V63.0140.4: Linear Algebra

Linear algebra is an essential tool for every scientist, engineer, economist, statistician, and mathematician. It is also useful for computer scientists and countless other numerical sciences. Linear problems are often the simplest models of the natural world that can be made, and arise in literally thousands of applications. Complexity can arise because large numbers of variables can be involved. With the rise in electronic computing power, huge linear algebra problems can be solved, and these can be used to model very complicated situations in the real world. In fact, the desire to solve such problems has driven computing technology. In this course you will learn the language, concepts, and techniques, from the ground up. You will also learn how to visualize geometrically, and manipulate abstract ideas which may not be visualizable.


SILV 507, Tuesday and Thursday, 6:20pm – 8:10pm. 110 mins, with 5–10 min break to get up and stretch! Attendance of lecture is important: there will often be activities such as worksheets which you will do in pairs or small groups. These will allow you to grapple with concepts and explain them to each other. For students lying on grade boundaries I will take lecture participation into account.

Required book

Linear Algebra and its Applications, 3rd Edition by David C. Lay, available at NYU Bookstore, Amazon, etc. (about $108). You may find the paperback Study Guide useful too. If the bookstore is sold out, or if you want to see what the Study Guide is like, you can get the first chapter for free at http://www.laylinalgebra.com. This website also has review notes.


Check blackboard website.


11 problem sets, almost all from the book, to be handed in at start of Thursday lectures. Permission to hand in late must be obtained from me the previous week, and otherwise, unless you have a doctor’s note or there are exceptional circumstances, late homework will not be counted in your grade. However you should still do HW even if late, since it forms an essential part of the learning process. Your lowest two HW scores will be dropped, so this allows you leeway.


I encourage you to study in groups. Get to know others in class. If you have no-one to study with, I can help you find someone. However you must write up homework in your own words, and understand what you write. Plagiarism (in homework or in exams) is a serious offense (see NYU CAS Academic Policies).

Exams and Grades

There will be 1 midterm and 1 final (dates given on below). There will also be about 4 quizzes, whose dates I will announce a week in advance, and which will happen during the last 30 minutes of lecture. Your lowest quiz grade will be dropped. Your overall grade will be determined using:

homework 25%, quizzes 20%, midterm 25%, final 30%.

The grades will not be curved. After the midterm I will settle on and announce a fixed grade boundary scheme. Office Hours: Tuesday 4-5pm, and Wednesday 2-3pm (rm 926 WWH).


Free math help has historically been available 4 days a week on a drop-in basis at rm 704 WWH. Hours were 1pm–8pm Mon+Wed, 11am–8pm Tues+Thurs. Tutors may vary in their knowledge of linear algebra. However I don’t know if the room has changed or if this is offered this year. Details to follow. . .

Timeline: (approximate)

Week DateDue SectionContent

1Tu Sep. 7 1.1 Systems of linear equations
Th 9 1.2+1.3 Solving linear equations, Vectors

2Tu 14 1.4+1.5 Matrix equations
Th 16HW1 1.6+1.7 Applications, Linear independence

3Tu 21Quiz 1 1.8 linear transformations
Th 23HW2 1.9+2.1 Matrix of Lin. Trans., Matrix operations

4Tu 28 2.2+2.3 Matrix inverses, Characterizations
Th 30HW3 2.4+2.5 Partitioned matrices, LU factorizations,

5Tu Oct. 5Quiz 2 2.6-2.7 Leontief I/O model,graphics applications
Th 7HW4 3.1-3.2 Determinants, Properties of Determinants

6Tu 12 3.3 Cramer’s rule
Th 14 Review

7Tu 19Midterm
Th 21HW5 4.1-4.2 vector spaces, subspaces

8Tu 26 4.3-4.4 bases, coordinate systems
Th 28HW6 4.5-4.6 dimension, rank

9Tu Nov. 2Quiz 3 4.7 Change of basis
Th 4HW7 4.8-4.9 Applications, Markov chains

10Tu 9 5.1-5.3 Eigenvectors
Th 11HW8 Eigenvalues

11Tu 16 5.4-5.6 linear transformations
Th 18HW9 eigenvalue applications

12Tu 23Quiz 4 6.1-6.2 inner product
Th 25
Thanksgiving Recess (Nov 25-27)

13Tu Nov. 30 6.3-6.4 orthogonal projection, Gram-Schmidt process
Th Dec. 2HW10 6.5, 6.7 least squares, inner product spaces

14Tu 7 6.6+7.1 applications, symmetric matrices
Th 9HW11 7.2+7.4 quadratic forms, singular value

15Tu 14
Last day of classes and Review

16TuDec. 21
Tentative Final Exam: 8:00pm-9:50pm