Linear Algebra

148 Chapter 2. Vector Spaces
however, linear algebra has been used [Zwicker] to argue that a tendency toward
cyclic preference is actually present in each voter’s list, and that it surfaces when
there is more adding of the tendency than cancelling.
For this argument, abbreviating the choices as D, R, andT , we can describe
how a voter with preference order D>R>T contributes to the above cycle
−1 voterD
1voter
T R
1voter
(the negative sign is here because the arrow describes T as preferred to D, but
this voter likes them the other way). The descriptions for the other preference
lists are in the table on page 150. Now, to conduct the election, we linearly
combine these descriptions; for instance, the Political Science mock election
D D D −1 voter 1voter −1 voter 1voter
1voter −1voter
5 · T R +4· T R + ··· +2· T R
1voter
−1 voter
−1 voter
yields the circular group preference shown earlier.
Of course, taking linear combinations is linear algebra. The above cycle notation
is suggestive but inconvienent, so we temporarily switch to using column
vectors by starting at the D and taking the numbers from the cycle in counterclockwise
order. Thus, the mock election and a single D>R>Tvote are
represented in this way.
⎛ ⎞ ⎛ ⎞
7
⎝1⎠
and
5
−1
⎝ 1 ⎠
1
We will decompose vote vectors into two parts, one cyclic and the other acyclic.
For the first part, we say that a vector is purely cyclic if it is in this subspace
of R3 .
⎛ ⎞
k
C = { ⎝k⎠
k
� ⎛ ⎞
1
� k ∈ R} = {k · ⎝1⎠
1
� � k ∈ R}
For the second part, consider the subspace (see Exercise 6) of vectors that are
perpendicular to all of the vectors in C.
C ⊥ ⎛
= { ⎝ c1
⎞
c2⎠
c3
� ⎛
� ⎝ c1
⎞ ⎛
c2⎠
⎝
c3
k
⎞
k⎠
= 0 for all k ∈ R}
k
⎛
= { ⎝ c1
⎞
⎠ � � c1 + c2 + c3 =0}
c2
c3
⎛
= {c2 ⎝ −1
⎞ ⎛
1 ⎠ + c3 ⎝
0
−1
⎞
0 ⎠
1
� � c2,c3 ∈ R}