# Fundamental theorem of linear algebra

From HandWiki

__: Name for certain results on linear maps between two finite-dimensional vector spaces__

**Short description**In mathematics, the **fundamental theorem of linear algebra** is a collection of statements regarding vector spaces and linear algebra, popularized by Gilbert Strang. The naming of these results is not universally accepted.

More precisely, let f be a linear map between two finite-dimensional vector spaces, represented by a *m*×*n* matrix M of rank r, then:

- r is the dimension of the column space of M, which represents the image of f;
*n*–*r*is the dimension of the null space of M, which represents the kernel of f;*m*–*r*is the dimension of the cokernel of f.

The transpose *M*^{T} of M is the matrix of the dual f^{*} of f. It follows that one has also:

- r is the dimension of the row space of M, which represents the image of f
^{*}; *m*–*r*is the dimension of the left null space of M, which represents the kernel of f^{*};*n*–*r*is the dimension of the cokernel of f^{*}.

The two first assertions are also called the rank–nullity theorem.

## References

- Strang, Gilbert.
*Linear Algebra and Its Applications*. 3rd ed. Orlando: Saunders, 1988. - Strang, Gilbert (1993), "The fundamental theorem of linear algebra",
*American Mathematical Monthly***100**(9): 848–855, doi:10.2307/2324660, http://www.dm.unibo.it/~regonati/ad0708/strang-FTLA.pdf - 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

## External links

- Gilbert Strang, MIT Linear Algebra Lecture on the Four Fundamental Subspaces, from MIT OpenCourseWare