Solution to Exercise 3.16.
The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.
—Jules Henri Poincaré  (1854–1912)

Exercise 3.16. Prove the relation used in defining lines is an equivalence relation.

Proof. To show the relation, ~, is an equivalence on the set of lines, we need to show that the relation is reflexive, symmetric, and transitive.
        Let [a₁, a₂, a₃] be a line. Choose k = 1. Then a_i = 1a_i for i = 1, 2, 3. Hence, [a₁, a₂, a₃] ~ [a₁, a₂, a₃]. Therefore, the relation is reflexive.
        Let [a₁, a₂, a₃] and [b₁, b₂, b₃] be two lines. Assume [a₁, a₂, a₃] ~ [b₁, b₂, b₃]. Then there exists a nonzero value k such that b_i = ka_i for i = 1, 2, 3. Thus, a_i = (1/k)b_i for i = 1, 2, 3 and 1/k is nonzero. Thus, [b₁, b₂, b₃] ~ [a₁, a₂, a₃]. Therefore, the relation is symmetric.
        Let [a₁, a₂, a₃], [b₁, b₂, b₃], and [c₁, c₂, c₃] be three lines. Assume [a₁, a₂, a₃] ~ [b₁, b₂, b₃] and [b₁, b₂, b₃] ~ [c₁, c₂, c₃]. Then there exist nonzero values k₁ and k₂ such that b_i = k₁a_i and c_i = k₂b_i for i = 1, 2, 3. Thus, c_i = k₂b_i = k₂(k₁a_i) = (k₂k₁)a_i for i = 1, 2, 3. Hence, [a₁, a₂, a₃] ~ [c₁, c₂, c₃]. Therefore, the relation is transitive.
         Hence, the relation, ~, is an equivalence relation on the set of lines.