<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml"><p>The rank+nullity theorem states that, if <italic>T</italic> is a linear transformation from a finite-dimensional vector space <italic>V</italic> to a finite-dimensional vector space <italic>W</italic>, then dim(<italic>V</italic>) = rank(<italic>T</italic>) + nullity(<italic>T</italic>), where rank(<italic>T</italic>) = dim(im(<italic>T</italic>)) and nullity(<italic>T</italic>) = dim(ker(<italic>T</italic>)). The proof treated here is standard; see, for example, [14]: take a basis <italic>A</italic> of ker(<italic>T</italic>) and extend it to a basis <italic>B</italic> of <italic>V</italic>, and then show that dim(im(<italic>T</italic>)) is equal to |<italic>B - A</italic>|, and that <italic>T</italic> is one-to-one on <italic>B - A.</italic></p></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.

The Rank+Nullity Theorem

Department of Philosophy, Stanford University, USA

Formalized Mathematics

{"article-title":"The Rank+Nullity Theorem"}

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...

The Rank+Nullity Theorem

Data publikacji: 09 cze 2008

Zakres stron: 137 - 142

DOI: https://doi.org/10.2478/v10037-007-0015-6

This content is open access.