Rank, nullity and the number of rows of a matrix












1












$begingroup$


I have this question here:




Let $A$ be a matrix with $10$ columns, $dim$ $Null(A)=5$ and $dim$ $Null(A^T)=3$. How many rows does $A$ have?



$a)$ $8$



$b)$ $3$



$c)$ $5$



$d)$ $10$



$e)$ This cannot be determined from the information given




I tried doing it. I know that:



$rank(A)+Nullity(A)=10$



So that means that



$rank(A)+5=10$



$rank(A)=5$



However, I am not really sure how that helps me find the number of rows. I know that $dim$ $Null(A^T)=3$ but how do I incorporate that into this?



Thanks!






linear-algebra matrix-rank transpose







share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    I have this question here:




    Let $A$ be a matrix with $10$ columns, $dim$ $Null(A)=5$ and $dim$ $Null(A^T)=3$. How many rows does $A$ have?



    $a)$ $8$



    $b)$ $3$



    $c)$ $5$



    $d)$ $10$



    $e)$ This cannot be determined from the information given




    I tried doing it. I know that:



    $rank(A)+Nullity(A)=10$



    So that means that



    $rank(A)+5=10$



    $rank(A)=5$



    However, I am not really sure how that helps me find the number of rows. I know that $dim$ $Null(A^T)=3$ but how do I incorporate that into this?



    Thanks!






    linear-algebra matrix-rank transpose







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      I have this question here:




      Let $A$ be a matrix with $10$ columns, $dim$ $Null(A)=5$ and $dim$ $Null(A^T)=3$. How many rows does $A$ have?



      $a)$ $8$



      $b)$ $3$



      $c)$ $5$



      $d)$ $10$



      $e)$ This cannot be determined from the information given




      I tried doing it. I know that:



      $rank(A)+Nullity(A)=10$



      So that means that



      $rank(A)+5=10$



      $rank(A)=5$



      However, I am not really sure how that helps me find the number of rows. I know that $dim$ $Null(A^T)=3$ but how do I incorporate that into this?



      Thanks!






      linear-algebra matrix-rank transpose







      share|cite|improve this question









      $endgroup$




      I have this question here:




      Let $A$ be a matrix with $10$ columns, $dim$ $Null(A)=5$ and $dim$ $Null(A^T)=3$. How many rows does $A$ have?



      $a)$ $8$



      $b)$ $3$



      $c)$ $5$



      $d)$ $10$



      $e)$ This cannot be determined from the information given




      I tried doing it. I know that:



      $rank(A)+Nullity(A)=10$



      So that means that



      $rank(A)+5=10$



      $rank(A)=5$



      However, I am not really sure how that helps me find the number of rows. I know that $dim$ $Null(A^T)=3$ but how do I incorporate that into this?



      Thanks!






      linear-algebra matrix-rank transpose




      linear-algebra matrix-rank transpose






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




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      asked Dec 19 '18 at 5:55









      Future Math personFuture Math person

      993817




      993817






















          1 Answer
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          $begingroup$

          $textbf{Note:}$ $text{Rank($A$) = dim(rowsp($A$)) = dim(colsp($A$))}$



          $text{Nullity($A^T$) + dim(rowsp($A$)) = Number of rows of $A$}$



          $text{Nullity($A$) + dim(colsp($A$)) = Number of columns of $A$}$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Yup I see how that works. I know that the 2nd equation is true but why is $text{Nullity($A^T$) + dim(rowsp($A$)) = Number of rows of $A$}$ true? How do you obtain that?
            $endgroup$
            – Future Math person
            Dec 19 '18 at 6:31










          • $begingroup$
            $text{Nullity($A^T$)}$ is the number of zero rows in RREF of $A^T$ and $text{dim(rowsp($A$))}$ is the number of pivot rows.
            $endgroup$
            – Yadati Kiran
            Dec 19 '18 at 6:36












          • $begingroup$
            In other words consider the third equation for $A^T$ then rows become columns and columns become rows. Then we have $text{Nullity($A^T$) + dim(colsp($A^T$)) = Number of columns of $A^T$=Number of rows of $A$}$
            $endgroup$
            – Yadati Kiran
            Dec 19 '18 at 6:42












