# The Rank+Nullity Theorem

Formalized Mathematics (2007)

• Volume: 15, Issue: 3, page 137-142
• ISSN: 1426-2630

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## Abstract

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The rank+nullity theorem states that, if T is a linear transformation from a finite-dimensional vector space V to a finite-dimensional vector space W, then dim(V) = rank(T) + nullity(T), where rank(T) = dim(im(T)) and nullity(T) = dim(ker(T)). The proof treated here is standard; see, for example, [14]: take a basis A of ker(T) and extend it to a basis B of V, and then show that dim(im(T)) is equal to |B - A|, and that T is one-to-one on B - A.

## How to cite

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Jesse Alama. "The Rank+Nullity Theorem." Formalized Mathematics 15.3 (2007): 137-142. <http://eudml.org/doc/267154>.

@article{JesseAlama2007,
abstract = {The rank+nullity theorem states that, if T is a linear transformation from a finite-dimensional vector space V to a finite-dimensional vector space W, then dim(V) = rank(T) + nullity(T), where rank(T) = dim(im(T)) and nullity(T) = dim(ker(T)). The proof treated here is standard; see, for example, [14]: take a basis A of ker(T) and extend it to a basis B of V, and then show that dim(im(T)) is equal to |B - A|, and that T is one-to-one on B - A.},
author = {Jesse Alama},
journal = {Formalized Mathematics},
language = {eng},
number = {3},
pages = {137-142},
title = {The Rank+Nullity Theorem},
url = {http://eudml.org/doc/267154},
volume = {15},
year = {2007},
}

TY - JOUR
AU - Jesse Alama
TI - The Rank+Nullity Theorem
JO - Formalized Mathematics
PY - 2007
VL - 15
IS - 3
SP - 137
EP - 142
AB - The rank+nullity theorem states that, if T is a linear transformation from a finite-dimensional vector space V to a finite-dimensional vector space W, then dim(V) = rank(T) + nullity(T), where rank(T) = dim(im(T)) and nullity(T) = dim(ker(T)). The proof treated here is standard; see, for example, [14]: take a basis A of ker(T) and extend it to a basis B of V, and then show that dim(im(T)) is equal to |B - A|, and that T is one-to-one on B - A.
LA - eng
UR - http://eudml.org/doc/267154
ER -

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