 Solution

ID: 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 that

a_{2 i - 1} = r, a_{2 i} = s (1 \leq i \leq n) .

One can also see the necessity directly by writing the nonnegative deficit as

n \sum_{i = 1}^{2 n} a_{i}^{2} - (\sum_{i = 1}^{n} a_{2 i})^{2} - (\sum_{i = 1}^{n} a_{2 i - 1})^{2} = n \sum_{i = 1}^{n} (a_{2 i} - \overset{a}{ˉ}_{e})^{2} + n \sum_{i = 1}^{n} (a_{2 i - 1} - \overset{a}{ˉ}_{o})^{2},

where

\overset{a}{ˉ}_{e}

and

\overset{a}{ˉ}_{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.

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