Rank–nullity theorem
In linear algebra, relation between 3 dimensions
Summary
The rank–nullity theorem is a theorem in linear algebra, which asserts:the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and the dimension of the domain of a linear transformation f is the sum of the rank of f and the nullity of f.
Originally created by Dysprosia
9/28/2003, 4:02:52 PM
Modified
5/31/2026, 6:34:35 PM
Recent revisions
Added the Linear algebra template.
Undid revision [[Special:Diff/1344209648|1344209648]] by [[Special:Contributions/Eteri byazrova|Eteri byazrova]] ([[User talk:Eteri byazrova|talk]]) this figure is incomprehensible in context: little of its content and notation is used in the article (either this section or anywhere else), and it is full of text that is completely illegible at the sizes it will be displayed to readers
/* Linear transformations */ Add diagram illustrating domain and codomain decompositions of fundamental subspaces
/* Reformulations and generalizations */
/* growthexperiments-addlink-summary-summary:2|0|0 */
Reverted 1 edit by [[Special:Contributions/192.42.89.170|192.42.89.170]] ([[User talk:192.42.89.170|talk]]): Unexplained change and [[MOS:VAR]]
/* Linear transformations */
/* Citations */ formatting
/* A third fundamental subspace */ Fixed inconsistent use of \operatorname{Im} and \operatorname{Image}, misleading as they are the same thing (chose Im because it is more prevalent in the article and in the literature)
/* Linear transformations */ Fixed inconsistent use of \operatorname{Im} and \operatorname{Image}, misleading as they are the same thing (chose Im because it is more prevalent in the article and in the literature)
Added condition to sentence that made the sentence true.
/* Proofs */Missing space
typo
/* top */
/* Proofs */ \mathbbb F -> F; rm notation Mat_{m,n}; [[MOS:NOTE]]
/* Matrices */ avoiding unnecessary pedantic notation
/* top */ prose
Added widely used corollary of theorem
merged so remove tag
