Projections from Direct Sum Theorem

Theorem

Suppose $V=W_1\oplus \cdots \oplus W_k$ Then, there exists $E_1,\dots ,E_k\in \mathcal{L}(V)$

1. $E_j^2=E_j,\forall j$
2. $E_i{E}_j=0$ if $i\ne j$
3. $I=E_1+\cdots +e_k$
4. $range(E_i)=W_i,\forall i$

Proofs

$$E_j({\alpha }_{ik})=\{\begin{array}{ll}0& i\ne j\\ a_{ik}& i=j\end{array}$$

Given this definition,

$$E_j({\alpha }_{ik}{)}^2=\{\begin{array}{ll}{E}_j(0)& i\ne j\\ E_j(a_{ik})& i=j\end{array}=\{\begin{array}{ll}0& i\ne j\\ a_{ik}& i=j\end{array}$$