bigger than m (n is then an argument of the problem). The use of existing language-dependent routines performing the same role is of course prohibited. |

Subject 2 |
Write a program which to a square matrix A returns its QR decomposition (or an error if the null space of A is not reduced to 0). As before the use of routines performing the same role is prohibited. |

Subject 3 |
Write a program which given a rectangular matrix A of size n*m with n>m, and a vector b, returns a vector x which is the least square approximation for the system Ax=b. The program will return an error if n<=m or if the null space of A is not reduced to 0. As before the use of routines performing the same role is prohibited. |

**Miscellaneous documents**

Approximate solution of the Laplace equation.

Teaching Statement (February 2012).