Can I please have help with this question? Thanks

Exercise 51 Why do each of the above equations determine exactly the same set of solutions 3;? Let r nr _ 'f\" U1,1 U1,2 U_ m( 0 0 ) where r is a rank of A and let ng = y = r (371,1 ) and L1P1b = r (bu ): b2,1 11,?\" 3211 m'r with some abuse of notation. Substituting back into the equation for x we obtain (U14 U13) (3'14) = (131,1) 0 0 y2,1 b2,1 ' We see that the last block equation requires that b2; = 0. Either this matrix has zero rows and the condition is trivially satised, or it does not, and then the validity of this equation depends entirely on the given b and L and P1. If b2; 7': 0 then there are no matrices x which satisfy the equation Ax = b. If b2; = 0 then we must look at the remaining rst block equation U1,1y1,1 + U1,2y2,1 = b1,1. Since we are guaranteed that U1,1 is invertible we see that the general solution is Y1,1 = Ufbm U1,2y2,1), where we are free to pick ygal freely