If $operatornamerank(AB)=operatornamerank(B)$ then do $A, B$ have the same row space?

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Is this true or false?




If $operatornamerank(AB)=operatornamerank(B)$ then $mathfrakR(AB)=mathfrakR(B)$, where $mathfrakR(A)$ is the row space of $A$.




I am unable to prove it. Is it false? Is there any example contradicting this?






linear-algebra matrix-rank







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  • Why am I getting downvotes instead of answers? @did I didn't violate any rule
    – user581912
    Aug 31 at 11:10










  • Probably because you’re not providing any motivation for the question, don’t show any effort of trying to solve the problem, and make it look like you just want others to do your homework for you.
    – Jendrik Stelzner
    Sep 7 at 12:13











  • @user581912 Hint: First examine what does the assumption $rank(AB)=rank(B)$ imply for matrices $A,B$. Second, what does it mean for $AB$ and $B$ having the same row spaces? Do you know of any equivalent conditions?
    – Test123
    Sep 7 at 13:42











  • This could help : math.stackexchange.com/questions/671665/…
    – Arnaud D.
    Sep 11 at 12:21














up vote
-3
down vote

favorite
1












Is this true or false?




If $operatornamerank(AB)=operatornamerank(B)$ then $mathfrakR(AB)=mathfrakR(B)$, where $mathfrakR(A)$ is the row space of $A$.




I am unable to prove it. Is it false? Is there any example contradicting this?






linear-algebra matrix-rank







share|cite|improve this question























  • Why am I getting downvotes instead of answers? @did I didn't violate any rule
    – user581912
    Aug 31 at 11:10










  • Probably because you’re not providing any motivation for the question, don’t show any effort of trying to solve the problem, and make it look like you just want others to do your homework for you.
    – Jendrik Stelzner
    Sep 7 at 12:13











  • @user581912 Hint: First examine what does the assumption $rank(AB)=rank(B)$ imply for matrices $A,B$. Second, what does it mean for $AB$ and $B$ having the same row spaces? Do you know of any equivalent conditions?
    – Test123
    Sep 7 at 13:42











  • This could help : math.stackexchange.com/questions/671665/…
    – Arnaud D.
    Sep 11 at 12:21












up vote
-3
down vote

favorite
1









up vote
-3
down vote

favorite
1






1





Is this true or false?




If $operatornamerank(AB)=operatornamerank(B)$ then $mathfrakR(AB)=mathfrakR(B)$, where $mathfrakR(A)$ is the row space of $A$.




I am unable to prove it. Is it false? Is there any example contradicting this?






linear-algebra matrix-rank







share|cite|improve this question















Is this true or false?




If $operatornamerank(AB)=operatornamerank(B)$ then $mathfrakR(AB)=mathfrakR(B)$, where $mathfrakR(A)$ is the row space of $A$.




I am unable to prove it. Is it false? Is there any example contradicting this?






linear-algebra matrix-rank




linear-algebra matrix-rank






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Sep 7 at 11:53









Jendrik Stelzner

7,69121137




7,69121137










asked Aug 31 at 5:27









user581912

856




856











  • Why am I getting downvotes instead of answers? @did I didn't violate any rule
    – user581912
    Aug 31 at 11:10










  • Probably because you’re not providing any motivation for the question, don’t show any effort of trying to solve the problem, and make it look like you just want others to do your homework for you.
    – Jendrik Stelzner
    Sep 7 at 12:13











  • @user581912 Hint: First examine what does the assumption $rank(AB)=rank(B)$ imply for matrices $A,B$. Second, what does it mean for $AB$ and $B$ having the same row spaces? Do you know of any equivalent conditions?
    – Test123
    Sep 7 at 13:42











  • This could help : math.stackexchange.com/questions/671665/…
    – Arnaud D.
    Sep 11 at 12:21
















  • Why am I getting downvotes instead of answers? @did I didn't violate any rule
    – user581912
    Aug 31 at 11:10










  • Probably because you’re not providing any motivation for the question, don’t show any effort of trying to solve the problem, and make it look like you just want others to do your homework for you.
    – Jendrik Stelzner
    Sep 7 at 12:13











  • @user581912 Hint: First examine what does the assumption $rank(AB)=rank(B)$ imply for matrices $A,B$. Second, what does it mean for $AB$ and $B$ having the same row spaces? Do you know of any equivalent conditions?
    – Test123
    Sep 7 at 13:42











  • This could help : math.stackexchange.com/questions/671665/…
    – Arnaud D.
    Sep 11 at 12:21















Why am I getting downvotes instead of answers? @did I didn't violate any rule
– user581912
Aug 31 at 11:10




Why am I getting downvotes instead of answers? @did I didn't violate any rule
– user581912
Aug 31 at 11:10












Probably because you’re not providing any motivation for the question, don’t show any effort of trying to solve the problem, and make it look like you just want others to do your homework for you.
– Jendrik Stelzner
Sep 7 at 12:13





Probably because you’re not providing any motivation for the question, don’t show any effort of trying to solve the problem, and make it look like you just want others to do your homework for you.
– Jendrik Stelzner
Sep 7 at 12:13













@user581912 Hint: First examine what does the assumption $rank(AB)=rank(B)$ imply for matrices $A,B$. Second, what does it mean for $AB$ and $B$ having the same row spaces? Do you know of any equivalent conditions?
– Test123
Sep 7 at 13:42





@user581912 Hint: First examine what does the assumption $rank(AB)=rank(B)$ imply for matrices $A,B$. Second, what does it mean for $AB$ and $B$ having the same row spaces? Do you know of any equivalent conditions?
– Test123
Sep 7 at 13:42













This could help : math.stackexchange.com/questions/671665/…
– Arnaud D.
Sep 11 at 12:21




This could help : math.stackexchange.com/questions/671665/…
– Arnaud D.
Sep 11 at 12:21















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