Exercise:
Let Axb S and A'xb' S' be two systems of m linear s in n-unknowns. Suppose the exted matrix A'|b' is row equivalent to A|b. Then LS'LS.

Solution:
Proof. We with the following claim: bf claim : If S' can be obtained from S by one elementary row operation then LS'subseteq LS. bf Proof of the first claim: Let x_...x_nin LS. we need to show x_...x_nin LS'. itemize item operation : c R_i rightarrow R_i c neq . All the s of S' coincide with the s of S except of i a_ix_+...+a_inx_n b_i longrightarrow c a_ix_+...+c a_inx_n c b_i If x_...x_n satisfies the left part then it satisfies also the right one. This proves the claim for the first operation. item operation : R_j+c R_i longrightarrow R_j. Again all the s of S and S' coincide except of j. a_jx_+...+a_jnx_n b_j longrightarrow a_jx_+...+a_jnx_n b_j+c b_i If x_...x_n satisfies the left part then it satisfies also the right one. This proves the claim for the second operation. item operation : R_i longleftrightarrow R_j. In this case we clearly have x_...x_nin LS Longrightarrow x_...x_nin LS' because the order of the s in the system does not have any effect on the set of solutions. itemize This concludes the proof of bf claim . bf claim : If S' is obtained from S by one elementary row operation then S can be obtained from S' by one elementary row operation. bf Proof of the second claim: Again there are three cases. itemize item operation : S rightarrow S' c R_i rightarrow R_i But then S' rightarrow S R_i rightarrow fraccR_i item operation : S rightarrow S' R_j+c R_i rightarrow R_j quad ineq j But then S' rightarrow S R_j-c R_i rightarrow R_j item operation : S rightarrow S' R_i leftrightarrow R_j But then S' rightarrow S R_j leftrightarrow R_i itemize This concludes the proof of bf claim . bf claim : If S' is obtained from S by one elementary row operation then LSLS'. bf Proof of the third claim: By bf claim LS subseteq LS'. By bf claim S is obtained from S' by one elementary row operation hence by bf claim again with the roles of S and S' reversed we have LS' subseteq LS. This concludes the proof of bf claim . We are now in position to prove the Theorem from the ning. By asption there is a finite sequence of elementary row operations SS_ rightarrow S_ rightarrow ... rightarrow S_kS'. By bf claim LSLS_LS_...LS_kLS'.