Dimension of nullspace and number of rows

Question

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?

Answer 1

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?

Answer 2

$A$ 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 $7 \leq m \leq 10$