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

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- Solution c

Solution

The proposed equivalence is false. On a nonzero vector space over any field of characteristic different from two, choose

E = I, F = I, G = 0.

These linear projections commute and satisfy

EFG = 0

. Nevertheless their sum is

2 I

, and

(2 I)^{2} - 2 I = 2 I \neq = 0.

Thus the proposed conditions are not sufficient. For commuting linear projections the actual obstruction is the sum of the pairwise products:

(E + F + G)^{2} - (E + F + G) = 2 (EF + EG + FG) .

The single condition on the triple product does not remove this obstruction.

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