Independent projections

Question

Suppose, we have a matrix $X\ (n\ \times\ n) $. There are r < n independent columns(therefore r is rank).And we project our vectors with $P\ (n\ \times\ n) $ operator = $I - E$ where E is the just matrix with ones in the first column and zeros in the another columns.I also know that both $\text{Im}(P)$ and $\text{Ker}(P)$ don't contain vectors $X$. The question is - What do I need to add to the conditions so the projections of the vectors will be independent? Or is it already true? I can add that the diagonal elements $X$ are greater than any other element and all elements are positive.Moreover, if necessary, we add that X is symmetric.