Rank-Nullity Theorem in Linear Algebra

Resumo

In this article we present a proof of the result known in Linear Algebra as the ``rank nullity Theorem'', which states that, given any linear form f from a finite dimensional vector space V to a vector space W, then the dimension of V is equal to the dimension of the kernel of f (which is a subspace of V) and the dimension of the range of f (which is a subspace of W). The proof presented here is based on the one given by Sheldon Axler in his book Linear Algebra Done Right. It makes use of the HOL-Multivariate-Analysis session of Isabelle, and of several of its results and definitions. As a corollary of the previous theorem, and taking advantage of the relationship between linear forms and matrices, we prove that, for every matrix A (which has associated a linear form between finite dimensional vector spaces), the sum of its null space and its column space (which is equal to the range of the linear form) is equal to the number of columns of A.