Dimension of nullspace and number of rows












1












$begingroup$



A matrix $A$ has $10$ columns and dim(Null($A^{T}$ ))$=7$. The smallest possible number of rows of $A$ is



$(A)$ $5$



$(B)$ $6$



$(C)$ $7$



$(D)$ $8$



$(E)$ $9$




I know that dim(Null($A^{T}$ ))$=7$ implies that there are $7$ rows of zeros and that:



Rank($A$)+Nullity($A^T$) $=$ # of rows



Rank($A$)+Nullity($A$) $=$ # of columns



I'm not really sure how to use all this information though... Can someone provide a hint?






linear-algebra matrices matrix-rank







share|cite|improve this question









$endgroup$

















    1












    $begingroup$



    A matrix $A$ has $10$ columns and dim(Null($A^{T}$ ))$=7$. The smallest possible number of rows of $A$ is



    $(A)$ $5$



    $(B)$ $6$



    $(C)$ $7$



    $(D)$ $8$



    $(E)$ $9$




    I know that dim(Null($A^{T}$ ))$=7$ implies that there are $7$ rows of zeros and that:



    Rank($A$)+Nullity($A^T$) $=$ # of rows



    Rank($A$)+Nullity($A$) $=$ # of columns



    I'm not really sure how to use all this information though... Can someone provide a hint?






    linear-algebra matrices matrix-rank







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$



      A matrix $A$ has $10$ columns and dim(Null($A^{T}$ ))$=7$. The smallest possible number of rows of $A$ is



      $(A)$ $5$



      $(B)$ $6$



      $(C)$ $7$



      $(D)$ $8$



      $(E)$ $9$




      I know that dim(Null($A^{T}$ ))$=7$ implies that there are $7$ rows of zeros and that:



      Rank($A$)+Nullity($A^T$) $=$ # of rows



      Rank($A$)+Nullity($A$) $=$ # of columns



      I'm not really sure how to use all this information though... Can someone provide a hint?






      linear-algebra matrices matrix-rank







      share|cite|improve this question









      $endgroup$





      A matrix $A$ has $10$ columns and dim(Null($A^{T}$ ))$=7$. The smallest possible number of rows of $A$ is



      $(A)$ $5$



      $(B)$ $6$



      $(C)$ $7$



      $(D)$ $8$



      $(E)$ $9$




      I know that dim(Null($A^{T}$ ))$=7$ implies that there are $7$ rows of zeros and that:



      Rank($A$)+Nullity($A^T$) $=$ # of rows



      Rank($A$)+Nullity($A$) $=$ # of columns



      I'm not really sure how to use all this information though... Can someone provide a hint?






      linear-algebra matrices matrix-rank




      linear-algebra matrices matrix-rank






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 21 '18 at 5:59









      Future Math personFuture Math person

      977817




      977817






















          2 Answers
          2






          active

          oldest

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          1












          $begingroup$

          Hint: You want Rank($A$)+Nullity($A^T$) to be as low as possible. You already know how large the right term is. What's the lowest possible the left term could theoretically be? What would the resulting matrix be?






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            How do I know how large the right side is though? I only know that there are 10 columns but how do I know the largest possible number of rows? Just because I have 7 rows of zeroes does not mean that the largest number of rows I have is 7.
            $endgroup$
            – Future Math person
            Dec 21 '18 at 7:18










          • $begingroup$
            The problem says, right out, that the nullity is $7$. That's the right term. You want the whole expression to be as low as possible, which means you want the remaining left term (the rank) to be as low as possible. What's the lowest possible rank a matrix can have, and what do you know about matrices with that rank?
            $endgroup$
            – Arthur
            Dec 21 '18 at 7:21










          • $begingroup$
            Ohh I see. By right term, you meant the Nullity($A^T$) not the number of rows... I thought by right side, you meant the number of columns.
            $endgroup$
            – Future Math person
            Dec 21 '18 at 7:23












          • $begingroup$
            I didn't mean the right side of the equation you wrote. I meant the right side of the expression I wrote, yes.
            $endgroup$
            – Arthur
            Dec 21 '18 at 7:24






          • 1




            $begingroup$
            Oops. Yes. $7 times 10$. Thanks!
            $endgroup$
            – Future Math person
            Dec 21 '18 at 8:18



















          1












          $begingroup$

          $A$ has $10$ columns implies you can view $A$ as a linear map $A:Bbb R^{10} to Bbb R^m$ and $A^T$ as a linear map $A^T:Bbb R^{m} to Bbb R^{10}$ where the $m$ we do not know. Here $dim (text{Null}( A^T))=7$ implies $7 leq m$ and $text{rank}(A^T) le 10$. Also $text{rank}(A^T)=text{rank}(A) leq m$. so $7 leq m leq 10$






          share|cite|improve this answer











          $endgroup$













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            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            1












            $begingroup$

            Hint: You want Rank($A$)+Nullity($A^T$) to be as low as possible. You already know how large the right term is. What's the lowest possible the left term could theoretically be? What would the resulting matrix be?






