Editing Rank–nullity theorem (section)

== A third fundamental subspace ==

When <math>T: V \to W</math> is a linear transformation between two finite-dimensional subspaces, with <math> n = \dim(V)</math> and <math> m = \dim(W)</math> (so can be represented by an <math>m \times n</math> matrix <math>M</math>), the rank–nullity theorem asserts that if <math>T</math> has rank <math>r</math>, then <math>n - r</math> is the dimension of the [[null space]] of <math>M</math>, which represents the [[kernel (linear algebra)|kernel]] of <math>T</math>.  In some texts, a third fundamental subspace associated to <math>T</math> is considered alongside its image and kernel: the [[cokernel]] of <math>T</math> is the  [[quotient space (linear algebra)|quotient space]] <math>W / \operatorname{Im}(T)</math>, and its dimension is <math>m - r</math>.  This dimension formula (which might also be rendered <math>\dim \operatorname{Im}(T) + \dim\operatorname{Coker}(T) = \dim(W)</math>) together with the rank–nullity theorem is sometimes called the ''fundamental theorem of linear algebra''.<ref>* [[Gilbert Strang|Strang, Gilbert]]. ''Linear Algebra and Its Applications''. 3rd ed. Orlando: Saunders, 1988.</ref><ref>{{Citation
| title = The fundamental theorem of linear algebra
| url = http://www.dm.unibo.it/~regonati/ad0708/strang-FTLA.pdf <!-- if that goes dead try one of these:
https://www.engineering.iastate.edu/~julied/classes/CE570/Notes/strangpaper.pdf
-->
| year = 1993
| author = Strang, Gilbert
| journal = American Mathematical Monthly
| volume = 100
| issue = 9
| pages = 848–855
| doi = 10.2307/2324660
| jstor = 2324660
| citeseerx = 10.1.1.384.2309
}}</ref>