Fundamental theorem of linear algebra

In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces. Those statements may be given concretely in terms of the rank r of an m × n matrix A and its singular value decomposition:

First, each matrix ( has rows and columns) induces four fundamental subspaces. These fundamental subspaces are as follows:

name of subspace definition containing space dimension basis
column space, range or image or (rank) The first columns of
nullspace or kernel or (nullity) The last columns of
row space or coimage or (rank) The first columns of
left nullspace or cokernel or (corank) The last columns of


  1. In , , that is, the nullspace is the orthogonal complement of the row space
  2. In , , that is, the left nullspace is the orthogonal complement of the column space.

The dimensions of the subspaces are related by the rank–nullity theorem, and follow from the above theorem.

Further, all these spaces are intrinsically defined—they do not require a choice of basis—in which case one rewrites this in terms of abstract vector spaces, operators, and the dual spaces as and : the kernel and image of are the cokernel and coimage of .

See also


  • Strang, Gilbert. Linear Algebra and Its Applications. 3rd ed. Orlando: Saunders, 1988.
  • Strang, Gilbert (1993), "The fundamental theorem of linear algebra" (PDF), American Mathematical Monthly, 100 (9): 848–855, CiteSeerX, doi:10.2307/2324660, JSTOR 2324660
  • Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC, ISBN 978-1420095388
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