This is a problem from the textbook Hoffman and Kunze (Sec 3.2, problem 12, page 84)
Let p, m and n be positive integers and F a field. Let V be the space of m by n matrices over F and W be the space of p by n matrices over F. Let B be a fixed p by m matrix and let T be the linear transformation, T(A)= BA. Prove that T is invertible if and only if p=m and B is an invertible m by m matrix.
I could prove the left pointing implication and I am trying to prove the right pointing implication. Suppose T is invertible, then this means that T is 1-1 and onto and thus, by the rank-nullity theorem we have that dim V = dim W and thus p=m (Is this a correct argument? ). I would be grateful if there is a slight hint about how to prove that B is invertible.