Rank, nullity and the number of rows of a matrix

Question

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!

Answer 1

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