Voting Paradoxes

Here is a reasonable way in which a voter could have a cyclic preference. Suppose that this voter ranks each candidate on each of three criteria.

• Draw up a table with the rows labeled ‘Democrat’, ‘Republican’, and ‘Third’, and the columns labeled ‘character’, ‘experience’, and ‘policies’. Inside each column, rank some candidate as most preferred, rank another as in the middle, and rank the remai…

Compute the values in the table of decompositions.

Let $U$ be a subspace of $\mathbb{R}^{3}$. Prove that the set $U^{\perp}=\text{$\{\hskip 1.00006pt\vec{v}\mid\text{$\vec{v}\mathbin{\smash{\bullet}}\vec{u}=0$ for all $\vec{u}\in U$}\hskip 1.00006pt\}$}$ of vectors that are perpendicular to each vector in $U$ is also subspace of $\mathbb{R}^{3}$. Does this hold if $U$ is not a subspace?