Proof: To see why this is true, let us first define rank to mean column rank. Since A = CF, it follows that A^\textsf{T} = F^\textsf{T}C^\textsf{T}. From the definition of matrix multiplication, this means that each column of A^\textsf{T} is a linear combination of the columns of F^\textsf{T}. Therefore, the column space of A^\textsf{T} is contained within the column space of F^\textsf{T} and, hence, \operatorname{rank}\left(A^\textsf{T}\right) \leq \operatorname{rank}\left(F^\textsf{T}\right).
