null space of an n-by-m matrix I have an $n$-by-$m$ ($n>m$) matrix named $J$. I wanted to find its null space so as I used matrix $M$ defined bellow:
$$JM=0\text{, when } M=I-J^\dagger J$$
$J^\dagger$ is the pseudo inverse of $J$.
Now my question is that am I allowed to choose any ($r=m-n$) columns of $M$ as null space of $J$ or not, in the other words, can I be sure that the columns of $M$ are linearly independent?
Thanks in advance.

Since $M$ is a square matrix, it has linearly independent columns if and only if it is invertible, which happens only if $J$ is the zero matrix. 
Indeed, the product $J^\dagger J$ is the orthogonal projection onto $(\ker J)^\perp$, see here. Hence, $M$ is the orthogonal projection onto $\ker J$. So, the columns of $M$ span $\ker J$. But selecting the minimal spanning set of columns is still a task up to you. 
Also, I don't think it's a good idea to compute $J^\dagger$ just to find a basis for $\ker M$. Gaussian elimination takes less work. 