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              How do I know how large the right side is though? I only know that there are 10 columns but how do I know the largest possible number of rows? Just because I have 7 rows of zeroes does not mean that the largest number of rows I have is 7.
              $endgroup$
              – Future Math person
              Dec 21 '18 at 7:18










            • $begingroup$
              The problem says, right out, that the nullity is $7$. That's the right term. You want the whole expression to be as low as possible, which means you want the remaining left term (the rank) to be as low as possible. What's the lowest possible rank a matrix can have, and what do you know about matrices with that rank?
              $endgroup$
              – Arthur
              Dec 21 '18 at 7:21










            • $begingroup$
              Ohh I see. By right term, you meant the Nullity($A^T$) not the number of rows... I thought by right side, you meant the number of columns.
              $endgroup$
              – Future Math person
              Dec 21 '18 at 7:23












            • $begingroup$
              I didn't mean the right side of the equation you wrote. I meant the right side of the expression I wrote, yes.
              $endgroup$
              – Arthur
              Dec 21 '18 at 7:24






            • 1




              $begingroup$
              Oops. Yes. $7 times 10$. Thanks!
              $endgroup$
              – Future Math person
              Dec 21 '18 at 8:18
















            1












            $begingroup$

            Hint: You want Rank($A$)+Nullity($A^T$) to be as low as possible. You already know how large the right term is. What's the lowest possible the left term could theoretically be? What would the resulting matrix be?






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              How do I know how large the right side is though? I only know that there are 10 columns but how do I know the largest possible number of rows? Just because I have 7 rows of zeroes does not mean that the largest number of rows I have is 7.
              $endgroup$
              – Future Math person
              Dec 21 '18 at 7:18










            • $begingroup$
              The problem says, right out, that the nullity is $7$. That's the right term. You want the whole expression to be as low as possible, which means you want the remaining left term (the rank) to be as low as possible. What's the lowest possible rank a matrix can have, and what do you know about matrices with that rank?
              $endgroup$
              – Arthur
              Dec 21 '18 at 7:21










            • $begingroup$
              Ohh I see. By right term, you meant the Nullity($A^T$) not the number of rows... I thought by right side, you meant the number of columns.
              $endgroup$
              – Future Math person
              Dec 21 '18 at 7:23












            • $begingroup$
              I didn't mean the right side of the equation you wrote. I meant the right side of the expression I wrote, yes.
              $endgroup$
              – Arthur
              Dec 21 '18 at 7:24






            • 1




              $begingroup$
              Oops. Yes. $7 times 10$. Thanks!
              $endgroup$
              – Future Math person
              Dec 21 '18 at 8:18














            1












            1








            1





            $begingroup$

            Hint: You want Rank($A$)+Nullity($A^T$) to be as low as possible. You already know how large the right term is. What's the lowest possible the left term could theoretically be? What would the resulting matrix be?






            share|cite|improve this answer









            $endgroup$



            Hint: You want Rank($A$)+Nullity($A^T$) to be as low as possible. You already know how large the right term is. What's the lowest possible the left term could theoretically be? What would the resulting matrix be?







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 21 '18 at 7:16









            ArthurArthur

            115k7116198




            115k7116198












            • $begingroup$
              How do I know how large the right side is though? I only know that there are 10 columns but how do I know the largest possible number of rows? Just because I have 7 rows of zeroes does not mean that the largest number of rows I have is 7.
              $endgroup$
              – Future Math person
              Dec 21 '18 at 7:18










            • $begingroup$
              The problem says, right out, that the nullity is $7$. That's the right term. You want the whole expression to be as low as possible, which means you want the remaining left term (the rank) to be as low as possible. What's the lowest possible rank a matrix can have, and what do you know about matrices with that rank?
              $endgroup$
              – Arthur
              Dec 21 '18 at 7:21










            • $begingroup$
              Ohh I see. By right term, you meant the Nullity($A^T$) not the number of rows... I thought by right side, you meant the number of columns.
              $endgroup$
              – Future Math person
              Dec 21 '18 at 7:23












            • $begingroup$
              I didn't mean the right side of the equation you wrote. I meant the right side of the expression I wrote, yes.
              $endgroup$
              – Arthur
              Dec 21 '18 at 7:24






            • 1




              $begingroup$
              Oops. Yes. $7 times 10$. Thanks!
              $endgroup$
              – Future Math person
              Dec 21 '18 at 8:18


















            • $begingroup$
              How do I know how large the right side is though? I only know that there are 10 columns but how do I know the largest possible number of rows? Just because I have 7 rows of zeroes does not mean that the largest number of rows I have is 7.
              $endgroup$
              – Future Math person
              Dec 21 '18 at 7:18










            • $begingroup$
              The problem says, right out, that the nullity is $7$. That's the right term. You want the whole expression to be as low as possible, which means you want the remaining left term (the rank) to be as low as possible. What's the lowest possible rank a matrix can have, and what do you know about matrices with that rank?
              $endgroup$
              – Arthur
              Dec 21 '18 at 7:21










            • $begingroup$
              Ohh I see. By right term, you meant the Nullity($A^T$) not the number of rows... I thought by right side, you meant the number of columns.
              $endgroup$
              – Future Math person
              Dec 21 '18 at 7:23












