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NYU Polytechnic School of Engineering
Mathematics
Course Outline for MA-UY 3113
Advanced Linear Algebra and Complex Variables
Spring 2015
Tuesday – Thursday 2:30 PM- 3:50 PM
MetroTech # 2: Rm 9.011
Classes: Monday Jan 26, 2015 – Monday May 4, 2015
{Monday Feb 16, 2015 : Presidients’ day: No classes}
{Spring Break : UG Monday March 16, 2015 – Sunday March 22, 2015}
{Last Day Add/Drop regular NYUPoly classes on Albert: Monday Feb 9, 2015}
{Friday April 10, 2015: Last day to withdraw NYU Poly courses}
UG last day classes: Monday May 4, 2015 UG NYUPoly classes only
Reading days: UG May 5 - May 7 UG NYUPoly classes only
Final Exams: UG May 8 - May 19 UG NYUPoly classes only
Course Coordinator: Professor Edward Y. Miller
Office: RH305J
Email: em1613@nyu.edu
Phone: 718-260-3386
Office Hours: Tuesday & Thursday noon-14:30 & by
appointment.
Student Disabilities:
If you are student with a disability who is requesting accommodations, please
contact New York University’s Moses Center for Students with Disabilities at 212998-4980 or mosescsd@nyu.edu.You must be registered with CSD to receive
accommodations. Information about the Moses Center can be found at
www.nyu.edu/csd. The Moses Center is located at 726 Broadway on the 2nd floor.
Religious Observation Policy As a nonsectarian, inclusive institution, NYU policy
permit members of any religious group to absent themselves form clasess without
penalty when required for compliance with their religious obligations. The policy
and principles to be followed by students and faculty may be found here universitycalendar-policy-on-religious-holidays
Course Website: NYU Classes and www.math.poly.edu
Course Pre-requisites: You are expected to have mastery of the concepts and
skills covered in MA0902,MA0912,MA1024/1324,MA1124/1424, and MA2034
Textbook The required texts for the course are
D. Lay Linear Algebra and Its Application, Addison-Wesley, 4-th edition,
ISBN-13: 978-0321385178 ; J. Ward & R. Churchill Complex Variables and
Applications, 9-th edition, McGraw-Hill, ISBN-13: 978-0073383170
Course Structure: The 3-credit, semester course meets for lecture two times a
week for one and a half hours each class. You are also expected to study outside of
class, a good ‘rule of thumb‘ is three hours of study for each hour of class.
Course Requirements: (The grading policy is detailed in a section below).
 Two Weekly Lectures
 Homework weekly. {work sheets due on Thursday}
 Midterm Exam ? in class.
 Final Exam ?
Examinations: Two 120-minute exams given during class time, and a 120-minute
cumulative Final Exam with emphasis on complex segment.
Homework:
Homework Homework is required and counts towards your grade. Collected
weekly, On Thursdays.
Information about the grading scale conversion to letter grades can be found on the
www.math.poly.edu website.
Course Lecture Syllabus Course Lecture Syllabus (The sections are quoted
from the 4th edition of Lay’s Linear Algebra and Its Application and the9th edition
of Churchill Complex Variables and Applications)
Linear Algebra Segment: [Snow day canceled first day = Jan 27 class]
[Lecture 1-2] Lay: Review [Chapter 1] of matrices, method of row reduction to
solve linear systems Ax=b. Matrix algebra [Chapter 2] matrix addition,
multiplication, and row reduction as A=LU decomposition. Symmetric group and its
properties, Determinants, inverses [Chapter 3] , and reintroduction to eigenvalues,
eigenvectors [Chapter 5].
Additionally, Churchill: [Chapter 1] Introduction to complex numbers.