          • $begingroup$
            I get it now! Thanks so much!
            $endgroup$
            – Future Math person
            Dec 19 '18 at 6:49












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          1 Answer
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          1 Answer
          1






          active

          oldest

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          active

          oldest

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          active

          oldest

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          2












          $begingroup$

          $textbf{Note:}$ $text{Rank($A$) = dim(rowsp($A$)) = dim(colsp($A$))}$



          $text{Nullity($A^T$) + dim(rowsp($A$)) = Number of rows of $A$}$



          $text{Nullity($A$) + dim(colsp($A$)) = Number of columns of $A$}$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Yup I see how that works. I know that the 2nd equation is true but why is $text{Nullity($A^T$) + dim(rowsp($A$)) = Number of rows of $A$}$ true? How do you obtain that?
            $endgroup$
            – Future Math person
            Dec 19 '18 at 6:31










          • $begingroup$
            $text{Nullity($A^T$)}$ is the number of zero rows in RREF of $A^T$ and $text{dim(rowsp($A$))}$ is the number of pivot rows.
            $endgroup$
            – Yadati Kiran
            Dec 19 '18 at 6:36












          • $begingroup$
            In other words consider the third equation for $A^T$ then rows become columns and columns become rows. Then we have $text{Nullity($A^T$) + dim(colsp($A^T$)) = Number of columns of $A^T$=Number of rows of $A$}$
            $endgroup$
            – Yadati Kiran
            Dec 19 '18 at 6:42












          • $begingroup$
            I get it now! Thanks so much!
            $endgroup$
            – Future Math person
            Dec 19 '18 at 6:49
















          2












          $begingroup$

          $textbf{Note:}$ $text{Rank($A$) = dim(rowsp($A$)) = dim(colsp($A$))}$



          $text{Nullity($A^T$) + dim(rowsp($A$)) = Number of rows of $A$}$



          $text{Nullity($A$) + dim(colsp($A$)) = Number of columns of $A$}$






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Yup I see how that works. I know that the 2nd equation is true but why is $text{Nullity($A^T$) + dim(rowsp($A$)) = Number of rows of $A$}$ true? How do you obtain that?
            $endgroup$
            – Future Math person
            Dec 19 '18 at 6:31










          • $begingroup$
            $text{Nullity($A^T$)}$ is the number of zero rows in RREF of $A^T$ and $text{dim(rowsp($A$))}$ is the number of pivot rows.
            $endgroup$
            – Yadati Kiran
            Dec 19 '18 at 6:36












          • $begingroup$
            In other words consider the third equation for $A^T$ then rows become columns and columns become rows. Then we have $text{Nullity($A^T$) + dim(colsp($A^T$)) = Number of columns of $A^T$=Number of rows of $A$}$
            $endgroup$
            – Yadati Kiran
            Dec 19 '18 at 6:42












          • $begingroup$
            I get it now! Thanks so much!
            $endgroup$
            – Future Math person
            Dec 19 '18 at 6:49














          2












          2








          2





          $begingroup$

          $textbf{Note:}$ $text{Rank($A$) = dim(rowsp($A$)) = dim(colsp($A$))}$



          $text{Nullity($A^T$) + dim(rowsp($A$)) = Number of rows of $A$}$



          $text{Nullity($A$) + dim(colsp($A$)) = Number of columns of $A$}$






          share|cite|improve this answer









          $endgroup$



          $textbf{Note:}$ $text{Rank($A$) = dim(rowsp($A$)) = dim(colsp($A$))}$



          $text{Nullity($A^T$) + dim(rowsp($A$)) = Number of rows of $A$}$



          $text{Nullity($A$) + dim(colsp($A$)) = Number of columns of $A$}$







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 19 '18 at 6:24









          Yadati KiranYadati Kiran

          2,1161622




          2,1161622












          • $begingroup$
            Yup I see how that works. I know that the 2nd equation is true but why is $text{Nullity($A^T$) + dim(rowsp($A$)) = Number of rows of $A$}$ true? How do you obtain that?
            $endgroup$
            – Future Math person
            Dec 19 '18 at 6:31










