The Four Fundamental Subspaces

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Taught by OCW
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Lesson Summary:

In this linear algebra lesson, the concept of the four fundamental subspaces is introduced. These subspaces include the column space, null space, row space, and left null space of a matrix. The dimension of each subspace is discussed, with the column space and row space having the same dimension and the null space and left null space having dimensions determined by the number of free variables. A basis for each subspace is also given, with the column space having a basis of pivot columns and the null space having a basis of special solutions.

Lesson Description:

The Four Fundamental Subspaces -- Lecture 10. Learn what the four fundamental subspaces are.

Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare), (Accessed November 19, 2008). License: Creative Commons BY-NC-SA.
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Questions answered by this video:
  • What are the four fundamental subspaces?
  • What is the column space?
  • What is the nullspace?
  • What is the row space?
  • What is the nullspace of A transpose?
  • What is the left nullspace?
  • Staff Review

    • Currently 4.0/5 Stars.
    This video explains the four fundamental subspaces: the column space, the nullspace of A, the row space, and the nullspace of A transpose (left nullspace of A). Another important and in-depth lecture in Linear Algebra.