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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!

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$\textbf{Note:}$ $\text{Rank($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 $A$}$

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  • $\begingroup$ 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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