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Give a matrix $A$ whose size e.g., $m$ x $n$, and $m<n$. I know the the null space of that matrix is $X$ such that $A*X = 0$. The nullity of matrix $X$ is e.g $r$. it means the null space I got is a matrix of size $m$ x $r$. As shown HERE, I can build the square $m$ x $m$ matrix $Y$ by taking the values of each column from the null space such that $A*Y = 0$.

My question, what is the maximum rank of $Y$ we can get by taking its value from the null space? Is it always equal to the nullity $r$? or it can be bigger?

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The rank of a matrix $M$ is by definition the dimension of the vector space generated by its columns. If the columns belong to a space of dimension $r$, they generate a space of dimension $\leq r$, whence $rank(M)\leq r$.

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  • $\begingroup$ you mean maximum rank of matrix $Y$ in my case must be $r$ since I built $Y$ based on a matrix $X$ whose rank is $r$? $\endgroup$
    – Fatima_Ali
    May 20, 2021 at 10:43
  • $\begingroup$ @Fatima_Ali Exactly. The maximum rank of $Y$ is the dimension of the null-space of $A$. $\endgroup$
    – Desperado
    May 20, 2021 at 10:44

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