Rank Equation for non-square Matrix

Question

If I have a $n\times d$ matrix $A$, can we say $rank(A) + null(A) = rank(A^T) + null(A^T)$? Also, is $rank (A) + null(A) = max(n,d)$? Thanks in advance.

Answer 1

$1$.Offcourse not. Take $A$ with $n\ne d$, then $null(A)\ne null(A^T)$, etc.

$2$. $rank(A)+null(A)=$no. of columns$=d$

Answer 2

The rank-nullity theorem implies $\text{rank}(A) + \text{null}(A) = d$ and $\text{rank}(A^\top) + \text{null}(A^\top) = n$.

So your first question is equivalent to $$d=n?$$ and your second question is equivalent to $$d = \max(n,d)?$$