Linear Algebra - Fall 2006

(last
updated 11/30/2006)

Instructor: S. M. Mintchev

Meeting Time: TR 8:55am - 10:45am

Location: Waverly Bldg., Rm. 435

Instructor
Information:

· office: Warren Weaver Hall, Rm. 718

· telephone: 212-998-3107

· office hours: M 10:00am - 11:00am; 2:00pm - 3:00pm; T 11:00am - 12:00pm

General
Information about the Course: This course is an
introduction to linear transformations of finite-dimensional real and
complex vector spaces. It begins with the study of real systems of
linear equations and their solution sets. We
will first develop the necessary matrix theory tools to solve linear
systems, then move toward understanding linear transformations through
their matrix representations. Throughout the semester we will see
real-world applications of this material, in the hope that students
develop an intuition for recognizing problems that fit into this
framework and solving them systematically.

Text: Linear
Algebra and Its Applications, third edition. David C.
Lay. Addison Wesley. 2003.

Grading Policy: There will be five,
approximately-bi-weekly homework assignments, posted on this website at
the beginning of the discussion period for the required material, and
due at the beginning of class on the posted date (see below for the
list of assignments). Usually, about 20 problems will be assigned at a
time. Of these, the grader for the course will choose about 8 at random
and grade them. Late homework will NOT be accepted, except with a
relevant doctor's note, etc. Some of the homework assignments (see
syllabus below) will be followed by a 15 min. in-class quiz consisting
of two or three problems related to the homework material. In addition
to these assessments, there will be two exams for the course. A midterm
exam will be given approximately in the third week of October. The
midterm will cover the first half of material for the course - in
particular, this will be the material from the first three homework
assignments. There will also be a final examination; this exam will
cover the second half of material taught in the course. The values of
these components towards the final grade in the course are presented
below:

Component |
Value |

Homework |
15 % |

Quiz Total |
25 % |

Midterm Exam |
30 % |

Final Exam |
30 % |

Students are advised to solve homework problems carefully and to write up neat presentations of their solutions. While collaboration is encouraged, students should make a strong individual (meaning: separated from each-other in space, and entirely independent) effort on a problem before attempting collaboration with peers. Such work will be key to placing the student in a position to perform well on the examinations. In addition, since there will be no examinations before the ADD/DROP deadline for most NYU programs, the first two homework assignments should be interpreted as being illustrative of the workload required for passing the course; the grades on these, as well as on any quizzes given during the first 2-4 weeks should be used to make the ADD/DROP decision if necessary.

Syllabus (please note that this schedule is tentative and will likely be adjusted as the semester progresses):

Date | Sections | Topic |

09/05/2006 | 1.1, 1.2 | Linear Systems; Row Reduction |

09/07/2006 | 1.3, 1.4 | Vector Equations; Matrix Equations |

09/12/2006 | 1.5, 1.6, 1.7 | Solution Sets; Applications; Linear Independence |

09/14/2006 | 1.8, 1.9 | Linear Transformations / Matrix Representation |

09/19/2006 | 2.1, 2.2, QUIZ 1 | Matrix Operations; Inverse of a Matrix |

09/21/2006 | 2.3, 2.4 | Invertible Matrices; Partitioned Matrices |

09/26/2006 | 2.5, 2.6 | Matrix Factorization; Leontief Input-Output Model |

09/28/2006 | 2.8, 2.9 | Subspaces; Dimension and Rank |

10/03/2006 | 3.1, 3.2 | Determinants and their Properties |

10/05/2006 | 3.3, 4.1, 4.2 | Cramer’s Rule, Vector Spaces, Subspaces |

10/10/2006 | 4.3, 4.4 | Linear Independence; Bases; Coordinate Systems |

10/12/2006 | 4.5, 4.6 | Dimension of a Vector Space; Rank |

10/17/2006 | 4.7, 4.8, 4.9 | Change of Basis; Difference Equations; Markov Chains |

10/19/2006 | 5.1, 5.2 | Eigenvectors, Eigenvalues; The Characteristic Equation |

10/24/2006 | 5.3, 5.4, QUIZ 2 | Diagonalization; Eigenvectors / Linear Transformations |

10/26/2006 | 5.5, 5.6 | Complex Eigenvalues; Discrete Dynamical Systems |

10/31/2006 | MIDTERM | EXAMINATION (covers through Homework 3) |

11/02/2006 | 6.1, 6.2 | Inner Product; Length; Orthogonality / Orthogonal Sets |

11/07/2006 | 6.3, 6.4 | Orthogonal Projections; Gram-Schmidt Process |

11/09/2006 | 6.5 | Least-Squares Problems |

11/14/2006 | 6.6 | Linear Models |

11/16/2006 | 7.1 | Diagonalization of Symmetric Matrices; Quadratic Forms |

11/21/2006 | 7.2 | Quadratic Forms; Constrained Optimization |

11/23/2006 | 7.2 | |

11/28/2006 | 7.2 | |

11/30/2006 | 7.2, 7.3 | |

12/05/2006 | 7.4, QUIZ 3 | Singular Value Decomposition |

12/07/2006 | Review | |

12/12/2006 | FINAL EXAM |

Homework Assignments (please see grading policy section for requirements):

Assigned |
Due |
Homework Exercises |

09/05/2006 |
09/19/2006 |
Section 1.1: 6, 10, 28 Section 1.2: 11, 16 Section 1.3: 23, 24, 28 Section 1.4: 31 Section 1.5: 12, 25 Section 1.6: 6, 12 Section 1.7: 22, 31 Section 1.8: 20, 21, 22 |

09/27/2006 |
10/10/2006 |
Section 1.9: 23 Section 2.1: 15, 16, 33 Section 2.2: 12, 30, 31 Section 2.3: 27, 38 Section 2.4: 13, 14 Section 2.5: 13, 19 Section 2.6: 5 Section 2.8: 23, 30 Section 2.9: 12, 27 Section 3.1: 13, 42 |

10/09/2006 |
10/24/2006 |
Section 3.2: 33, 34 Section 3.3: 16, 17 Section 4.1: 19, 26, 33 Section 4.2: 34 Section 4.3: 31, 32 Section 4.4: 18, 23 Section 4.5: 26 Section 4.6: 3 Section 4.7: 6, 13 Section 4.8: 25, 35 |

11/09/2006 |
11/21/2006 |
Section 5.1: 16, 20, 33 Section 5.2: 18, 19 Section 5.3: 25, 31, 32 Section 5.4: 19, 20, 21, 22 Section 5.5: 4, 6 Section 5.6: 1, 3 Section 6.1: 17, 18, 30 Section 6.2: 23, 24, 29 Section 6.3: 9, 24 Section 6.4: 3, 19 |

11/30/2006 |
12/05/2006 |
Section 6.5: 12, 19, 21 Section 6.6: 15,16 Section 7.1: 23, 36 Section 7.2: 6, 8 Section 7.3: 13 |