The Four Fundamental Subspaces

MIT has put up the course materials for a number of their courses on their OpenCourseWare website–including a linear algebra course taught by Gilbert Strang.  There are video lectures, plus assignments, exams, and other study materials.

Strang is an MIT math professor who has written another widely used linear algebra textbook (CUNY Library/Amazon), and so the course follows his book. One of the sections of his book is called “The Four Fundamental Subspace.”  You can find a pdf paper by Strang about these four subspaces here, which introduces the subspaces in the first paragraph:

“The expression “Four Fundamental Subspaces” has become familiar
to thousands of linear algebra students. Those subspaces are the column space and the nullspace of A and A^T [i.e., transpose of A]. They lift the understanding of [the matrix equation] Ax = b to a higher level—–a subspace level. The first step sees Ax (matrix times vector) as a combination of the columns of A. Those vectors Ax fill the column space C(A) [in Lay’s notation, Col A].  When we move from one combination to all combinations (by allowing every x), a subspace appears. Ax = b has a solution exactly when b is in the column space of A.”

from Strang (1993), Introduction to Linear Algebra

Here is Strang’s lecture–it’d be worth setting aside an hour to watch it:

Posted in Uncategorized | Leave a comment

The Invertible Matrix Theorem

The Invertible Matrix Theorem ties together just about all the concepts we’ve covered in the course so far. Here are some resources to supplement the textbook and our discussion in class:

Posted in Uncategorized | Tagged , , | Leave a comment

Hello MAT 2580 world! Welcome & Get Started

Hi all,

I’ll be using this site and the associated course profile page on OpenLab for communicating with you about the course, and occasionally directing you to online resources that might help you.  You can also communicate with me & your classmates via the comments to posts here, and via the discussion forums on the course profile page.

The first thing you should do is set up an account on OpenLab if you haven’t already done so.  Here are some instructions pulled from this post by another math prof on how to do so:

You will need to do two things:

  1. If you have not used the openlab before, you must first create an account.  You will need access to your citytech email address  for this.  Detailed instructions for signing up on the OpenLab can be found here.
  2. Once you have created an account on the OpenLab, log in and then join this particular course, Spring 2013 – MAT 1272 Statistics – Reitz [for our course, join MAT 2580 Linear Algebra – Spring 2013 – Ganguli].  To do this, first click the “Course Profile” link at the top left of this page (just under the picture) [which I’ve also linked to above].  Then click the “Join Now” button, which should appear just underneath the picture of the blackboard.

Problems with the OpenLab or with your CityTech email:

Please let me know if you run into any problems registering or joining our course (send me an email,  I also wanted to give you two resources to help out in the process:

1.  For problems with your citytech email account, contact the Student Computing Helpdesk, either in person, by phone, or by email:

Student Computing Helpdesk
Location: Namm First Floor – Information Booth
Hours: Monday, Wednesday & Friday: 9:00am – 5:00pm
Tuesday & Thursday: 9:00am – 6:00pm
Phone: 718.260.4900

2. For problems registering for the OpenLab, contact the OpenLab admin team, either by email at, or by following this link.


Posted in Uncategorized | 1 Comment