four fundamental subspaces

(2.4 hours to learn)


The four fundamental subspaces of a matrix A are the column space, nullspace, row space, and left nullspace. The bases of all four spaces can be obtained using Gaussian elimination, and certain of them are orthogonal to one another. There are close relationships between the dimensions of all four spaces, and the dimensions of the row and column spaces both equal the rank of A.


This concept has the prerequisites:

Core resources (read/watch one of the following)


Khan Academy: Linear Algebra
  • Lecture "Null space and column space basis"
  • Lecture "Visualizing a column space as a plane in R^3"
  • Lecture "Dimension of the null space or nullity"
  • Lecture "Dimension of the column space or rank"
  • Lecture "Showing relation between basis cols and pivot cols"
  • Lecture "Showing that the candidate basis does span C(A)"
MIT Open Courseware: Linear Algebra (2011)
Videos for an introductory linear algebra course focusing on numerical methods.
Author: Gilbert Strang


See also