 Solution

ID: past-exam-of-the-algebra-1-and-differential-equations-1-course-of-the-university-of-oxford/2015/1/b/i/solution

Expand the square using

E^{2} = E

and

F^{2} = F

(E + F)^{2} - (E + F) = EF + FE .

Thus

E + F is a linear projection ⟺ EF = - FE .

This sum of two linear projections criterion holds in every characteristic; it uses an equality of linear maps, not cancellation of factors in the noncommutative associative algebra of linear endomorphisms.

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