spaces, then The rank–nullity theorem for finite-dimensional vector spaces may also be formulated in terms of the index of a linear map. The index of a linear map , where and are finite-dimensional, is defined by Intuitively, is the number of independent solutions of the equation , and is the number of independent restrictions that have to be put on to make solvable. The rank–nullity theorem for finite-dimensional vector spaces is equivalent to the statement We see that we can easily read off the index of the linear map from the involved spaces, without any need to analyze in detail. This effect also occurs in a much deeper result: the Atiyah–Singer index theorem states that the index of certain differential operators can be read off the geometry of the involved spaces. Citations References . Theorems in linear algebra Isomorphism theorems Articles containing proofs