Let U, V, and W be vector spaces over a field F. Suppose that T : U --> V and S : V --> W are linear transformations and that Im(T) = Ker(S). If T is injective and S is surjective, prove that
dim(V) = dim(U) + dim(W).© BrainMass Inc. brainmass.com March 22, 2019, 12:09 am ad1c9bdddf
For ease of notation I will drop reference to the field F. We assume throughout that F is fixed and that dim refers to dimension of a vector space over F. This problem boils down to the following fundamental property of linear transformations, called the rank-nullity theorem:
Let X and Y be vector spaces and let L : X --> Y be a ...
The rank-nullity theorem is used to derive a relationship between the vector spaces involved in the composition of an injective and surjective linear map. The step-by-step solution to this problem illustrates the use of fundamental concepts in linear algebra.