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.
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Welcome to MathSE! I see someone already edited this post of yours, but please learn to format your posts according to the rules of the site. – evaristegd Jul 10 '19 at 03:32
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The ranks of a matrix and its transpose are equal. – amd Jul 10 '19 at 07:53
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$1$.Offcourse not. Take $A$ with $n\ne d$, then $null(A)\ne null(A^T)$, etc.
$2$. $rank(A)+null(A)=$no. of columns$=d$
Nitin Uniyal
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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 $$d = \max(n,d)?$$
angryavian
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