1. The problem statement, all variables and given/known data Suppose T:V→U is linear and V has finite dimension. Prove that Im Tt = (Ker T)0 2. Relevant equations dim(W)+dim(W0)=dim(V) where W is a subspace of V and V has finite dimension. 3. The attempt at a solution I first proved Im Tt ⊆ (Ker T)0. Let u be an arbitrary element of Ker T and Tt(Φ )∈ Im Tt, then Tt(Φ)(u)=Φ(T(u))=Φ(0)=0. Thus, since all elements of Im Tt map all elements of Ker T into 0, Im Tt ⊆ (Ker T)0. I now only need to proof dim((Ker T)0) = dim Im Tt. Since V has finite dimension, dim((Ker T)0) = dim(V)-dim(Ker T), but dim(Im T) = dim(V)-dim(Ker T). Thus, dim((Ker T)0)=dim(Im T). This is where I got stuck, how do I prove dim (Im T) = dim ((Im T)0)?