Let $E$ be a finite dimensional vector space. Find a basis of $\mathcal{L}(E)$ formed by projections. a.k.a : endomorphisms p such that $p^2=p$.
My solution was to let $(e_1,...,e_n)$ be a basis for $E$, then construct the family $(p_{i,j})$ such that : $\forall (i,j)\in\{1,...,n\}^2: p_{i,j}$ is the linear map such that: $$p_{i,j}(e_i)=e_j,\ \ p_{i,j}(e_j)=e_j$$ and $p_{i,j}(e_k)=0$ if $k\notin\{i,j\}$ .
there are $n^2$ of them, they are projections, and are linearly independent. Am i right?