          • $begingroup$
            $text{Nullity($A^T$)}$ is the number of zero rows in RREF of $A^T$ and $text{dim(rowsp($A$))}$ is the number of pivot rows.
            $endgroup$
            – Yadati Kiran
            Dec 19 '18 at 6:36












          • $begingroup$
            In other words consider the third equation for $A^T$ then rows become columns and columns become rows. Then we have $text{Nullity($A^T$) + dim(colsp($A^T$)) = Number of columns of $A^T$=Number of rows of $A$}$
            $endgroup$
            – Yadati Kiran
            Dec 19 '18 at 6:42












          • $begingroup$
            I get it now! Thanks so much!
            $endgroup$
            – Future Math person
            Dec 19 '18 at 6:49


















          • $begingroup$
            Yup I see how that works. I know that the 2nd equation is true but why is $text{Nullity($A^T$) + dim(rowsp($A$)) = Number of rows of $A$}$ true? How do you obtain that?
            $endgroup$
            – Future Math person
            Dec 19 '18 at 6:31










          • $begingroup$
            $text{Nullity($A^T$)}$ is the number of zero rows in RREF of $A^T$ and $text{dim(rowsp($A$))}$ is the number of pivot rows.
            $endgroup$
            – Yadati Kiran
            Dec 19 '18 at 6:36












          • $begingroup$
            In other words consider the third equation for $A^T$ then rows become columns and columns become rows. Then we have $text{Nullity($A^T$) + dim(colsp($A^T$)) = Number of columns of $A^T$=Number of rows of $A$}$
            $endgroup$
            – Yadati Kiran
            Dec 19 '18 at 6:42












          • $begingroup$
            I get it now! Thanks so much!
            $endgroup$
            – Future Math person
            Dec 19 '18 at 6:49
















          $begingroup$
          Yup I see how that works. I know that the 2nd equation is true but why is $text{Nullity($A^T$) + dim(rowsp($A$)) = Number of rows of $A$}$ true? How do you obtain that?
          $endgroup$
          – Future Math person
          Dec 19 '18 at 6:31




          $begingroup$
          Yup I see how that works. I know that the 2nd equation is true but why is $text{Nullity($A^T$) + dim(rowsp($A$)) = Number of rows of $A$}$ true? How do you obtain that?
          $endgroup$
          – Future Math person
          Dec 19 '18 at 6:31












          $begingroup$
          $text{Nullity($A^T$)}$ is the number of zero rows in RREF of $A^T$ and $text{dim(rowsp($A$))}$ is the number of pivot rows.
          $endgroup$
          – Yadati Kiran
          Dec 19 '18 at 6:36






          $begingroup$
          $text{Nullity($A^T$)}$ is the number of zero rows in RREF of $A^T$ and $text{dim(rowsp($A$))}$ is the number of pivot rows.
          $endgroup$
          – Yadati Kiran
          Dec 19 '18 at 6:36














          $begingroup$
          In other words consider the third equation for $A^T$ then rows become columns and columns become rows. Then we have $text{Nullity($A^T$) + dim(colsp($A^T$)) = Number of columns of $A^T$=Number of rows of $A$}$
          $endgroup$
          – Yadati Kiran
          Dec 19 '18 at 6:42






          $begingroup$
          In other words consider the third equation for $A^T$ then rows become columns and columns become rows. Then we have $text{Nullity($A^T$) + dim(colsp($A^T$)) = Number of columns of $A^T$=Number of rows of $A$}$
          $endgroup$
          – Yadati Kiran
          Dec 19 '18 at 6:42














          $begingroup$
          I get it now! Thanks so much!
          $endgroup$
          – Future Math person
          Dec 19 '18 at 6:49




          $begingroup$
          I get it now! Thanks so much!
          $endgroup$
          – Future Math person
          Dec 19 '18 at 6:49


















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