I need a clarification about the correct way to compute the projection of a vector into the nullspace of a matrix.
For sake of clarity, let's call $A$ the matrix, $N(A)$ it's kernel and $A^\sharp$ it's pseudoinverse. Let's call $v$ the vector.
It was told me that $(I-A^\sharp A)\cdot v$ corresponds to the projection of $v$ into the kernel of $A$.
However, given the matrix $A$, I can find a basis for it's kernel. Let's denote with $N$ the matrix which has as columns the vectors of this basis. Why I can't easily multiply $N\cdot v$ to get the projection of $v$ into the kernel of $A$?
Thanks for the explanation