A is an mxn matrix and P is an mxm matrix if LA is a left mu

A is an mxn matrix and P is an mxm matrix
if L_A is a left multiplication transformation from F^n to F^m
How do you prove Rank (PA)=Rank (A)

Solution

Let A is an m x n matrix and

     P is an m x m matrix and

     L_A is a left multiplication transformation from Fⁿ to F^m.

Note that rank(P_A) = rank (L_PA)

                 L_PA = L_P O L_A

   And R(L_PA) = (L_P O L_A)(Fⁿ)

But, dim(L_P(L_A(Fⁿ))) = dim (L_A(Fⁿ))

Then Rank (P_A) = Rank (L_PA) = dim (R(L_PA))

                 dim (L_A(Fⁿ)) = dim (R(L_A)) rank (L_A) = rank (A).