Row-equivalence and row space
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Exercise:
If AB in M_mtimes nK are row-equivalent matrices then Row SA Row SB. The statement does NOT hold for cols!
Solution:
Proof. Let Min M_mtimes nK. When we exchange two rows of M the span of the rows is not affected. So Row SM doesn't change under such an operation. The operation R_i longrightarrow lambda R_i where lambdain Kbackslash does NOT change the span of the rows of M. So Row SM is left unchanged after such an operation. The operation R_i+lamba R_j longrightarrow R_i where ineq j lambdain K. We claim that Spw_...w_mSpw_...w_i+lambda w_j...w_m. Indeed Spw_...w_i+lambda w_j...w_m subseteq Spw_...w_m because every linear combination of the left side is obviously also a linear combination of the right side. But we also have Spw_...w_msubseteq Spw_...w_i+lambda w_j...w_m because w_iw_i+lambda w_j-lambda w_j and since ineq j when we write a linear combination of w_...w_n we will get a linear combination of w_...w_i+lambda w_j...w_m. This proves the claim. From the claim it follows that the operation R_i+lambda R_j longrightarrow R_i ineq j lambda in K does not change Row SM. Now if A B are row equivalent then by definition exists afinite sequence of matrices A_AA_...A_r- A_rB s.t. A_i+ is obtained from A_i by an elementary row operation forall leq i leq r-. AA_longrightarrow A_ longrightarrow ... longrightarrow A_rB. By what we proved earlier we have Row SA Row SA_ Row SA_ ... Row SA_r Row SB
If AB in M_mtimes nK are row-equivalent matrices then Row SA Row SB. The statement does NOT hold for cols!
Solution:
Proof. Let Min M_mtimes nK. When we exchange two rows of M the span of the rows is not affected. So Row SM doesn't change under such an operation. The operation R_i longrightarrow lambda R_i where lambdain Kbackslash does NOT change the span of the rows of M. So Row SM is left unchanged after such an operation. The operation R_i+lamba R_j longrightarrow R_i where ineq j lambdain K. We claim that Spw_...w_mSpw_...w_i+lambda w_j...w_m. Indeed Spw_...w_i+lambda w_j...w_m subseteq Spw_...w_m because every linear combination of the left side is obviously also a linear combination of the right side. But we also have Spw_...w_msubseteq Spw_...w_i+lambda w_j...w_m because w_iw_i+lambda w_j-lambda w_j and since ineq j when we write a linear combination of w_...w_n we will get a linear combination of w_...w_i+lambda w_j...w_m. This proves the claim. From the claim it follows that the operation R_i+lambda R_j longrightarrow R_i ineq j lambda in K does not change Row SM. Now if A B are row equivalent then by definition exists afinite sequence of matrices A_AA_...A_r- A_rB s.t. A_i+ is obtained from A_i by an elementary row operation forall leq i leq r-. AA_longrightarrow A_ longrightarrow ... longrightarrow A_rB. By what we proved earlier we have Row SA Row SA_ Row SA_ ... Row SA_r Row SB