Past exam of the algebra 1 and differential equations 1 course of the University of Oxford / 2015 / 1 / b / i

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- Solution 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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