            • $begingroup$
              I didn't mean the right side of the equation you wrote. I meant the right side of the expression I wrote, yes.
              $endgroup$
              – Arthur
              Dec 21 '18 at 7:24






            • 1




              $begingroup$
              Oops. Yes. $7 times 10$. Thanks!
              $endgroup$
              – Future Math person
              Dec 21 '18 at 8:18
















            $begingroup$
            How do I know how large the right side is though? I only know that there are 10 columns but how do I know the largest possible number of rows? Just because I have 7 rows of zeroes does not mean that the largest number of rows I have is 7.
            $endgroup$
            – Future Math person
            Dec 21 '18 at 7:18




            $begingroup$
            How do I know how large the right side is though? I only know that there are 10 columns but how do I know the largest possible number of rows? Just because I have 7 rows of zeroes does not mean that the largest number of rows I have is 7.
            $endgroup$
            – Future Math person
            Dec 21 '18 at 7:18












            $begingroup$
            The problem says, right out, that the nullity is $7$. That's the right term. You want the whole expression to be as low as possible, which means you want the remaining left term (the rank) to be as low as possible. What's the lowest possible rank a matrix can have, and what do you know about matrices with that rank?
            $endgroup$
            – Arthur
            Dec 21 '18 at 7:21




            $begingroup$
            The problem says, right out, that the nullity is $7$. That's the right term. You want the whole expression to be as low as possible, which means you want the remaining left term (the rank) to be as low as possible. What's the lowest possible rank a matrix can have, and what do you know about matrices with that rank?
            $endgroup$
            – Arthur
            Dec 21 '18 at 7:21












            $begingroup$
            Ohh I see. By right term, you meant the Nullity($A^T$) not the number of rows... I thought by right side, you meant the number of columns.
            $endgroup$
            – Future Math person
            Dec 21 '18 at 7:23






            $begingroup$
            Ohh I see. By right term, you meant the Nullity($A^T$) not the number of rows... I thought by right side, you meant the number of columns.
            $endgroup$
            – Future Math person
            Dec 21 '18 at 7:23














            $begingroup$
            I didn't mean the right side of the equation you wrote. I meant the right side of the expression I wrote, yes.
            $endgroup$
            – Arthur
            Dec 21 '18 at 7:24




            $begingroup$
            I didn't mean the right side of the equation you wrote. I meant the right side of the expression I wrote, yes.
            $endgroup$
            – Arthur
            Dec 21 '18 at 7:24




            1




            1




            $begingroup$
            Oops. Yes. $7 times 10$. Thanks!
            $endgroup$
            – Future Math person
            Dec 21 '18 at 8:18




            $begingroup$
            Oops. Yes. $7 times 10$. Thanks!
            $endgroup$
            – Future Math person
            Dec 21 '18 at 8:18











            1












            $begingroup$

            $A$ has $10$ columns implies you can view $A$ as a linear map $A:Bbb R^{10} to Bbb R^m$ and $A^T$ as a linear map $A^T:Bbb R^{m} to Bbb R^{10}$ where the $m$ we do not know. Here $dim (text{Null}( A^T))=7$ implies $7 leq m$ and $text{rank}(A^T) le 10$. Also $text{rank}(A^T)=text{rank}(A) leq m$. so $7 leq m leq 10$






            share|cite|improve this answer











            $endgroup$


















              1












              $begingroup$

              $A$ has $10$ columns implies you can view $A$ as a linear map $A:Bbb R^{10} to Bbb R^m$ and $A^T$ as a linear map $A^T:Bbb R^{m} to Bbb R^{10}$ where the $m$ we do not know. Here $dim (text{Null}( A^T))=7$ implies $7 leq m$ and $text{rank}(A^T) le 10$. Also $text{rank}(A^T)=text{rank}(A) leq m$. so $7 leq m leq 10$






              share|cite|improve this answer











              $endgroup$
















                1












                1








                1





                $begingroup$

                $A$ has $10$ columns implies you can view $A$ as a linear map $A:Bbb R^{10} to Bbb R^m$ and $A^T$ as a linear map $A^T:Bbb R^{m} to Bbb R^{10}$ where the $m$ we do not know. Here $dim (text{Null}( A^T))=7$ implies $7 leq m$ and $text{rank}(A^T) le 10$. Also $text{rank}(A^T)=text{rank}(A) leq m$. so $7 leq m leq 10$






                share|cite|improve this answer











                $endgroup$



                $A$ has $10$ columns implies you can view $A$ as a linear map $A:Bbb R^{10} to Bbb R^m$ and $A^T$ as a linear map $A^T:Bbb R^{m} to Bbb R^{10}$ where the $m$ we do not know. Here $dim (text{Null}( A^T))=7$ implies $7 leq m$ and $text{rank}(A^T) le 10$. Also $text{rank}(A^T)=text{rank}(A) leq m$. so $7 leq m leq 10$







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Dec 21 '18 at 7:31

























                answered Dec 21 '18 at 7:25









                Chinnapparaj RChinnapparaj R

                5,5072928




                5,5072928






























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