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.
@article{bwmeta1.element.doi-10_2478_v10037-007-0015-6,
author = {Jesse Alama},
title = {The Rank+Nullity Theorem},
journal = {Formalized Mathematics},
volume = {15},
year = {2007},
pages = {137-142},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10037-007-0015-6}
}
Jesse Alama. The Rank+Nullity Theorem. Formalized Mathematics, Tome 15 (2007) pp. 137-142. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10037-007-0015-6/
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