 Solution

Suppose first that $E+F$ is a linear projection. Part (i) gives $EF=-FE$. Multiplying on the left by $E$ gives $EF=-EFE$, and multiplying on the right by $E$ gives $EFE=-FE=EF$. Combining them yieldsIn a field of characteristic different from two, the nonzero scalar $2$ is invertible. Hence $EF=0$, and $FE=-EF=0$ too. Conversely these two zero products make the expansion in part (i) equal zero. ThereforeThe argument cancels a nonzero scalar, never a possibly singular linear map.