The Rank+Nullity Theorem

Abstract

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 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.