[Lectures 3-4] Vector Spaces: , [Chapter 2]: Basic Constructions on Vector spaces:
addition, subspaces, quotient spaces, duals, tensor products. Reintroduction to
Linear maps [Examples, Hom(V,W), GL(V), O(n), U(n), Sp(2n,R),Lorentz group,
representation theory, Group ring], rank, column space, row space, nullity, Linear
dependence, Span, subspace in Rn; Vector spaces, dimension, change of basis
[Chapter 4]
[Lectures 5-6] [Chapter 5] Applications: Models for Circuits, [p82, p86 via
Kirchoff’s Laws], heat flow [p87, p131], computer graphics, homogeneous
coordinates [p139-141], Projective Geometry, Signals, filtering [p244], Markov
chains [p253],
[Lectures 6-7] [Chapter 5] Eigenvalues, characteristic polynomial, eigenvectors,
Hermitian, real Symmetric matrices, diagonalization, Decomposition theorems,
Jordan blocks, An , eTA , power methods,
[Lecture 7-8] [Chapter 6] Inner products, angles and lengths, Gramm-Schmidt
method, mini-max principle, orthogonal projection, unitary and orthogonal groups,
Least squares methods in applications, linear approximation, curve fitting, weighted
least squares. [Chapter 7] Real symmetric and complex Hermitian matrices and
their spectral decompositions, quadratic forms applications to constrained
optimization.
[Lecture 9] [Chapter 7] Singular Value decomposition, [Chapter 8] Bezier curves
and surfaces in computer graphics. Application of SU(2) to computer game
graphics. The Kirchoff and Trent spanning tree theorem for the Laplacian on
graphs, decomposition of unitary representations into irreducibles.
[Lecture 10] Review for comprehensive test on Linear Algebra Component
[Meeting 11 = Test on Linear Algebra Segment ] Mar 5, 2015
Complex Variables Segment: Churchill
[Lecture 12] Some standard methods and representations Section 7 Exponential
form, products and quotients, algebraically and geometrically. Section 8 Powers
and roots of complex numbers via exponential form. Section 9 Examples,
geometric representations, n-th roots of unity and their geometry. Section 10
Regions in the complex plane, bounded, open, and connected.
[Lecture 13] Chapter 2-Introduction to Analytic Functions, Examples Section 11
Definition: Functions of a complex variable. Two approaches: Real, Imaginary &
Modulus, Argument. Section 12 Mappings, translation, reflection, dilation, z->z^2 in
deta
[Lecture 14] Limits and Continuity Section 14, Limits in the complex setting: limit
of f(z) as z→a. Section 15, 17 Theorems on limits and continuity. Section 16 Limits
involving the point at infinity, stereographic projection, the projective plane.
[Lecture 15] Derivatives of complex functions, analytic functions Section 18, 19
Derivatives, differentiation formulas. Section 20, 21 Cauchy Riemann equations, a
sufficient condition. Section 23, 24, 25 Analytic functions, examples, Harmonic
functions.
[Lecture 16] Chapter 3-Elementary Functions Section 28 Exponential Function
exp(z). Section 29, 30, 31 Logarithm Function Log(z), branches and derivatives of
the logarithm function, identities. Section 32 Complex exponents, a^z. Section 33
Trigonometric functions, sine, cosine, tangent, etc. Section 34 Hyperbolic
functions sinh, cosh, tanh etc. Section 35 Inverse Trigonometric Functions.
[Lecture 17] Computation of elementary functions, Examples, Review of chapters 13
[Lecture 18] Exam: Complex Variables; April 7, 2015
[Lecture 19] Contour Integration Section 36 Derivatives of w(t) complex function
of real variable t. Section 37 Review: Definite integral. Section 38, 39
Parametrized paths, contour integration. Section 40, 41 Examples, Basic inequality
for contour integration. Section 42,43 Antiderivatives, examples, the complex
analog of the fundamental theorem of calculus.
[Lecture 20] Cauchy-Goursat Theorem for simply closed curves Section 44
Statement of The Cauchy Goursat theorem. SecTion 46 Simply and multiply
connected regions, the main trick in evaluation integrals in This special analytic
setting, Examples. Section 47 Cauchy Integral Theorem, an application of The
Cauchy Goursat Theorem. Section 48 Formulas for all derivatives, an application of
The Cauchy Goursat Theorem. Section 49 Liouville's Theorem, an application of
The Cauchy Goursat Theorem.
