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?
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 $rank(M)\leq r$.