Past exam of the algebra and analysis II for mathematics and philosophy course of the University of Oxford / 2013 / 2 / c / Solution

Equality in part (a) requires $a$ to lie in the span of the two chosen unit vectors. Equivalently, there are real constants $r,s$ such thatOne can also see the necessity directly by writing the nonnegative deficit aswhere ${\bar{a}}_e$ and ${\bar{a}}_o$ are the means of the even and odd entries. The deficit vanishes exactly when each group is constant; the two group constants need not be equal.