[Lecture 21] Applications of the Cauchy Integral Theorem Section 50 Cauchy's
Inequality, the Maximal modulus principle. Section 51 Expansion of an analytic
function converging on a disk as a convergent power series. Examples. Section
52,65 Laurant expansions, three Types of isolated singular points. Section 62, 63,
66 Integrals by method of residues. Residues at poles.
[Lecture 22] Computation of integral via method of residues Section 71, 72
Evaluation of some improper integrals. Review by example of the method of
residues to do integrals. ] Examples of Conformal Mapping
[Lecture 23] Product and Series expansions, sin(z), cotan(z), csc(z)2 ,the theorem
of Weirstrass, the Gamma function
[
[Lecture 24]: Review for Final
Final Examination scheduled during finals May 18 – May 19.
Grading Policy
Course Grade: Final grades will be calculated according to the rules below. The
course grade is determined by the best of your course averages using the table
below.
Midterm
Final Exam
Homework
Average 1
35 %
50 %
15 %
Average 2
15 %
85 %
Average 3
40 %
60 %
Additional Learning Resources: General Evening Math WorkshopsDays Hours
Location Mon-Thurs 6PM-9PM JAB 373 Room 2C
General Math Workshops Days Hours Location Friday 9PM-6PM RH 315
Saturday 10AM-2PM RH 707
Internet Resources:
Math Department Website: www.math.poly.edu. This comprehensive website has
the course policies as well as old exam and practice materials for both the
midterm and final exam.
http://web.mit.edu/18.06/www/Video/video-fall-99.html
http://tutorial.math.lamar.edu/ Paul’s Online Math Notes,
Choose Class Notes and then the course you want.
www.Youtube.com There are many good lectures, the Khan Academy is a favourite
for many students
Additional Learning Resources:
General Math Workshops
Days
Hours
Location
M--Th 6PM-9PM JAB373, PTC
Fri
11AM-6PM RH304
Internet Resources
Math Department Website: www.math.poly.edu This comprehensive website has
the course policies as well as old exam and practice materials for both the midterm
and final exam.
YouTube: Kahn Academy
http://web.mit.edu/18.06/www/Video/video-fall-99.html
http://tutorial.math.lamar.edu/ Paul’s Online Math Notes,
Choose Class Notes and then the course you want.
Important: General Exam Policies
Valuables (especially your laptop!):
Please do not bring your laptop or any other valuable items to the exam. You are
required to leave your bags and books at the front of the exam room.
Time and Place:
It is your responsibility to consult the web site to know when and where an exam is
being held. You will not receive any special consideration for being late or missing
an exam by mistake.
Identification:
You are required to bring your Polytechnic ID to the exam. If for any reason you
are unable to do so, another photo ID, such as a drivers license, is acceptable.
Before the Exam:
You must wait outside the exam room before the start of an exam. You must sit
only in seats where there is an exam for your course. You must not move the exam
to a different seat.
Neatness and Legibility:
You are expected to write as neatly and legibly on your exam. Your final answer
must be clearly identified (by placing a box around it). Points will be deducted if
the grader has difficulty reading or finding your answer.
Missed Exams:
If you missed an exam due to a medical reason, then University policy requires
you to provide written documentation to the Office of Student Development
(JB158). It is University policy that the Mathematics Department may not
give make-up exams without prior authorization by the Office of Student
Development.
Academic Integrity:
Any incident of cheating or dishonesty will be dealt with swiftly and severely. The
University does not tolerate cheating. (There is no such thing as "a little bit of
cheating.") During an exam you are not allowed to borrow or lend a calculator;
borrowing or lending a calculator will be considered cheating.
Useful and Interesting math-physics Web Sites, etc.
See http://math.poly.edu/reference/external.phtml