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:

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